Table mf_boxes:
This table stores metadata describing sets of newspaces determined by constraints on level N, weight k, character order o, Nk^2, dimension D, and for each set lists counts of newspaces, newforms (when known), and embeddings in the set, along with a series of boolean flags indicating what data is available for newforms in the set.
| Column | Type | Notes |
|---|---|---|
| Nmin | integer | lower bound on the level N |
| Nmax | integer | upper bound on the level N |
| kmin | integer | lower bound on the weight k |
| kmax | integer | upper bound on the weight k |
| omin | integer | lower bound on the character order o |
| omax | integer | upper bound on the character order o |
| Nk2min | integer | lower bound on Nk^2 |
| Nk2max | integer | upper bound on Nk^2 |
| Dmin | integer | lower bound on newspace Q-dimension |
| Dmax | integer | upper bound on newspace Q-dimension |
| newspace_count | integer | total number of newspaces in this box |
| nonzero_newspace_count | integer | total number of nonzero newspaces in this box |
| newform_count | integer | total number of newforms in this box (if known, may be null) |
| embedding_count | bigint | total number of complex embeddings of newforms in this box |
| straces | boolean | set if space trace forms are stored |
| split | boolean | set if list of dimensions of irreducible subspaces (newforms) are stored |
| traces | boolean | set if newform trace forms are stored |
| eigenvalues | boolean | set if eigenvalue data of newforms of small dimension are stored |
| embeddings | boolean | set if complex embeddings are stored |
| lfunctions | boolean | set if lfunctions have been computed |
Table mf_newspaces:
This table represents (Galois orbits of) spaces of newforms S_k^new(N, [\chi]), where \chi is a Dirichlet character of modulus N and [\chi] denotes its conjugacy class under the action of G_Q. Character orbits are sorted by order and traces of values on [1..N] (lexicographically), so that 1 is the index of the orbit of the trivial character.
| Column | Type | Notes |
|---|---|---|
| label | text | Format is N.k.i where N = level, k = weight, a = base26 encoding of character orbit index ('a'=1) |
| level | integer | the level N of the modular form |
| level_radical | integer | product of prime divisors of N |
| level_primes | integer[] | list of primes divisors of N |
| level_is_prime | boolean | true if N is prime (1 is not prime) |
| level_is_prime_power | boolean | true if N is a prime power (1 is not a prime power, primes are prime powers) |
| level_is_squarefree | boolean | true if N is squarefree (1 is squarefree) |
| level_is_square | boolean | true if N is a square (1 is a square) |
| weight | smallint | the weight k of the modular form |
| weight_parity | smallint | (-1)^k |
| analytic_conductor | double precision | N*(Exp(Psi((k)/2))/(2*pi))^2 where Psi(t) := Gamma'(t)/Gamma(t) |
| Nk2 | integer | N*k^2 |
| char_orbit_index | smallint | the index a of the galois orbit of the character for this space. Galois orbits of Dirichlet characters of modulus N are sorted by the character order and then lexicographically by traces (to Q) of values on 1...N-1. |
| char_orbit_label | text | base26 encoding of char_orbit_index with a=1 |
| conrey_indexes | integer[] | Sorted list of Conrey indexes of characters in this Galois orbit |
| char_order | integer | the order of the character |
| char_conductor | integer | Conductor of the Dirichlet character |
| prim_orbit_index | smallint | char_orbit for the primitive version of this character |
| char_degree | integer | the degree of the (cyclotomic) character field |
| char_parity | smallint | 1 or -1, depending on the parity of the character |
| char_is_real | boolean | whether the character takes only real values (trivial or quadratic) |
| char_values | jsonb | quadruple <N,n,u,v> where N is the level, n is the order of the character, u is a list of generators for the unit group of Z/NZ, and v is a corresponding list of integers for which chi(u[i]) = zeta_n^v[i] |
| sturm_bound | integer | \floor(k*Index(Gamma0(N))/12) |
