Implementation of QuasiModularFormsElement class. Fixed some inconsis…

…tencies in the code. The default precision is now 6 (to be consistent with classical ModularForms)

src/sage/modular/quasimodform/all.py

@@ -1 +1 @@
-from .quasimodform import QuasiModularForms, QuasiModularFormsElement
+from .quasimodform import QuasiModularForms

src/sage/modular/quasimodform/quasimodform.py

@@ -26,13 +26,97 @@
 
 
 from sage.modular.modform.eis_series import eisenstein_series_qexp
+from sage.modular.modform.constructor import ModularForms
 from sage.modular.arithgroup.all import Gamma0, is_CongruenceSubgroup
+
 from sage.rings.all import Integer, QQ, ZZ
+
 from sage.structure.sage_object  import SageObject
 from sage.structure.parent import Parent
 from sage.structure.element import Element
 from sage.structure.unique_representation import UniqueRepresentation
+
 from sage.categories.graded_modules import GradedModules
+from sage.categories.hecke_modules import HeckeModules
+from sage.categories.rings import Rings
+
+class QuasiModularFormsElement(Element):
+    def __init__(self, parent, mflist):
+        r"""
+        An element of a graded ring of quasimodular form.
+
+        INPUTS:
+
+        - ``parent`` - QuasiModularForms
+        - ``mflist`` - a list `[f0, f1, ..., fn]` of modular forms
+
+        OUTPUT:
+
+        - ``QuasiModularFormsElement`` - a quasimodular form defined be `f_0 + f_1*E_2 + ... + f_n*(E_2)^n` 
+        where `E_2` is the weight 2 Eisenstein serie.
+
+        EXAMPLES::
+
+            sage: M = QuasiModularForms()
+            sage: B = M.generators(); B
+            [(2,
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)),
+            (4,
+            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)),
+            (6,
+            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))]
+        """
+        self.__mflist = mflist
+        self.__base_ring = parent.base_ring()
+        Element.__init__(self, parent)
+
+    def q_expansion(self, prec=6):
+        r"""
+        Computes the `q`-expansion of self to precision `prec`.
+
+        EXAMPLES:::
+
+            sage: M = QuasiModularForms()
+            sage: B = M.generators()
+            sage: E2 = B[0][1]
+            sage: E2.q_expansion()
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
+            sage: E2.q_expansion(prec=10)
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 - 288*q^6 - 192*q^7 - 360*q^8 - 312*q^9 + O(q^10)
+        """
+        #we compute the q-expansion of every forms and sums them.
+        qexp=0
+        E2 = eisenstein_series_qexp(2, prec=prec, K=self.__base_ring, normalization='constant') #normalization -> to force integer coefficients
+        for idx, f in enumerate(self.__mflist):
+            if f.parent().category() is HeckeModules(self.__base_ring):
+                qexp += f.q_expansion(prec=prec)*E2**idx
+            else:
+                qexp += f*E2**idx
+        return qexp
+
+    def _repr_(self):
+        r"""
+        String representation of self.
+
+        EXAMPLES::
+
+            sage: M = QuasiModularForms()
+            sage: B = M.generators()
+            sage: E2 = B[0][1]; E4 = B[1][1]; E6 = B[2][1];
+            sage: E2
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
+            sage: E4
+            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)
+            sage: E6
+            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6)
+        """
+        return "%s"%(self.q_expansion())
+
+    def _richcmp_(self, other, op):
+        from sage.structure.richcmp import richcmp
+        return richcmp(self.q_expansion(), other.q_expansion(), op)
+
+
 
 class QuasiModularForms(Parent, UniqueRepresentation):
     Element = QuasiModularFormsElement
@@ -55,23 +139,23 @@ def __init__(self, group=1, base_ring=QQ):
             Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Rational Field
             sage: B = M.generators(); B
             [(2,
-            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 - 288*q^6 - 192*q^7 - 360*q^8 - 312*q^9 + O(q^10)),
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)),
             (4,
-            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + 140400*q^8 + 181680*q^9 + O(q^10)),
+            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)),
             (6,
-            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 - 4058208*q^6 - 8471232*q^7 - 17047800*q^8 - 29883672*q^9 + O(q^10))]
+            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))]
             sage: P = B[0][1]; Q = B[1][1]
             sage: D = M.differentiation_operator
             sage: D(P)
-            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
-            sage: (P^2 - Q)/12
-            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 + O(q^6)
+            sage: (P.q_expansion()^2 - Q.q_expansion())/12
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 + O(q^6)
             sage: M = QuasiModularForms(1, Integers(5)); M
             Ring of quasimodular forms for Modular Group SL(2,Z) with coefficients in Ring of integers modulo 5
             sage: B = M.generators(); B
-            [(2, 1 + q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + 2*q^6 + 3*q^7 + 3*q^9 + O(q^10)),
-            (4, 1 + O(q^10)),
-            (6, 1 + q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + 2*q^6 + 3*q^7 + 3*q^9 + O(q^10))]
+            [(2, 1 + q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + O(q^6)),
+            (4, 1 + O(q^6)),
+            (6, 1 + q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + O(q^6))]
 
