gh-35517: implement the depth of a quasimodular form
    
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This PR implements the depth of a quasimodular form. Recall that the
depth of a quasimodular form $f$ of weight $k$ is the integer $p$ such
that:

$$
f = f_0 + f_1 E_2 + \cdots + f_p E_2^p
$$

where $f_i$ is of weight $k - 2*i$ and $f_p\neq 0$.

CC: @videlec

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URL: #35517
Reported by: David Ayotte
Reviewer(s): David Ayotte, grhkm21

Author: Release Manager

src/sage/modular/quasimodform/element.py

@@ -31,16 +31,20 @@
 
 class QuasiModularFormsElement(ModuleElement):
     r"""
-    A quasimodular forms ring element. Such an element is describbed by SageMath
-    as a polynomial
+    A quasimodular forms ring element. Such an element is described by
+    SageMath as a polynomial
 
     .. MATH::
 
-        f_0 + f_1 E_2 + f_2 E_2^2 + \cdots + f_m E_2^m
+        F = f_0 + f_1 E_2 + f_2 E_2^2 + \cdots + f_m E_2^m
 
     where each `f_i` a graded modular form element
     (see :class:`~sage.modular.modform.element.GradedModularFormElement`)
 
+    For an integer `k`, we say that `F` is homogeneous of weight `k` if
+    it lies in an homogeneous component of degree `k` of the graded ring
+    of quasimodular forms.
+
     EXAMPLES::
 
         sage: QM = QuasiModularForms(1)
@@ -295,6 +299,45 @@ def __bool__(self):
         """
         return bool(self._polynomial)
 
+    def depth(self):
+        r"""
+        Return the depth of this quasimodular form.
+
+        Note that the quasimodular form must be homogeneous of weight
+        `k`. Recall that the *depth* is the integer `p` such that
+
+        .. MATH::
+
+            f = f_0 + f_1 E_2 + \cdots + f_p E_2^p,
+
+        where `f_i` is a modular form of weight `k - 2i` and `f_p` is
+        nonzero.
+
+        EXAMPLES::
+
+            sage: QM = QuasiModularForms(1)
+            sage: E2, E4, E6 = QM.gens()
+            sage: E2.depth()
+            1
+            sage: F = E4^2 + E6*E2 + E4*E2^2 + E2^4
+            sage: F.depth()
+            4
+            sage: QM(7/11).depth()
+            0
+
+        TESTS::
+
+            sage: QM = QuasiModularForms(1)
+            sage: (QM.0 + QM.1).depth()
+            Traceback (most recent call last):
+            ...
+            ValueError: the given graded quasiform is not an homogeneous element
+        """
+        if not self.is_homogeneous():
+            raise ValueError("the given graded quasiform is not an "
+                             "homogeneous element")
+        return self._polynomial.degree()
+
     def is_zero(self):
         r"""
         Return whether the given quasimodular form is zero.