def eis_phipsi(phi, psi, k, prec=10, t=1, cmplx=False):
r"""
Return Fourier expansion of Eisenstein series at the cusp oo.
INPUT:
- ``phi`` -- Dirichlet character.
- ``psi`` -- Dirichlet character.
- ``k`` -- integer, the weight of the Eistenstein series.
- ``prec`` -- integer (default: 10).
- ``t`` -- integer (default: 1).
OUTPUT:
The Fourier expansion of the Eisenstein series $E_k^{\phi,\psi, t}$ (as
defined by [Diamond-Shurman]).
EXAMPLES:
sage: phi = DirichletGroup(3)[1]
sage: psi = DirichletGroup(5)[1]
sage: E = eisenstein_series_at_inf(phi, psi, 4)
"""
N1, N2 = phi.level(), psi.level()
N = N1 * N2
#The Fourier expansion of the Eisenstein series at infinity is in the field Q(zeta_Ncyc)
Ncyc = lcm([euler_phi(N1), euler_phi(N2)])
if cmplx == True:
CC = ComplexField(53)
pi = ComplexField().pi()
I = ComplexField().gen()
R = CC
zeta = CC(exp(2 * pi * I / Ncyc))
else:
R = CyclotomicField(Ncyc)
zeta = R.zeta(Ncyc)
phi, psi = phi.base_extend(R), psi.base_extend(R)
Q = PowerSeriesRing(R, 'q')
q = Q.gen()
s = O(q**prec)
#Weight 2 with trivial characters is calculated separately
if k == 2 and phi.conductor() == 1 and psi.conductor() == 1:
if t == 1:
raise TypeError('E_2 is not a modular form.')
s = 1 / 24 * (t - 1)
for m in srange(1, prec):
for n in srange(1, prec / m + 1):
s += n * (q**(m * n) - t * q**(m * n * t))
return s + O(q**prec)
if psi.level() == 1 and k == 1:
s -= phi.bernoulli(k) / k
elif phi.level() == 1:
s -= psi.bernoulli(k) / k
for m in srange(1, prec / t):
for n in srange(1, prec / t / m + 1):
s += 2 * phi(m) * psi(n) * n**(k - 1) * q**(m * n * t)
return s + O(q**prec)
