# Monomial expansion of $$(aX + bY)^i (cX + dY)^{j-i}$$¶

class sage.modular.modsym.apply.Apply

Bases: object

sage.modular.modsym.apply.apply_to_monomial(i, j, a, b, c, d)

Return a list of the coefficients of

$(aX + bY)^i (cX + dY)^{j-i},$

where $$0 \leq i \leq j$$, and $$a$$, $$b$$, $$c$$, $$d$$ are integers.

One should think of $$j$$ as being $$k-2$$ for the application to modular symbols.

INPUT:

• i, j, a, b, c, d – all ints

OUTPUT:

list of ints, which are the coefficients of $$Y^j, Y^{j-1}X, \ldots, X^j$$, respectively.

EXAMPLES:

We compute that $$(X+Y)^2(X-Y) = X^3 + X^2Y - XY^2 - Y^3$$:

sage: from sage.modular.modsym.apply import apply_to_monomial
sage: apply_to_monomial(2, 3, 1,1,1,-1)
[-1, -1, 1, 1]
sage: apply_to_monomial(5, 8, 1,2,3,4)
[2048, 9728, 20096, 23584, 17200, 7984, 2304, 378, 27]
sage: apply_to_monomial(6,12, 1,1,1,-1)
[1, 0, -6, 0, 15, 0, -20, 0, 15, 0, -6, 0, 1]