| trace_bound | integer | nonnegative integer n such that the traces from 1 up to n distinguish all forms in this space (0 if space has one form, 1 if more than 1 form but dimensions are all distinct), only set when known |
| dim | integer | Q-dimension of the newspace S_k^new(N,[chi]) |
| relative_dim | integer | Q(chi)-dimension of the newspace S_k^new(N,[chi]), equal to dim/degree(chi) |
| num_forms | smallint | number of (Hecke/Galois orbits of ) newforms, only set when known |
| hecke_orbit_dims | integer[] | Sorted list of Q-dimensions of Hecke orbits, only set when known |
| eis_dim | integer | Q-dimension of the eisenstein subspace of M_k(N, \chi) |
| eis_new_dim | integer | Q-dimension of the new eisenstein subspace of M_k(N, \chi) |
| cusp_dim | integer | Q-dimension of the cuspidal space S_k(N, \chi) |
| mf_dim | integer | Q-dimension of M_k(N, \chi) |
| mf_new_dim | integer | Q-dimension of the new subspace of M_k(N,\chi) |
| AL_dims | jsonb | For spaces with trivial character, this is a lists of triples [AL_eigs, d, n], where AL_eigs is a list of pairs [p, ev] where p is a prime dividing N and ev=+/-1 is an Atkin-Lehner eigevnalue at p, while d and n record the total dimension and number of newforms that lie in the intersection of the corresponding eigenspaces. |
| plus_dim | integer | For spaces with tirival character, dimension of the subspace with Fricke-eigevalue +1 |
| trace_display | numeric[] | list of integer traces tr(a_2), tr(a_3), tr(a_5), tr(a_7), only set when dim > 0 and not yet computed in every case. |
| traces | numeric[] | integer coefficients a_n of the trace form (sum of all newforms in the space) for n from 1 to 1000, only set when dim > 0 and not yet computed in every case. |
| hecke_cutter_primes | integer[] | list of primes that appear in the hecke cutters for the newforms in this space (empty list if num_forms=1, not set for wt1 spaces or if we don't store exact eigenvalues for any forms in the space); only includes primes not dividing the level, minimal in the sense that each successive prime distinguishes forms not distinguished by any previous prime (so the length is always less than num_forms). |
| dihedral_dim | integer | total dimension of dihedral Hecke orbits (only set for weight 1) |
| a4_dim | integer | total dimension of A4 Hecke orbits (only set for weight 1) |
| s4_dim | integer | total dimension of S4 Hecke orbits (only set for weight 1) |
| a5_dim | integer | total dimension of A5 Hecke orbits (only set for weight 1) |
| hecke_orbit_code | bigint | Encoding of the tuple (N.k.i) into 64 bits, used as a key in mf_hecke_newspace_traces. N + (k<<24) + ((i-1)<<36) this is the same as the Hecke orbit code of the first newform in the space. |
Table mf_gamma1:
This table contains data for spaces of newforms S_k^new(Gamma1(N)), most of which is computed by summing the corresponding rows in mf_newspaces.
| Column | Type | Notes |
|---|---|---|
| label | text | (N.k) |
| level | integer | the level N of the modular form |
| level_radical | integer | product of prime divisors of N |
| level_primes | integer[] | list of primes divisors of N |
| level_is_prime | boolean | true if N is prime (1 is not prime) |
| level_is_prime_power | boolean | true if N is a prime power (1 is not a prime power, primes are prime powers) |
| level_is_squarefree | boolean | true if N is squarefree (1 is squarefree) |
| level_is_square | boolean | true if N is a square (1 is a square) |
| weight | smallint | the weight k of the modular form |
| weight_parity | smallint | (-1)^k |
| analytic_conductor | double precision | N*(Exp(Psi((k)/2))/(2*pi))^2 where Psi(t) := Gamma'(t)/Gamma(t) |
| Nk2 | integer | N*k^2 |
| sturm_bound | integer | floor(k*Index(Gamma1(N))/12) |
| trace_bound | integer | nonnegative integer n such that the traces from 1 up to n distinguish all forms in this space (0 if space has 1 form, 1 if more than 1 form but dimensions are all distinct) |
| dim | integer | Q-dimension of S_k^new(Gamma1(N)) |
| num_forms | integer | number of (Hecke/Galois orbits of) newforms, only set when known |
| hecke_orbit_dims | integer[] | Sorted list of Q-dimensions of Hecke orbits, only set when known |
