Trac #13445: Cuspidal subspace of modular forms over finite field con…

…tains forms that are not cuspidal

Executing:
{{{
M=ModularForms(Gamma1(29),base_ring=GF(29))
S=M.cuspidal_subspace()
S.basis()
}}}
gives:
{{{
[
1 + O(q^6),
q + O(q^6),
q^2 + O(q^6),
q^3 + O(q^6),
q^4 + O(q^6),
q^5 + O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6),
O(q^6)
]
}}}

The first element is clearly not cuspidal

URL: http://trac.sagemath.org/13445
Reported by: mderickx
Ticket author(s): Alex Ghitza
Reviewer(s): Peter Bruin

src/sage/modular/modform/ambient.py

@@ -182,9 +182,9 @@ def change_ring(self, base_ring):
             sage: M3 = M.change_ring(GF(3))
             sage: M3.basis()
             [
-            1 + q^3 + q^4 + 2*q^5 + O(q^6),
             q + q^3 + q^4 + O(q^6),
-            q^2 + 2*q^3 + q^4 + q^5 + O(q^6)
+            q^2 + 2*q^3 + q^4 + q^5 + O(q^6),
+            1 + q^3 + q^4 + 2*q^5 + O(q^6)
             ]
         """
         import constructor

src/sage/modular/modform/ambient_R.py

@@ -85,21 +85,60 @@ def _compute_q_expansion_basis(self, prec=None):
         """
         Compute q-expansions for a basis of self to precision prec.
 
-        EXAMPLES:
+        EXAMPLES::
+
             sage: M = ModularForms(23,2,base_ring=GF(7))
-            sage: M._compute_q_expansion_basis(5)
-            [1 + 5*q^3 + 5*q^4 + O(q^5),
-            q + 6*q^3 + 6*q^4 + O(q^5),
-            q^2 + 5*q^3 + 6*q^4 + O(q^5)]
+            sage: M._compute_q_expansion_basis(10)
+            [q + 6*q^3 + 6*q^4 + 5*q^6 + 2*q^7 + 6*q^8 + 2*q^9 + O(q^10),
+            q^2 + 5*q^3 + 6*q^4 + 2*q^5 + q^6 + 2*q^7 + 5*q^8 + O(q^10),
+            1 + 5*q^3 + 5*q^4 + 5*q^6 + 3*q^8 + 5*q^9 + O(q^10)]
+
+        TESTS:
+
+        This checks that :trac:`13445` is fixed::
+
+            sage: M = ModularForms(Gamma1(29), base_ring=GF(29))
+            sage: S = M.cuspidal_subspace()
+            sage: 0 in [f.valuation() for f in S.basis()]
+            False
+            sage: len(S.basis()) == dimension_cusp_forms(Gamma1(29), 2)
+            True
         """
         if prec == None:
             prec = self.prec()
-        if self.base_ring().characteristic() == 0:
+        R = self._q_expansion_ring()
+        c = self.base_ring().characteristic()
+        if c == 0:
             B = self.__M.q_expansion_basis(prec)
+            return [R(f) for f in B]
+        elif c.is_prime_power():
+            K = self.base_ring()
+            p = K.characteristic().prime_factors()[0]
+            from sage.rings.all import GF
+            Kp = GF(p)
+            newB = [f.change_ring(K) for f in list(self.__M.cuspidal_subspace().q_integral_basis(prec))]
+            A = Kp**prec
+            gens = [f.padded_list(prec) for f in newB]
+            V = A.span(gens)
+            B = [f.change_ring(K) for f in self.__M.q_integral_basis(prec)]
+            for f in B:
+                fc = f.padded_list(prec)
+                gens.append(fc)
+                if not A.span(gens) == V:
+                    newB.append(f)
+                    V = A.span(gens)
+            if len(newB) != self.dimension():
+                raise RuntimeError("The dimension of the space is %s but the basis we computed has %s elements"%(self.dimension(), len(newB)))
+            lst = [R(f) for f in newB]
+            return [f/f[f.valuation()] for f in lst]
         else:
+            # this returns a basis of q-expansions, without guaranteeing that 
+            # the first vectors form a basis of the cuspidal subspace
+            # TODO: bring this in line with the other cases
+            # simply using the above code fails because free modules over
+            # general rings do not have a .span() method
             B = self.__M.q_integral_basis(prec)
-        R = self._q_expansion_ring()
-        return [R(f) for f in B]
+            return [R(f) for f in B]
 
     def cuspidal_submodule(self):
         r"""