Introduction

Consider a newform $f\in S_2=S_2(\Gamma_0(N),\mathbf{C})$, and denote by A_f the corresponding optimal quotient of J₀(N); thus A_f=J₀(N)/I_f J₀(N)with I_f the annihilator, in the Hecke algebra, of f. As a complex torus, A_f is the cokernel of the map $\Phi:H_1(X_0(N),\mathbf{Z})\rightarrow\mbox{\rm Hom}(S_2[I_f],\mathbf{C})$ arising from the integration pairing between H₁(X₀(N),Z) and S₂. Let $\mathbf{e}=\{0,\infty\}\in H_1(X_0(N),\mathbf{Q})$ be the winding element, and consider the lattice index

$$L_f := [\Phi(H_1(X_0(N),\mathbf{Z})^+) : \Phi(\mathbf{T}e)]\in\mathbf{Q}.$$

There is a modular symbols algorithm to compute this rational number; this paper contains the result of such a computation. Let $\Omega_f^0$ be the measure of the identity component of A_f(R) with respect to a Z-basis for $S_f = S_2(\Gamma_0(N),\mathbf{Z})[I_f]$. Let c_f be the Manin constant; it is the absolute value of the determinant of a change of basis matrix relating S_f to a basis of integral differentials on the Néron model of A_f.

Theorem 1.1  

$$L_f = \frac{L(A_f,1)}{\Omega_f^0}\cdot c_f$$

Conjecture 1.2 (Birch, Swinnerton-Dyer, Tate)  

$$L_f = \frac{\char93 \mbox{\cyr X}(A_f) \cdot c_\infty\cdot c_... ... {\char93 A_f(\mathbf{Q})\cdot \char93 A_f^{\vee}(\mathbf{Q})}$$

If A_f(Q) is infinite, then the right hand side is 0.