| num_spaces | integer | number of nozero newspaces S_k^new(N,[\chi]) in S_k^{new}(Gamma1(N)) |
| newspace_dims | integer[] | list of Q-dimensions of newspaces S_k^new(N,\chi) in S_k^new(Gamma1(N)) ordered by character orbit index |
| eis_dim | integer | Q-dimension of the eisenstein subspace of M_k(Gamma1(N)) |
| eis_new_dim | integer | Q-dimension of the new eisenstein subspace ofM_k(Gamma1(N)) |
| cusp_dim | integer | Q-dimension of the cuspidal space S_k(Gamma1(N)) |
| mf_dim | integer | Q-dimension of the full spaceM_k(Gamma1(N)) |
| mf_new_dim | integer | Q-dimension of the new subspace of M_k(N,\chi) |
| trace_display | numeric[] | list of integer traces tr(a_2), tr(a_3), tr(a_5), tr(a_7), only set when dim > 0, not yet computed in every case. |
| traces | numeric[] | integer coefficients a_n of the trace form (sum of all newforms) for n from 1 to 1000, only set when dim > 0, not yet computed in every case. |
| dihedral_dim | integer | total dimension of dihedral Hecke orbits (only set for weight 1) |
| a4_dim | integer | total dimension of A4 Hecke orbits (only set for weight 1) |
| s4_dim | integer | total dimension of S4 Hecke orbits (only set for weight 1) |
| a5_dim | integer | total dimension of A5 Hecke orbits (only set for weight 1) |
Table mf_newspace_portraits:
| Column | Type | Notes |
|---|---|---|
| label | text | label (N.k.a) of the newspace |
| level | integer | level N |
| weight | smallint | weight k |
| char_orbit_index | smallint | index of the character orbit [\chi] n the sorted list of character orbits of modulus N |
| portrait | text | base-64 encoded image of the newspace (plot created by portrait.sage) to display in the properties box |
Tablemf_gamma1_portraits:
| Column | Type | Notes |
|---|---|---|
| label | text | label N.k for the space S_k^new(Gamma1(N)) |
| level | integer | level N |
| weight | smallint | weight k |
| portrait | text | base-64 encoded image of the newspace (plot created by portrait.sage) to display in the properties box |
Table mf_subspaces:
This table represents embeddings of newspaces at level M into cusp spaces at level N (these will be old at level N except when M=N).
| Column | Type | Notes |
|---|---|---|
| label | text | label N.k.a for the cuspidal space S_k(N, [\chi]) (same as the label for S_k^{new}(N, [\chi])) |
| level | integer | level N of the cuspidal space S_k(N, [\chi]) |
| weight | smallint | weight k of the cuspidal space S_k(N, [\chi]) |
| char_orbit_index | smallint | index a of the character orbit [\chi] in the sorted list of character orbits of modulus N |
| char_orbit_label | text | base-26 encoding (1='a') of index a of the character orbit that appears in label |
| conrey_indexes | integer[] | list of Conrey indexes n of the characters N.n in the Galois orbit indexed by a |
| sub_label | text | The label of the newspace S_k^{new}(M, [\psi]) that appears as a non-trivial subspace ofS_k(N, [\chi]) |
| sub_level | integer | (M) |
| sub_char_orbit_index | smallint | index j of [\psi] in sorted list of character orbits of modulus M |
| sub_char_orbit_label | text | base-26 encoding (1='a') of index j of the subspace character orbit that appears in sub_label |
| sub_conrey_indexes | integer[] | list of Conrey indexes n of the characters M.n in the Galois orrbit indexed by j. |
| sub_dim | integer | the dimension of S_k^{new}(M, [\psi]) |
| sub_mult | integer | Multiplicity ofS_k^{new}(M, [\psi]) as a direct summand of S_k(N, [\chi]) (this is just the number of divisors of N/M). Summing dimensions of embedded newspaces with multiplicity gives the dimension of the cusp space. |
Table mf_gamma1_subspaces:
| Column | Type | Notes |
|---|---|---|
| label | text | label N.k for the cuspidal space S_k(Gamma1(N)) (same as the label for S_k^{new}(Gamma1(N)) |
| level | integer | level N of the space S_k(Gamma_1(N)) |
| weight | smallint | weight k of the space S_k(Gamma_1(N)) |
| sub_level | integer | level M of the newspace S_k^{new}(Gamma_1(M)) that embed in S^k(Gamma_1(N)) |
| sub_dim | integer | dimension of S_k^{new}(Gamma_1(M)) |
| sub_mult | integer | multiplicity of S_k^{new}(Gamma_1(M)) as a direct summand of S_k^{Gamma_1(N)). Summing dimensions of embedded newspaces S_k^{new}(Gamma_1(M)) with multiplicity gives the dimension of the cusp space S_k(Gamma_1(N). |
