> 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

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