         .. TESTS:
 
@@ -80,9 +164,6 @@ def __init__(self, group=1, base_ring=QQ):
             Modular Group SL(2,Z)
             sage: M.base_ring()
             Rational Field
-            sage: M = QuasiModularForms(1, ZZ)
-            sage: M.base_ring()
-            Integer Ring
             sage: M = QuasiModularForms(1, Integers(5))
             sage: M.base_ring()
             Ring of integers modulo 5
@@ -93,10 +174,14 @@ def __init__(self, group=1, base_ring=QQ):
             sage: QuasiModularForms(Integers(5))
             Traceback (most recent call last):
             ...
-            ValueError: Group (=Ring of integers modulo 5) should be a congruence subgroup           
+            ValueError: Group (=Ring of integers modulo 5) should be a congruence subgroup 
+            sage: M2 = QuasiModularForms(1, GF(7))
+            sage: M == M2
+            False          
 
 
         """
+        #check if the group is SL2(Z)
         if isinstance(group, (int, Integer)):
             if group>1:
                 raise NotImplementedError("space of quasimodular forms have only been implemented for the full modular group")
@@ -106,6 +191,11 @@ def __init__(self, group=1, base_ring=QQ):
         elif group is not Gamma0(1):
             raise NotImplementedError("space of quasimodular forms have only been implemented for the full modular group")
 
+        #Check if the base ring is a field
+        #For some reasons, there is a problem when computing a basis of ModularForms
+        if not base_ring.is_field():
+            raise ValueError("The base ring must be a field")
+
         self.__group = group
         self.__base_ring = base_ring
         Parent.__init__(self, base=base_ring, category=GradedModules(base_ring))
@@ -140,8 +230,6 @@ def base_ring(self):
 
             sage: QuasiModularForms(1).base_ring()
             Rational Field
-            sage: QuasiModularForms(1, base_ring=ZZ).base_ring()
-            Integer Ring
             sage: QuasiModularForms(1, base_ring=Integers(5)).base_ring()
             Ring of integers modulo 5
         """
@@ -160,11 +248,39 @@ def _repr_(self):
         """
         return "Ring of quasimodular forms for %s with coefficients in %s" % (self.group(), self.base_ring())
 
-    def generators(self, prec=10):
+    def _create_element(self, mflist):
         r"""
+        Element constructor of self.
+
+        INPUT:
+
+        - `mflist` (list) -- a list of classical modular forms or elements of the base ring of self
+
+        OUTPUT:
+
+        QuasiModularFormsElement
 
-        TODO: implement element class
+        TESTS::
 
+            sage: M = QuasiModularForms()
+            sage: S = CuspForms(1, 12)
+            sage: f = S.0; f
+            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
+            sage: type(f)
+            <class 'sage.modular.modform.cuspidal_submodule.CuspidalSubmodule_level1_Q_with_category.element_class'>
+            sage: g = M._create_element([f]); g
+            q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 + O(q^6)
+            sage: type(g)
+            <class 'sage.modular.quasimodform.quasimodform.QuasiModularForms_with_category.element_class'>
+            sage: E2 = M.generators()[0][1]; E2
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)
+            sage: type(E2)
+            <class 'sage.modular.quasimodform.quasimodform.QuasiModularForms_with_category.element_class'>
+        """
+        return self.element_class(self, mflist)
+
+    def generators(self, prec=10):
+        r"""
         If `R` is the base ring of self, then this method returns a set of
         quasimodular forms which generate the `R`-algebra of all quasimodular forms.
 