Table mf_newforms:
| Column | Type | Notes |
|---|---|---|
| label | text | newform label N.k.a.x where N=level, k=weight, i=char_orbit_label, x=hecke_orbit_label (both a and x are base26 indexes with 'a'=1) |
| space_label | text | (newspace label N.k.a |
| level | integer | the level N of the modular form |
| level_radical | integer | product of prime divisors of N |
| level_primes | integer[] | list of prime divisors of N |
| level_is_prime | boolean | true if N is prime (1 is not prime) |
| level_is_prime_power | boolean | true if N is a prime power (1 is not a prime power, primes are prime powers) |
| level_is_squarefree | boolean | true if N is squarefree (1 is squarefree) |
| level_is_square | boolean | true if N is a square (1 is a square) |
| weight | smallint | the weight k of the modular form |
| weight_parity | smallint | (-1)^k |
| analytic_conductor | double precision | N*(Exp(Psi((k)/2))/(2*pi))^2 where Psi(t) := Gamma'(t)/Gamma(t) |
| Nk2 | integer | N*k^2 |
| char_orbit_index | smallint | The index i of the Galois orbit of this form in the sorted list of character orbits, as described above. |
| char_orbit_label | text | base26 encodeing of char_orbit_index a (with a=1). |
| char_conductor | integer | Conductor of the Dirichlet character |
| prim_orbit_index | smallint | char_orbit_index for the primitive version of this character |
| char_order | integer | the order of the character |
| conrey_indexes | integer[] | Sorted list of Conrey indexes of characters in this Galois orbit |
| char_degree | integer | Degree of the (cyclotomic) character field |
| char_parity | smallint | 1 or -1, depending on the parity of the character |
| char_is_real | boolean | whether the character takes only real values (trivial or quadratic) |
| char_values | jsonb | quadruple <N,n,u,v> where N is the level, n is the order of the character, u is a list of generators for the unit group of Z/NZ, and v is a corresponding list of integers for which chi(u[i]) = zeta_n^v[i] |
| hecke_orbit | integer | the index of this newform in its newpace (lex-ordered by traces), its base26 endoding is the x in the newofrm lable (a=1) |
| hecke_orbit_code | bigint | encoding of the tuple (N.k.a.x) into 64 bits, used in eigenvalue tables. N + (k<<24) + ((a-1)<<36) + ((x-1)<<52). |
| dim | integer | the Q-dimension of this Galois orbit |
| relative_dim | integer | the Q(chi)-dimension of this Hecke orbit (=dim/char_degree) |
| field_disc | numeric | discriminant of the coefficient field, if known |
| field_disc_factorization | numeric[] | factorization of field discriminant stored as ordered list of pairs [p,e] |
| field_poly | numeric[] | list of integers giving defining polynomial for the Hecke field (standard Sage order of coefficients) |
| field_poly_is_cyclotomic | boolean | true if field_poly is a cylcotomic polynomial (the field might be Q(zeta_n) even when this flage is not set if we haven't chosen a cyclotomic polynomial to define it) |
| field_poly_is_real_cyclotomic | boolean | true if field_poly is the minimal polynomial of zeta_n + zeta_n^-1 for some n (the field might be Q(zeta_n)^+ even when this flage is not set if we haven't chosen a cyclotomic polynomial to define it) |
| field_poly_root_of_unity | integer | the value of n if either field_poly_is_cylotomic of field_poly_is_real_cyclotomic is set |
| is_polredabs | boolean | whether the polynomial has been reduced by Pari's polredabs |
| nf_label | text | LMFDB label for the corresponding number field (can be NULL) |
| is_self_dual | boolean | true if L-func is self-dual (coeff field is totally real) |
| hecke_ring_generator_nbound | integer | minimal integer m such that a_1,...,a_m generate the Hecke ring |
| hecke_ring_index | numeric | (a divisor of) the index of the order generated by the Hecke eigenvalues in the maximal order. |
| hecke_ring_index_factorization | numeric[] | Factorization of hecke_ring_index stored as ordered list of pairs [p,e]. |
| hecke_ring_index_proved | boolean | whether the index has been proved correct (computing the maximal order may not be possible) |
