void
acb_modular_lambda(acb_t r, const acb_t tau, long prec)
{
psl2z_t g;
arf_t one_minus_eps;
acb_t tau_prime, q;
acb_struct thetas[4];
int R[4], S[4], C;
int Rsum, qpower;
psl2z_init(g);
arf_init(one_minus_eps);
acb_init(tau_prime);
acb_init(q);
acb_init(thetas + 0);
acb_init(thetas + 1);
acb_init(thetas + 2);
acb_init(thetas + 3);
arf_set_ui_2exp_si(one_minus_eps, 63, -6);
acb_modular_fundamental_domain_approx(tau_prime, g, tau,
one_minus_eps, prec);
acb_modular_theta_transform(R, S, &C, g);
acb_exp_pi_i(q, tau_prime, prec);
acb_modular_theta_const_sum(thetas + 1, thetas + 2, thetas + 3, q, prec);
acb_zero(thetas + 0);
/* divide the transformation factors */
Rsum = 4 * (R[1] - R[2]);
/* possible factor [q^(+/- 1/4)]^4 needed for theta_1^4 or theta_2^4 */
qpower = (S[1] == 0 || S[1] == 1) - (S[2] == 0 || S[2] == 1);
acb_div(r, thetas + S[1], thetas + S[2], prec);
acb_mul(r, r, r, prec);
acb_mul(r, r, r, prec);
if ((Rsum & 7) == 4)
acb_neg(r, r);
if (qpower == 1)
acb_mul(r, r, q, prec);
else if (qpower == -1)
acb_div(r, r, q, prec);
psl2z_clear(g);
arf_clear(one_minus_eps);
acb_clear(tau_prime);
acb_clear(q);
acb_clear(thetas + 0);
acb_clear(thetas + 1);
acb_clear(thetas + 2);
acb_clear(thetas + 3);
}
void
acb_modular_delta(acb_t z, const acb_t tau, long prec)
{
psl2z_t g;
arf_t one_minus_eps;
acb_t tau_prime, t1, t2, t3, t4, q;
psl2z_init(g);
arf_init(one_minus_eps);
acb_init(tau_prime);
acb_init(t1);
acb_init(t2);
acb_init(t3);
acb_init(t4);
acb_init(q);
arf_set_ui_2exp_si(one_minus_eps, 63, -6);
acb_modular_fundamental_domain_approx(tau_prime, g, tau,
one_minus_eps, prec);
acb_exp_pi_i(q, tau_prime, prec);
acb_modular_theta_const_sum(t2, t3, t4, q, prec);
/* (t2 t3 t4) ^ 8 * q^2 */
acb_mul(t1, t2, t3, prec);
acb_mul(t1, t1, t4, prec);
acb_mul(t1, t1, t1, prec);
acb_mul(t1, t1, t1, prec);
acb_mul(t1, t1, q, prec);
acb_mul(t1, t1, t1, prec);
acb_mul_2exp_si(t1, t1, -8);
if (!fmpz_is_zero(&g->c))
{
acb_mul_fmpz(t2, tau, &g->c, prec);
acb_add_fmpz(t2, t2, &g->d, prec);
acb_pow_ui(t2, t2, 12, prec);
acb_div(t1, t1, t2, prec);
}
acb_set(z, t1);
psl2z_clear(g);
arf_clear(one_minus_eps);
acb_clear(tau_prime);
acb_clear(t1);
acb_clear(t2);
acb_clear(t3);
acb_clear(t4);
acb_clear(q);
}
void
acb_modular_eisenstein(acb_ptr r, const acb_t tau, slong len, slong prec)
{
psl2z_t g;
arf_t one_minus_eps;
acb_t tau_prime, t1, t2, t3, t4, q;
slong m, n;
if (len < 1)
return;
psl2z_init(g);
arf_init(one_minus_eps);
acb_init(tau_prime);
acb_init(t1);
acb_init(t2);
acb_init(t3);
acb_init(t4);
acb_init(q);
arf_set_ui_2exp_si(one_minus_eps, 63, -6);
acb_modular_fundamental_domain_approx(tau_prime, g, tau,
one_minus_eps, prec);
acb_exp_pi_i(q, tau_prime, prec);
acb_modular_theta_const_sum(t2, t3, t4, q, prec);
/* fourth powers of the theta functions (a, b, c) */
acb_mul(t2, t2, t2, prec);
acb_mul(t2, t2, t2, prec);
acb_mul(t2, t2, q, prec);
acb_mul(t3, t3, t3, prec);
acb_mul(t3, t3, t3, prec);
acb_mul(t4, t4, t4, prec);
acb_mul(t4, t4, t4, prec);
/* c2 = pi^4 * (a^8 + b^8 + c^8) / 30 */
/* c3 = pi^6 * (b^12 + c^12 - 3a^8 * (b^4+c^4)) / 180 */
/* r = a^8 */
acb_mul(r, t2, t2, prec);
if (len > 1)
{
/* r[1] = -3 a^8 * (b^4 + c^4) */
acb_add(r + 1, t3, t4, prec);
acb_mul(r + 1, r + 1, r, prec);
acb_mul_si(r + 1, r + 1, -3, prec);
}
/* b^8 */
acb_mul(t1, t3, t3, prec);
acb_add(r, r, t1, prec);
/* b^12 */
if (len > 1)
acb_addmul(r + 1, t1, t3, prec);
/* c^8 */
acb_mul(t1, t4, t4, prec);
acb_add(r, r, t1, prec);
/* c^12 */
if (len > 1)
acb_addmul(r + 1, t1, t4, prec);
acb_const_pi(t1, prec);
acb_mul(t1, t1, t1, prec);
acb_mul(t2, t1, t1, prec);
acb_mul(r, r, t2, prec);
acb_div_ui(r, r, 30, prec);
if (len > 1)
{
acb_mul(t2, t2, t1, prec);
acb_mul(r + 1, r + 1, t2, prec);
acb_div_ui(r + 1, r + 1, 189, prec);
}
/* apply modular transformation */
if (!fmpz_is_zero(&g->c))
{
acb_mul_fmpz(t1, tau, &g->c, prec);
acb_add_fmpz(t1, t1, &g->d, prec);
acb_inv(t1, t1, prec);
acb_mul(t1, t1, t1, prec);
acb_mul(t2, t1, t1, prec);
acb_mul(r, r, t2, prec);
if (len > 1)
{
acb_mul(t2, t1, t2, prec);
acb_mul(r + 1, r + 1, t2, prec);
}
}
/* compute more coefficients using recurrence */
for (n = 4; n < len + 2; n++)
{
acb_zero(r + n - 2);
m = 2;
for (m = 2; m * 2 < n; m++)
acb_addmul(r + n - 2, r + m - 2, r + n - m - 2, prec);
acb_mul_2exp_si(r + n - 2, r + n - 2, 1);
if (n % 2 == 0)
acb_addmul(r + n - 2, r + n / 2 - 2, r + n / 2 - 2, prec);
acb_mul_ui(r + n - 2, r + n - 2, 3, prec);
acb_div_ui(r + n - 2, r + n - 2, (2 * n + 1) * (n - 3), prec);
}
/* convert c's to G's */
for (n = 0; n < len; n++)
acb_div_ui(r + n, r + n, 2 * n + 3, prec);
psl2z_clear(g);
arf_clear(one_minus_eps);
acb_clear(tau_prime);
acb_clear(t1);
acb_clear(t2);
acb_clear(t3);
acb_clear(t4);
acb_clear(q);
}
