gh-40083: using _an_element_ in modular/
    
in two files about Multiple Zeta Values

### 📝 Checklist

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- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: #40083
Reported by: Frédéric Chapoton
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/modular/multiple_zeta.py

@@ -420,7 +420,7 @@ class MultizetaValues(Singleton):
         sage: parent(M((2,3,4,5), prec=128))
         Real Field with 128 bits of precision
     """
-    def __init__(self):
+    def __init__(self) -> None:
         """
         When first called, pre-compute up to weight 8 at precision 1024.
 
@@ -443,7 +443,7 @@ def __repr__(self) -> str:
         """
         return f"Cached multiple zeta values at precision {self.prec} up to weight {self.max_weight}"
 
-    def reset(self, max_weight=8, prec=1024):
+    def reset(self, max_weight=8, prec=1024) -> None:
         r"""
         Reset the cache to its default values or to given arguments.
 
@@ -464,7 +464,7 @@ def reset(self, max_weight=8, prec=1024):
         self.max_weight = int(max_weight)
         self._data = pari.zetamultall(self.max_weight, precision=self.prec)
 
-    def update(self, max_weight, prec):
+    def update(self, max_weight, prec) -> None:
         """
         Compute and store more values if needed.
 
@@ -550,7 +550,7 @@ def __call__(self, index, prec=None, reverse=True):
 Values = MultizetaValues()
 
 
-def extend_multiplicative_basis(B, n) -> Iterator:
+def extend_multiplicative_basis(B, n) -> Iterator[tuple]:
     """
     Extend a multiplicative basis into a basis.
 
@@ -647,7 +647,7 @@ class Multizetas(CombinatorialFreeModule):
         sage: z
         ζ(1,2,3)
     """
-    def __init__(self, R):
+    def __init__(self, R) -> None:
         """
         TESTS::
 
@@ -726,7 +726,7 @@ def some_elements(self) -> tuple:
         """
         return self([]), self([2]), self([3]), self([4]), self((1, 2))
 
-    def an_element(self):
+    def _an_element_(self):
         r"""
         Return an element of the algebra.
 
@@ -1253,7 +1253,7 @@ def _richcmp_(self, other, op) -> bool:
                 raise TypeError('invalid comparison for multizetas')
             return self.iterated()._richcmp_(other.iterated(), op)
 
-        def __hash__(self):
+        def __hash__(self) -> int:
             """
             Return the hash of ``self``.
 
@@ -1404,7 +1404,7 @@ class Multizetas_iterated(CombinatorialFreeModule):
         sage: M((1,0))*M((1,0,0))
         6*I(11000) + 3*I(10100) + I(10010)
     """
-    def __init__(self, R):
+    def __init__(self, R) -> None:
         """
         TESTS::
 
@@ -2067,7 +2067,7 @@ class All_iterated(CombinatorialFreeModule):
         sage: x.regularise()
         -I(10)
     """
-    def __init__(self, R):
+    def __init__(self, R) -> None:
         """
         TESTS::
 

src/sage/modular/multiple_zeta_F_algebra.py

@@ -266,7 +266,7 @@ class F_algebra(CombinatorialFreeModule):
         sage: s = f2*f3+f5; s
         f5 + f2*f3
     """
-    def __init__(self, R, start=3):
+    def __init__(self, R, start=3) -> None:
         r"""
         Initialize ``self``.
 
@@ -460,7 +460,7 @@ def gen(self, i):
         B *= ZZ(2)**(3 * i - 1) * ZZ(3)**i / ZZ(2 * i).factorial()
         return B * f2**i
 
-    def an_element(self):
+    def _an_element_(self):
         """
         Return a typical element.
 
@@ -473,7 +473,7 @@ def an_element(self):
         """
         return self("253") + 3 * self("235")
 
-    def some_elements(self):
+    def some_elements(self) -> list:
         """
         Return some typical elements.