A063198 Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 10 ).
0, 1, 3, 1, 3, 5, 3, 5, 7, 5, 7, 9, 7, 9, 11, 9, 11, 13, 11, 13, 15, 13, 15, 17, 15, 17, 19, 17, 19, 21, 19, 21, 23, 21, 23, 25, 23, 25, 27, 25, 27, 29, 27, 29, 31, 29, 31, 33, 31, 33
Offset: 1
Links
- R. J. Mathar, Table of n, a(n) for n = 1..1000
- William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N))
- William A. Stein, The modular forms database
- Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Crossrefs
Cf. A063942.
Programs
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Maple
s0star := proc(n) local pf,a,p,e ; if n = 1 then 1; else a :=1 ; for pf in ifactors(n)[2] do p := op(1,pf) ; e := op(2,pf) ; if e =1 then a := a*(1-1/p) ; elif e = 2 then a := a*(1-1/p-1/p^2) ; else a := a*(1-1/p)*(1-1/p^2) ; end if; end do: a ; end if; end proc: nuInfstar := proc(n) local pf,a,p,e ; if n = 1 then 1; else a :=1 ; for pf in ifactors(n)[2] do p := op(1,pf) ; e := op(2,pf) ; if type(e,'odd') then return 0; elif e = 2 then a := a*(p-2) ; else a := a*(p-1)^2*p^(e/2-2) ; end if; end do: a ; end if; end proc: nu2star := proc(n) local pf,a,p,e ; if n = 1 then 1; else a :=1 ; for pf in ifactors(n)[2] do p := op(1,pf) ; e := op(2,pf) ; if p = 2 then if e =1 or e =2 then a := -a ; elif e =3 then ; else return 0 ; end if; elif modp(p,4) = 1 then if e = 2 then a := -a ; else return 0; end if; else if e = 1 then a := -2*a ; elif e = 2 then ; else return 0; end if; end if; end do: a ; end if; end proc: nu3star := proc(n) local pf,a ; if n = 1 then 1; else a :=1 ; for pf in ifactors(n)[2] do p := op(1,pf) ; e := op(2,pf) ; if p = 3 then if e =1 or e =2 then a := -a ; elif e =3 then ; else return 0 ; end if; elif modp(p,3) = 1 then if e = 2 then a := -a ; else return 0; end if; else if e = 1 then a := -2*a ; elif e = 2 then ; else return 0; end if; end if; end do: a ; end if; end proc: c2 := proc(k) 1/4+floor(k/4)-k/4 ; end proc: c3 := proc(k) 1/3+floor(k/3)-k/3 ; end proc: g0star := proc(k,N) local a; a := (k-1)/12*N*s0star(N) -nuInfstar(N)/2 +c2(k)*nu2star(N)+c3(k)*nu3star(N) ; if k/2 = 1 then a := a+numtheory[mobius](N) ; end if; a; end proc: A063198 := proc(n) g0star(2*n,10) ; end proc: A063199 := proc(n) g0star(2*n,11) ; end proc: A063200 := proc(n) g0star(2*n,15) ; end proc: A063201 := proc(n) g0star(2*n,18) ; end proc: A063205 := proc(n) g0star(2*n,29) ; end proc: # R. J. Mathar, Jul 19 2024
Formula
G.f.: x^2*(1+2*x-2*x^2+x^3) / ( (1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Jul 15 2015
For n>1, a(n) = (6*n-3+12*cos(2*n*Pi/3)-4*sqrt(3)*sin(2*n*Pi/3))/9. - Wesley Ivan Hurt, Sep 30 2017
Comments