def Weyl_law_N(self,T,T1=None):
r"""
The counting function for this space. N(T)=#{disc. ev.<=T}
INPUT:
- ``T`` -- double
EXAMPLES::
sage: M=MaassWaveForms(MySubgroup(Gamma0(1))
sage: M.Weyl_law_N(10)
0.572841337202191
"""
(c1,c2,c3,c4,c5)=self._Weyl_law_const
cc1=RR(c1); cc2=RR(c2); cc3=RR(c3); cc4=RR(c4); cc5=RR(c5)
#print "c1,c2,c3,c4,c5=",cc1,cc2,cc3,cc4,cc5
t=sqrt(T*T+0.25)
try:
lnt=ln(t)
except TypeError:
lnt=mpmath.ln(t)
#print "t,ln(t)=",t,lnt
NT=cc1*t*t-cc2*t*lnt+cc3*t+cc4*t+cc5
if(T1<>None):
t=sqrt(T1*T1+0.25)
NT1=cc1*(T1*T1+0.25)-cc2*t*ln(t)+cc3*t+cc4*t+cc5
return RR(abs(NT1-NT))
else:
return RR(NT)
def _Weyl_law_consts(self):
r"""
Compute constants for the Weyl law on self._G
OUTPUT:
- tuple of real numbers
EXAMPLES::
sage: M=MaassWaveForms(MySubgroup(Gamma0(1)))
sage: M._Weyl_law_consts()
(0, 2/pi, (log(pi) - log(2) + 2)/pi, 0, -2)
"""
import mpmath
pi=mpmath.fp.pi
ix=Integer(self._G.index())
nc=Integer(len(self._G.cusps()))
if(self._G.is_congruence()):
lvl=Integer(self._G.level())
else:
lvl=0
n2=Integer(self._G.nu2())
n3=Integer(self._G.nu3())
c1=ix/Integer(12)
c2=Integer(2)*nc/pi
c3=nc*(Integer(2)-ln(Integer(2))+ln(pi))/pi
if(lvl<>0):
A=1
for q in divisors(lvl):
num_prim_dc=0
DG=DirichletGroup(q)
for chi in DG.list():
if(chi.is_primitive()):
num_prim_dc=num_prim_dc+1
for m in divisors(lvl):
if(lvl % (m*q) == 0 and m % q ==0 ):
fak=(q*lvl)/gcd(m,lvl/m)
A=A*Integer(fak)**num_prim_dc
c4=-ln(A)/pi
else:
c4=Integer(0)
# constant term
c5=-ix/144+n2/8+n3*2/9-nc/4-1
return (c1,c2,c3,c4,c5)
def Weyl_law_Np(self,T,T1=None):
r"""
The derviative of the counting function for this space. N(T)=#{disc. ev.<=T}
INPUT:
- ``T`` -- double
EXAMPLES::
sage: M=MaassWaweForms(MySubgroup(Gamma0(1))
sage: M.Weyl_law_Np(10)
"""
(c1,c2,c3,c4,c5)=self._Weyl_law_const
cc1=RR(c1); cc2=RR(c2); cc3=RR(c3); cc4=RR(c4); cc5=RR(c5)
#print "c1,c2,c3,c4,c5=",c1,c2,c3,c4,c5
NpT=2.0*cc1*T-cc2*(ln(T)+1.0)+cc3+cc4
return RR(NpT)