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)[source]#

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

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]

Python

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