Function: elleisnum
Section: elliptic_curves
C-Name: elleisnum
Prototype: GLD0,L,p
Help: elleisnum(E,k,{flag=0}): E being an elliptic curve (or, alternatively,
 given by a 2-component vector representing its periods)
 and k an even positive integer, computes the
 numerical value of the Eisenstein series of weight k. When flag is non-zero
 and k=4 or 6, this gives g2 or g3 with the correct normalization.
Doc:
 $E$ being an elliptic curve as
 output by \kbd{ellinit} (or, alternatively, given by a 2-component vector
 $[\omega_1,\omega_2]$ representing its periods), and $k$ being an even
 positive integer, computes the numerical value of the Eisenstein series of
 weight $k$ at $E$, namely
 $$
 (2i \pi/\omega_2)^k
 \Big(1 + 2/\zeta(1-k) \sum_{n\geq 0} n^{k-1}q^n / (1-q^n)\Big),
 $$
 where $q = \exp(2i\pi \tau)$ and $\tau:=\omega_1/\omega_2$ belongs to the
 complex upper half-plane.

 When \fl\ is non-zero and $k=4$ or 6, returns the elliptic invariants $g_2$
 or $g_3$, such that
 $$y^2 = 4x^3 - g_2 x - g_3$$
 is a Weierstrass equation for $E$.