| trace_hash | bigint | linear combination of the a_p between 2^12 and 2^13 reduced mod 2^61-1 as defined in BSSVY, only guaranteed for wt > 1 and dim <= 20 |
| trace_zratio | double precision | proportion of zero a_p values for p <= 2^13 (rounded to three decimal places) |
| trace_moments | numeric[] | list of moments of a_p/p^((k-1)/2) computed over p <= 2^13 (rounded to three decimal places) |
| related_objects | text[] | list of text URLs of related objects (e.g. elliptic curve isogeny class, Artin rep, ...), e.g. ["EllipticCurve/Q/11/a"] |
| embedded_related_objects | text[] | list of lists of text URLs of related objects (e.g. Artin reps), indexed by embedding_m (so first entry is a list of friends for the first embeddeded newform) |
| analytic_rank | smallint | order of vanishing of L-function at s=1 (an upper bound, tight if analytic_rank_proved is set) |
| analytic_rank_proved | boolean | true if analytic rank is provably correct (it is always an upper bound) |
| self_twist_type | smallint | 0=none, 1=cm, 2=rm, 3=both |
| is_self_twist | boolean | whether this form is a self twist |
| minimal_twist | text | label of the designated twist-minimal rep of the twist-class of this newform |
| is_twist_minimal | boolean | true if level N is the same as the level of minimal_twist |
| is_twist_rep | boolean | true if label == minimal_twist, i.e this newform is the designated rep of its twist class |
| is_cm | boolean | whether the form has CM |
| is_rm | boolean | whether the form has RM |
| self_twist_discs | integer[] | list of discriminants giving self twists (either 0,1,or 3 quadratic discriminants) |
| cm_discs | integer[] | list of CM discriminants (the negative discriminants listed in self_twist_discs) |
| rm_discs | integer[] | list of RM discriminants (the positive discriminants listed in self_twist_discs) |
| has_non_self_twist | smallint | 1 if form admits a non-trivial inner twist, 0 if it does not, -1 if unknown |
| inner_twists | integer[] | List of septuples of integers [b,m,M,o,parity,order,disc] where <M,o> identifies the Galois orbit of a Dirichlet character, m is the number of characters in this orbit that give rise to an inner twist, and b is 1 if the inner twists is proved. All inner twists are guaranteed to be included in the list, but those without proved set could be false positives. |
| inner_twist_count | integer | number of inner twists (includes proved and unproved), -1 if inner twists have not been computed (this applies to all forms of dimension > 20 and weight > 1) |
| atkin_lehner_eigenvals | integer[] | a list of pairs [p, ev] where ev is 1 or -1, the Atkin-Lehner eigenvalue for each p dividing N (NULL overall if nontrivial character, an empty list for level 1 and trivial character) |
| atkin_lehner_string | text | list of signs +/- of Atkin-Lehner eigenvalues ordered by p (facilitates lookups) |
| fricke_eigenval | smallint | product of the Atkin-Lehner eigenvalues (NULL if nontrivial character) |
| hecke_cutters | jsonb | a list of pairs [p, F_p] where F_p is a list of integers encoding a polynomial; the intersection of the kernels of F_p(T_p) is this Hecke orbit |
| qexp_display | text | latexed string for display on search page results |
| trace_display | numeric[] | list of traces tr(a_2), tr(a_3), tr(a_5), tr(a_7) for display on search page results |
| traces | numeric[] | full list of traces tr(a_n) for n from 1 to 1000 (or more) |
| projective_image_type | text | for weight 1 forms, one of "Dn", "A4", "S4", "A5" |
| projective_image | text | for weight 1 forms, isomorphism class of project image (e.g. which Dn) |
| projective_field | numeric[] | for weight 1 forms, list of integer coefficients of polynomial whose splitting field is the fixed field of the kernel of the projective Galois rep (subfield of the artin field fixed be the center of its Galois group) |
| projective_field_label | text | LMFDB label of projective field (if present) |
| artin_degree | integer | for weight 1 forms, order of the image of the Galois rep, equivalently, the degree of the Artin field |
