sagemathgh-41194: Fix printing of Tate series

The bigoh of a Tate series with a nonzero log-radius is currently
incorrectly printed.

```
sage: S.<x> = TateAlgebra(Qp(5), log_radii=[-1])
sage: S(x, 10)
(1 + O(5^9))*x + O(5^10 * <5*x>)
```

This should be `O(5^10 * <x/5>)`. This PR fixes this.

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
- [x] I have updated the documentation and checked the documentation
preview.

URL: sagemath#41194
Reported by: Xavier Caruso
Reviewer(s): Rubén Muñoz--Bertrand

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,3 +1,3 @@
 tarball=configure-VERSION.tar.gz
-sha1=8b7aba4104cd77481cfc7602c26382d1788ecd2e
-sha256=99c4fd081a93073924a90d7b8af2dbb6fe23d8b4602115511fa1399080667023
+sha1=760ea281be72aa2e57309e5f0f9544dea2d6759d
+sha256=b4edef2c928a0c929b1c5d09f8c2fb108a5f350e941b8dbe7b510d5ea3a4a819

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-4119163dc533ea5446f33e48918fd637a8e856e2
+070bec01dbddc7c8980fbe0ed0746ff39802d219

src/sage/rings/tate_algebra_element.pyx

@@ -1123,7 +1123,7 @@ cdef class TateAlgebraElement(CommutativeAlgebraElement):
             sage: S.<x,y> = TateAlgebra(Qp(5), log_radii=(1,0))
             sage: f = 5*x
             sage: f.add_bigoh(1)
-            (5 + O(5^2))*x + O(5 * <x/5, y>)
+            (5 + O(5^2))*x + O(5 * <5*x, y>)
         """
         self._is_normalized = True
         if self._prec is Infinity:
@@ -1169,6 +1169,15 @@ cdef class TateAlgebraElement(CommutativeAlgebraElement):
 
             sage: A(x + 2*x^2 + x^3, prec=5)
             ...00001*x^3 + ...00001*x + ...00010*x^2 + O(2^5 * <x, y>)
+
+        TESTS::
+
+            sage: S.<x> = TateAlgebra(R, log_radii=[-1])
+            sage: S(x, 5)
+            ...0001*x + O(2^5 * <x/2>)
+            sage: S.<x> = TateAlgebra(R, log_radii=[1])
+            sage: S(x, 5)
+            ...000001*x + O(2^5 * <2*x>)
         """
         vars = self._parent.variable_names()
         s = ""
@@ -1191,14 +1200,14 @@ cdef class TateAlgebraElement(CommutativeAlgebraElement):
             for i in range(len(vars)):
                 if lr[i] == 0:
                     sv.append(vars[i])
-                elif lr[i] == -1:
-                    sv.append("%s*%s" % (su, vars[i]))
                 elif lr[i] == 1:
+                    sv.append("%s*%s" % (su, vars[i]))
+                elif lr[i] == -1:
                     sv.append("%s/%s" % (vars[i], su))
-                elif lr[i] < 0:
-                    sv.append("%s^%s*%s" % (su, -lr[i], vars[i]))
+                elif lr[i] > 0:
+                    sv.append("%s^%s*%s" % (su, lr[i], vars[i]))
                 else:
-                    sv.append("%s/%s^%s" % (vars[i], su, lr[i]))
+                    sv.append("%s/%s^%s" % (vars[i], su, -lr[i]))
             sv = ", ".join(sv)
             if self._prec == 0:
                 s += "O(<%s>)" % sv
@@ -2534,8 +2543,8 @@ cdef class TateAlgebraElement(CommutativeAlgebraElement):
         However `\log(1+x)` converges on a smaller disk::
 
             sage: f.restriction(-1).log()
-            ...000000001*x + ...0000000.1*x^3 + ...111111*x^2 + ...
-             + O(3^10 * <3*x, 3*y>)
+            ...000000001*x + ...0000000.1*x^3 + ...11111111*x^2 + ...
+             + O(3^10 * <x/3, y/3>)
 
         TESTS::
 
@@ -2692,8 +2701,8 @@ cdef class TateAlgebraElement(CommutativeAlgebraElement):
         However `\exp(x)` converges on a smaller disk::
 
             sage: f.restriction(-1).exp()
-            ...0000000001 + ...000000001*x + ...1111111.2*x^3 + ...111112*x^2
-             + ... + O(3^10 * <3*x, 3*y>)
+            ...0000000001 + ...000000001*x + ...1111111.2*x^3 + ...11111112*x^2
+             + ... + O(3^10 * <x/3, y/3>)
 
         TESTS::