Binary quadratic forms of negative discriminant describe positive definite lattices in the complex plane, with integral-valued inner product. As such, it is possible to apply modular and elliptic functions to the form, interpreting this as an element of the upper half plane.
Given a binary quadratic form f = ax2 + bxy + cy2 of negative discriminant, returns the rank two lattice of f having Gram matrix(( a b/2atop b/2 c )).
Note that the lattice L is the half-integral lattice such that integral representations f(x, y) = n are in bijection with vectors (x, y) of norm n, which will be a rational number.
Returns the Gram matrix of the binary quadratic form f, which need not be of negative discriminant. The matrix will be half-integral and defined over the rationals.
The integral theta series of the binary quadratic form f to precision n.
The nth representation number of the form f of negative discriminant.
For a binary quadratic form f = ax2 + bxy + cy2 with negative discriminant, return the j--invariant of f, equal to the j--invariant of τ = ( - b + Sqrt(b2 - 4ac))/2a.
Given a positive even integer k = 2n and a binary quadratic form f = ax2 + bxy + cy2, return the value of the Eisenstein series Ek(L) at the complex lattice L = < a, ( - b + Sqrt(b2 - 4ac)) /2 >.
Given a complex power series z with positive valuation and a binary quadratic form f = ax2 + bxy + cy2, returns the q--expansion of the Weierstrass wp-function at the complex lattice L = < a, ( - b + Sqrt(b2 - 4ac) )/2 >.
> Q := QuadraticForms(-163); > f := PrimeForm(Q,41); > CC<i> := ComplexField(); > PC<z> := LaurentSeriesRing(CC); > x := WeierstrassSeries(z,f); > y := -Derivative(x)/2; > A := -Eisenstein(4,f)/48; > B := Eisenstein(6,f)/864; > Evaluate(y^2 - (x^3 + A*x + B),1/2); 1.384608660824596881000000000 E-26 - 1.305091481190174818000000000 E-26*i