@@ -184,20 +300,20 @@ def generators(self, prec=10):
             sage: M = QuasiModularForms(1)
             sage: M.generators()
             [(2,
-            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 - 288*q^6 - 192*q^7 - 360*q^8 - 312*q^9 + O(q^10)),
+            1 - 24*q - 72*q^2 - 96*q^3 - 168*q^4 - 144*q^5 + O(q^6)),
             (4,
-            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + 60480*q^6 + 82560*q^7 + 140400*q^8 + 181680*q^9 + O(q^10)),
+            1 + 240*q + 2160*q^2 + 6720*q^3 + 17520*q^4 + 30240*q^5 + O(q^6)),
             (6,
-            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 - 4058208*q^6 - 8471232*q^7 - 17047800*q^8 - 29883672*q^9 + O(q^10))]
-            sage: QuasiModularForms(1, Integers(17)).generators(prec=6)
+            1 - 504*q - 16632*q^2 - 122976*q^3 - 532728*q^4 - 1575504*q^5 + O(q^6))]
+            sage: QuasiModularForms(1, Integers(17)).generators()
             [(2, 1 + 10*q + 13*q^2 + 6*q^3 + 2*q^4 + 9*q^5 + O(q^6)),
             (4, 1 + 2*q + q^2 + 5*q^3 + 10*q^4 + 14*q^5 + O(q^6)),
             (6, 1 + 6*q + 11*q^2 + 2*q^3 + q^4 + 5*q^5 + O(q^6))]
         """
-        E2 = eisenstein_series_qexp(2, prec=prec, K=self.base_ring(), normalization='constant')
-        E4 = eisenstein_series_qexp(4, prec=prec, K=self.base_ring(), normalization='constant')
-        E6 = eisenstein_series_qexp(6, prec=prec, K=self.base_ring(), normalization='constant')
-        return [(2, E2), (4, E4), (6, E6)]
+        #Create Eisenstein series of weight 4 and 6
+        E4 = ModularForms(1, 4, base_ring=self.base_ring()).gen(0)
+        E6 = ModularForms(1, 6, base_ring=self.base_ring()).gen(0)
+        return [(2, self._create_element([Integer(0), Integer(1)])), (4, self._create_element([E4])), (6, self._create_element([E6]))]
 
     def differentiation_operator(self, f):
         r"""
@@ -209,7 +325,7 @@ def differentiation_operator(self, f):
 
         OUTPUT:
 
-        The power serie `q\frac{d}{dq}(f)`
+        The power series `q\frac{d}{dq}(f)`
 
         EXAMPLES::
 
@@ -218,17 +334,19 @@ def differentiation_operator(self, f):
             sage: B = M.generators()
             sage: P = B[0][1]; Q = B[1][1]; R = B[2][1]
             sage: D(P)
-            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
-            sage: (P^2 - Q)/12
-            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 - 1728*q^6 - 1344*q^7 - 2880*q^8 - 2808*q^9 + O(q^10)
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 + O(q^6)
+            sage: (P.q_expansion()^2 - Q.q_expansion())/12
+            -24*q - 144*q^2 - 288*q^3 - 672*q^4 - 720*q^5 + O(q^6)
             sage: D(Q)
-            240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + 362880*q^6 + 577920*q^7 + 1123200*q^8 + 1635120*q^9 + O(q^10)
-            sage: (P*Q - R)/3
-            240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + 362880*q^6 + 577920*q^7 + 1123200*q^8 + 1635120*q^9 + O(q^10)
+            240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + O(q^6)
+            sage: (P.q_expansion()*Q.q_expansion() - R.q_expansion())/3
+            240*q + 4320*q^2 + 20160*q^3 + 70080*q^4 + 151200*q^5 + O(q^6)
             sage: D(R)
-            -504*q - 33264*q^2 - 368928*q^3 - 2130912*q^4 - 7877520*q^5 - 24349248*q^6 - 59298624*q^7 - 136382400*q^8 - 268953048*q^9 + O(q^10)
-            sage: (P*R - Q^2)/2
-            -504*q - 33264*q^2 - 368928*q^3 - 2130912*q^4 - 7877520*q^5 - 24349248*q^6 - 59298624*q^7 - 136382400*q^8 - 268953048*q^9 + O(q^10)
+            -504*q - 33264*q^2 - 368928*q^3 - 2130912*q^4 - 7877520*q^5 + O(q^6)
+            sage: (P.q_expansion()*R.q_expansion() - Q.q_expansion()^2)/2
+            -504*q - 33264*q^2 - 368928*q^3 - 2130912*q^4 - 7877520*q^5 + O(q^6)
+
+        TODO:: This method need some work. It should return a QuasiModularFormsElement (not a power series in q)
         """
-        q = f.parent().gen()
-        return q*f.derivative()
+        q = f.q_expansion().parent().gen()
+        return q*f.q_expansion().derivative()