| artin_image | text | for weight 1 forms, small group label of the image of the Galois rep (and the Galois group of the artin field) |
| artin_field | numeric[] | for weight 1 forms, list of integer coefficients of polynomial whose splitting field is the fixed field of the Galois rep (equivalently, a defining polynomial for the 2-dim Artin rep corresponding to this weight 1 form) |
| artin_field_label | text | LMFDB label of artin field (if present) |
| sato_tate_group | text | LMFDB label of Sato-Tate group (currently only present for weight k > 1) |
Table mf_newform_portraits:
| Column | Type | Notes |
|---|---|---|
| label | text | label (N.k.a.x) of the newform |
| level | integer | level N |
| weight | smallint | weight k |
| char_orbit_index | smallint | character orbit index (base26 encoding a appears in label) |
| hecke_orbit | integer | Hecke orbit index (base26 endcoding x appears in label) |
| portrait | text | base-64 encoded image of the newform (plot created by portrait.sage) to display in the properties box |
Table mf_twists_nf:
| Column | Type | Notes |
|---|---|---|
| source_label | text | label (N.k.a.x) of the newform being twisted |
| target_label | text | label (N.k.a.x) of the twisted newform |
| twisting_char_label | text | label M.a of the twisting character orbit [psi] (psi is always primitive) |
| multiplicity | smallint | # of twists by distinct psi in [psi] for each embedded newform |
| conductor | integer | conductor of psi (equal to the modulus M in its label since psi is primitive) |
| order | integer | order of psi |
| degree | integer | degree of psi = [Q(psi):Q] = phi(order) = cardinality of character orbit [psi] |
| parity | smallint | parity of psi |
| self_twist_disc | integer | for self twists the discriminant of the Kronecker character (1 for trivial char), 0 otherwise |
| source_level | integer | level of the source newform |
| target_level | integer | level of the target newform |
| source_dim | integer | dimension of the source newform |
| target_dim | integer | dimension of the target newform |
| source_char_orbit | smallint | character orbit index of source newform (numeric value of i in source_label) |
| target_char_orbit | smallint | character orbit index of target newform (numeric value of i in target_label) |
| twisting_char_orbit | smallint | character orbit index of the twisting character (numeric value of i in twisting_char_label) |
| source_hecke_orbit | integer | Hecke orbit index of source newform (numeric value of x in source label) |
| target_hecke_orbit | integer | Hecke orbit index of target newform (numeric value of x in target label) |
| twist_class_label | text | newform label N.k.a.x of the designated twist class representative (of minimal level and character) |
| twist_class_level | integer | level of the twist-minimal newforms in this twist class (N in twist_class_label) |
| source_is_minimal | boolean | true if source newform is twist-minimal and has minimal character |
| target_is_minimal | boolean | true if target newform is twist-minimal and has minimal character |
| weight | smallint | weight k of source and target newforms (and all newforms in the twist equivalence class) |
Table mf_twists_cc:
| Column | Type | Notes |
|---|---|---|
| source_label | text | label (N.k.a.x.n.i) of the embedded newform being twisted |
| target_label | text | label (N.k.a.x.n.i) of the twisted embedded newform |
| twisting_char_label | text | Conrey label M.n of the twisting character psi (psi is always primitive) |
| conductor | integer | conductor of psi (equal to the modulus M in its label since psi is primitive) |
| order | integer | order of psi |
| degree | integer | degree of psi = [Q(psi):Q] = phi(order) = cardinality of character orbit [psi] |
| parity | smallint | parity of psi |
| source_level | integer | level of the source newform |
| target_level | integer | level of the target newform |
| source_dim | integer | dimension of the source newform orbit |
| target_dim | integer | dimension of the target newform orbit |
| source_conrey_index | integer | Conrey index of source newform (numeric value of n in source_label) |
| target_conrey_index | integer | Conrey index of target newform character (numeric value of n in target_label) |
