For any congruence subgroup 
, the direct sum
is a ring, since the product of modular forms
and 
is
an element 
. Sage can compute
likely generators for rings of modular forms, but currently doesn’t
prove any of these results.
We verify the statement proved in Serre’s “A Course in Arithmetic”
that 
and 
generate the space of level
one modular forms.
sage: from sage.modular.modform.find_generators import modform_generators
sage: modform_generators(1)
[(4, 1 + 240*q + 2160*q^2 + 6720*q^3 + O(q^4)),
(6, 1 - 504*q - 16632*q^2 - 122976*q^3 + O(q^4))]
Have you ever wondered which forms generate the ring
? it turns out a form of weight 2 and two
forms of weight 4 together generate.
sage: modform_generators(2)
[(2, 1 + 24*q + 24*q^2 + ... + 288*q^11 + O(q^12)),
(4, 1 + 240*q^2 + .. + 30240*q^10 + O(q^12)),
(4, q + 8*q^2 + .. + 1332*q^11 + O(q^12))]
Here’s generators for 
. Notice that
elements of weight 
are now required, in addition to
weights 
and 
.
sage: modform_generators(3)
[(2, 1 + 12*q + 36*q^2 + .. + 168*q^13 + O(q^14)),
(4, 1 + 240*q^3 + 2160*q^6 + 6720*q^9 + 17520*q^12 + O(q^14)),
(4, q + 9*q^2 + 27*q^3 + 73*q^4 + .. + O(q^14)),
(6, q - 6*q^2 + 9*q^3 + 4*q^4 + .. + O(q^14)),
(6, 1 - 504*q^3 - 16632*q^6 .. + O(q^14)),
(6, q + 33*q^2 + 243*q^3 + .. + O(q^14))]
-series