Let f=sum a_n q^n be a weight 2 newform with rational Fourier
coefficients. Then the integrals
Int_gamma f(z) dz
as gamma varies through paths in H_1(X_0(N),Z) form a lattice
Lambda_f \subset C.
Conjecturally, the Weierstrass invariants c4 and c6 of Lambda_f lie in
Z, and an elliptic curve E/Q with L-series the Mellin transform of f is:
y^2 = x^3 - 27*c_4*x - 54*c_6.
For all this and an algorithm to compute c_4 and c_6 see Cremona,
Algorithms for Modular Elliptic Curves, sections 2.10--2.14. In this
talk I will give a generalization of the algorithm described in Cremona
which can be used to numerically compute the period lattice Lambda_f of
any newform f of weight k>=2.
When the weight is greater than 2 the meaning of the "intermediate
Jacobian"
C^d/Lambda_f
is mysterious, but when f has rational Fourier coefficient it must be an
elliptic curve E_f. Can we determine E_f? Is the j-invariant of E_f
transcendental or algebraic?

When k=2 this algorithm has applications to the numerical verification of
the Birch and Swinnerton-Dyer conjecture for quotients of J_0(N) of
dimension >= 1.