| twisting_conrey_index | integer | Conrey index of twisting character psi (numeric value of n in twisting_char_label) |
| source_hecke_orbit_code | bigint | Hecke orbit code of source newform (64-bit encoding of N.k.a.x in source label) |
| target_hecke_orbit_code | bigint | Hecke orbit code of target newform (64-bit encoding of N.k.a.x in target label) |
| twist_class_label | text | embedded newform label N.k.a.x.n.i of the designated twist class representative (of minimal level and character) |
| twist_class_level | integer | level of the minimal twist equivalent embedded newform |
| source_is_minimal | boolean | true if source newform is twist-minimal and has minimal character |
| target_is_minimal | boolean | true if target newform is twist-minimal and has minimal character |
| weight | smallint | weight k of source and target newforms (and all newforms in the twist equivalence class) |
Table mf_hecke_nf:
| Column | Type | Notes |
|---|---|---|
| label | text | label of newform (N.k.a.x) |
| level | integer | level N |
| weight | smallint | weight k |
| char_orbit_index | smallint | character orbit index (base26 encodes as a in newform label, 'a'=1) |
| hecke_orbit_code | bigint | encoding of the tuple (N.k.a.x) into 64 bits |
| field_poly | numeric[] | list of integers of defining polynomial for Hecke field |
| hecke_ring_rank | integer | rank of Hecke ring as a free Z-module (same as dimension of form, degree of field_poly) |
| hecke_ring_power_basis | boolean | if true the chanage of basis matrix is the (implicit) identity matrix, in which case hecke_ring_numerators, ..., hecke_ring_inverse_denominators are set to null |
| hecke_ring_cyclotomic_generator | integer | either zero or an integer m such that the an and ap are encoded as sparse integer polynomials in zeta_m (typically same as field_poly_root_of_unity but this is not required) |
| hecke_ring_numerators | numeric[] | List of lists of integers, giving the numerators of a basis for the Hecke order in terms of the field generator specified by the field polynomial |
| hecke_ring_denominators | numeric[] | List of integers, giving the denominators of the basis |
| hecke_ring_inverse_numerators | numeric[] | List of lists of integers, giving the numerators of the inverse basis that specifies powers of nu in terms of the betas |
| hecke_ring_inverse_denominators | numeric[] | List of integers, giving the denominators of the inverse basis |
| hecke_ring_character_values | jsonb | list of pairs [[m1,[a11,...a1n]],[m2,[a12,...,a2n]],...] where [m1,m2,...,mr] are generators for Z/NZ and [ai1,...,ain] is the value of chi(mi) expressed in terms of the Hecke ring basis or in cyclotomic representation [[c,e]] encoding c x zeta_m^e where m is hecke_ring_cyclotomic_generator |
| an | jsonb | list of a1,...,a100, where each an is either a list of integers encoding an in the Hecke ring basis described above or a list of pairs [[c1,e1],...,[cr,er]] encoding an = c1 x zeta_m^e1 + ... + cr x zeta_m^er, where m is the value of hecke_ring_cyclotomic_generator (if nonzero) |
| ap | jsonb | list of lists of integers expressing a_p for p=2,3,5,...,pmax in same format as an |
| maxp | integer | largest prime p for which ap is stored |
Table mf_hecke_traces:
| Column | Type | Notes |
|---|---|---|
| hecke_orbit_code | bigint | encoding of the tuple (N.k.a.x) into 64 bits |
| n | integer | index of a_n |
| trace_an | numeric | trace of a_n down to Z |
Table mf_hecke_charpolys:
| Column | Type | Notes |
|---|---|---|
| hecke_orbit_code | bigint | encoding of the tuple (N.k.a.x) into 64 bits |
| p | integer | prime identifying L-poly L_p(T) = prod_(sigma in Gal(Q(f)/Q) (1 - sigma(a_p(f))T + chi(p)p^(k-1)T^2)) |
| charpoly_factorization | jsonb | list consisting of a single pair [fac,e] where fac is a list of coefficients of the minpoly and e is [Q(f):Q(a_p)] |
Table mf_hecke_newspace_traces:
| Column | Type | Notes |
|---|---|---|
| hecke_orbit_code | bigint | encoding of the tuple (N.k.a) into 64 bits (this is the same as the Hecke orbit code for the first newform in the space, but in this table the traces are sums over the entire newspace N.k.a) |
