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Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 10 ).
1

%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