On Jul 31, 10:35 pm, "D. S. McNeil" <dsm...@gmail.com> wrote:
> > We would like to know if certain sums of modular symbols span the
> > space.
>
> Is this the sort of thing you had in mind?
>
> sage: M=ModularSymbols(11,2);M
> Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with
> sign 0 over Rational Field
> sage: b = M.basis()
> sage:
> sage: s1 = 2*b[1] - b[2]
> sage: s2 = -b[0] + b[2]
> sage: s3 = -b[0] + b[1]
> sage: s1, s2, s3
> (2*(1,8) - (1,9), -(1,0) + (1,9), -(1,0) + (1,8))
> sage: s1.list()
> [0, 2, -1]
> sage:
> sage: V = span([s.list() for s in s1, s2, s3],QQ); V
> Vector space of degree 3 and dimension 3 over Rational Field
> Basis matrix:
> [1 0 0]
> [0 1 0]
> [0 0 1]
> sage:
> sage: s3 = s2
> sage: V = span([s.list() for s in s1, s2, s3],QQ); V
> Vector space of degree 3 and dimension 2 over Rational Field
> Basis matrix:
> [ 1 0 -1]
> [ 0 1 -1/2]
>
> Doug
Thanks for this. There still seems to be a manual step in going from,
say,
s1 = 2*(1,8) - (1,9)
to
s1 = 2*b[1] - b[2]
Now consider
sage: M=ModularSymbols(389,2);M
Modular Symbols space of dimension 65 for Gamma_0(389) of weight 2
with sign 0 over Rational Field
This manual step could become slow and error prone.
Any suggestions for automating this step?
Best,
Jack Fearnley
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