| n | integer | index of a_n |
| trace_an | numeric | trace of a_n down to Z, where a_n is the sum of a_n over all newforms in the space |
Table mf_hecke_cc:
| Column | Type | Notes |
|---|---|---|
| label | text | label N.k.a.x.n.i of the embedded newform |
| level | integer | the level N of the embedded newform |
| weight | smallint | weight k of the embedded newform |
| char_orbit_index | smallint | orbit index a of the character N.n of the embedded newform |
| hecke_orbit | integer | the integer value of x in the Hecke orbit code |
| hecke_orbit_code | bigint | encoding of the tuple (N.k.a.x) into 64 bits |
| conrey_index | integer | Conrey index n of the character N.n of the embedded newform |
| embedding_index | integer | index of embedding among the embeddings for the given conrey_index |
| embedding_m | integer | enumeration of which embedding over all conrey labels in the specified hecke orbit. Ordering is the same as lexicographic on (conrey_index, embedding_index). 1-indexed. |
| dual_conrey_index | integer | Conrey index of the dual character |
| dual_embedding_index | integer | embedding_index of dual (complex conjugate) embedded newform |
| dual_embedding_m | integer | embedding_m for dual embedded newform |
| embedding_root_real | double precision | real part of the root corresponding to this embedding |
| embedding_root_imag | double precision | imaginary part of the root corresponding to this embedding |
| an_normalized | double precision[] | list of pairs {x,y} of doubles x, y so that a_n = n^{k-1)/2} * (x + iy) for n \in [1,1000] |
| angles | double precision[] | list of \theta_p, where '\theta_p' is Null if p is bad, and for good p we have a_p = p^{(k-1)/2} (e^{2\pi i \theta_p} + chi(p)e^{-2\pi i \theta_p}); indexed by increasing primes p less than 1000, where theta_p lie in (-0.5,0.5]. Furthermore, we store the minimum value of the two options for \theta_p in that interval. |
Table char_dir_orbits:
| Column | Type | Notes |
|---|---|---|
| label | text | character orbit label N.a, where N is the modulus and a is the orbit index base 26 encoded |
| orbit_label | text | character orbit label N.a, where N is the modulus and a is the orbit index (currently not base26 encoded, this should really be fixed!) |
| orbit_index | smallint | Index in the list of traces down to Q of the values of all characters of modulus N |
| modulus | integer | |
| conductor | integer | |
| prim_orbit_index | integer | Orbit index for the primitive character inducing this one (note that this index identifies a Galois orbit of characters of modulus M = conductor) |
| order | integer | order of the character |
| parity | smallint | +1 = even, -1 = odd |
| galois_orbit | integer[] | sorted list of conrey indexes of characters in the same galois orbit |
| is_real | boolean | if quadratic or trivial |
| is_primitive | boolean | if modulus = conductor |
| is_minimal | boolean | if the character is minimal (as defined in the character.dirichlet.minimal knowl) |
| char_degree | integer | degree of the cyclotomic field containing the image, ie Euler phi of the order; this is the same as the size of the Galois orbit |
| kernel_field_poly | numeric[] | coefficients of defining polynomial for kernel field |
Table char_dir_values:
Note that the values in this table are stored as integers m so that the actual value is e^{2\pi i m/d} where d is the order.
| Column | Type | Notes |
|---|---|---|
| label | text | N.n where N is the modulus and n is the conrey index |
| modulus | smallint | the modulis, i.e., the N in the label |
| conrey_index | smallint | the n in the Conrey label N.n |
| orbit_label | text | N.a where N is the modulus and a is the orbit_index |
| prim_label | text | the label of primitive character inducing this one |
| modulus | integer | |
| conrey_index | integer | |
| order | integer | |
| values | integer[] | list of pairs [x,m] giving first twelve values e(m/n) on x=-1,1, then the next ten integers relatively prime to the modulus, where n is the order of the character |
| values_gens | integer[] | list of pairs [x, m] giving values on generators x of the unit group |