def hurwitz_kronecker_class_no_x( n, D):
r"""
OUTPUT
The generalized Hurwitz-Kronecker class number $H_n(D)$
as defined in [S-Z].
INPUT
n -- an integer $\ge 1$
D -- an integer $\le 0$
"""
n = Integer(n)
D = Integer(D)
if n <= 0:
raise ValueError, "%s: must be an integer >=1" % n
if D > 0:
raise ValueError, "%s: must be an integer <= 0" % D
g = D.gcd( n)
a,b = _ab(g)
h = g*b
if 0 != D%h:
return 0
Dp = D//h
H1 = Rational( gp.qfbhclassno( -Dp))
if 0 == H1:
return H1
return g * Dp.kronecker( n//g) * H1
def __check( k, m, l, n):
k = Integer(k)
m = Integer(m)
l = Integer(l)
n = Integer(n)
if m <= 0:
raise ValueError, "%s: m must be a positive integer" % m
if l < 1 or 1 != l.gcd( m):
raise ValueError, "%s: l must be a positive integer prime to %d" \
% ( l, m)
if n <=0 or 0 != m%n or 1 != n.gcd( m//n):
raise ValueError, "%s: n must be an exact divisor of %d" \
( n, m)
return k, m, l, n
def _sz_s( k, m, l, n):
r"""
OUTPUT
The function $s_{k,m}(l,n)$, i.e.~the trace
of $T(l) \circ W_n$ on the "certain" space
$\mathcal{M}_{2k-2}^{\text{cusp}}(m)$ of modular forms
as defined in [S-Z].
INPUT
l -- index of the Hecke operator, rel. prime to the level $m$
n -- index of the Atkin-Lehner involution (exact divisor of $m$)
k -- integer ($2k-2$ is the weight)
m -- level
"""
k = Integer(k)
m = Integer(m)
l = Integer(l)
n = Integer(n)
if k < 2:
return 0
if 1 != m.gcd(l):
raise ValueError, \
"gcd(%s,%s) != 1: not yet implemented" % (m,l)
ellT = 0
for np in n.divisors():
ellT -= sum( l**(k-2) \
* _gegenbauer_pol( k-2, s*s*np/l) \
* hurwitz_kronecker_class_no_x( m/n, s*s*np*np - 4*l*np) \
for s in range( -(Integer(4*l*np).isqrt()//np), \
1+(Integer(4*l*np).isqrt()//np)) \
if (n//np).gcd(s*s).is_squarefree())/2
parT = -sum( min( lp, l//lp)**(2*k-3) \
* _ab(n)[0].gcd( lp+l//lp) \
* _ab(m/n)[0].gcd( lp-l//lp) \
for lp in l.divisors())/2
if 2 == k and (m//n).is_square():
corT = sigma(n,0)*sigma(l,1)
else:
corT = 0
return Integer( ellT + parT + corT)
