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

Author

N. J. A. Sloane, Jul 10 2001

Keywords

• nonn
• easy

Comments

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

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

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