%I #16 Jul 19 2024 11:10:47
%S 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,
%T 19,21,19,21,23,21,23,25,23,25,27,25,27,29,27,29,31,29,31,33,31,33
%N Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 10 ).
%C The dimension of weight n is apparently given by 0, 0, 2, 1, 0, 3, 2, 1, 4,... etc as in A063942. - _R. J. Mathar_, Jul 14 2015
%H R. J. Mathar, <a href="/A063198/b063198.txt">Table of n, a(n) for n = 1..1000</a>
%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>
%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F 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
%F 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
%p s0star := proc(n)
%p local pf,a,p,e ;
%p if n = 1 then
%p 1;
%p else
%p a :=1 ;
%p for pf in ifactors(n)[2] do
%p p := op(1,pf) ;
%p e := op(2,pf) ;
%p if e =1 then
%p a := a*(1-1/p) ;
%p elif e = 2 then
%p a := a*(1-1/p-1/p^2) ;
%p else
%p a := a*(1-1/p)*(1-1/p^2) ;
%p end if;
%p end do:
%p a ;
%p end if;
%p end proc:
%p nuInfstar := proc(n)
%p local pf,a,p,e ;
%p if n = 1 then
%p 1;
%p else
%p a :=1 ;
%p for pf in ifactors(n)[2] do
%p p := op(1,pf) ;
%p e := op(2,pf) ;
%p if type(e,'odd') then
%p return 0;
%p elif e = 2 then
%p a := a*(p-2) ;
%p else
%p a := a*(p-1)^2*p^(e/2-2) ;
%p end if;
%p end do:
%p a ;
%p end if;
%p end proc:
%p nu2star := proc(n)
%p local pf,a,p,e ;
%p if n = 1 then
%p 1;
%p else
%p a :=1 ;
%p for pf in ifactors(n)[2] do
%p p := op(1,pf) ;
%p e := op(2,pf) ;
%p if p = 2 then
%p if e =1 or e =2 then
%p a := -a ;
%p elif e =3 then
%p ;
%p else
%p return 0 ;
%p end if;
%p elif modp(p,4) = 1 then
%p if e = 2 then
%p a := -a ;
%p else
%p return 0;
%p end if;
%p else
%p if e = 1 then
%p a := -2*a ;
%p elif e = 2 then
%p ;
%p else
%p return 0;
%p end if;
%p end if;
%p end do:
%p a ;
%p end if;
%p end proc:
%p nu3star := proc(n)
%p local pf,a ;
%p if n = 1 then
%p 1;
%p else
%p a :=1 ;
%p for pf in ifactors(n)[2] do
%p p := op(1,pf) ;
%p e := op(2,pf) ;
%p if p = 3 then
%p if e =1 or e =2 then
%p a := -a ;
%p elif e =3 then
%p ;
%p else
%p return 0 ;
%p end if;
%p elif modp(p,3) = 1 then
%p if e = 2 then
%p a := -a ;
%p else
%p return 0;
%p end if;
%p else
%p if e = 1 then
%p a := -2*a ;
%p elif e = 2 then
%p ;
%p else
%p return 0;
%p end if;
%p end if;
%p end do:
%p a ;
%p end if;
%p end proc:
%p c2 := proc(k)
%p 1/4+floor(k/4)-k/4 ;
%p end proc:
%p c3 := proc(k)
%p 1/3+floor(k/3)-k/3 ;
%p end proc:
%p g0star := proc(k,N)
%p local a;
%p a := (k-1)/12*N*s0star(N) -nuInfstar(N)/2 +c2(k)*nu2star(N)+c3(k)*nu3star(N) ;
%p if k/2 = 1 then
%p a := a+numtheory[mobius](N) ;
%p end if;
%p a;
%p end proc:
%p A063198 := proc(n)
%p g0star(2*n,10) ;
%p end proc:
%p A063199 := proc(n)
%p g0star(2*n,11) ;
%p end proc:
%p A063200 := proc(n)
%p g0star(2*n,15) ;
%p end proc:
%p A063201 := proc(n)
%p g0star(2*n,18) ;
%p end proc:
%p A063205 := proc(n)
%p g0star(2*n,29) ;
%p end proc: # _R. J. Mathar_, Jul 19 2024
%Y Cf. A063942.
%K nonn,easy
%O 1,3
%A _N. J. A. Sloane_, Jul 10 2001