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%I #22 Aug 22 2019 12:24:02
%S 0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,1,0,1,1,1,0,2,1,0,2,1,0,2,1,2,1,1,1,
%T 3,1,2,2,3,1,3,1,3,1,1,1,4,1,1,2,3,1,4,2,3,2,3,2,5,0,4,3,3,1,5,3,5,2,
%U 3,1,6,1,5,4,3,1,5,1,6,2,2,3,7,2,5,4,5,3,7,3,7,2,5,3,7,2,7,3,4,1,8,3
%N Dimension of the space of newforms of weight 2 and level n.
%C "Newform" is meant in the sense of Atkin-Lehner, that is, a primitive Hecke eigenform relative to the subgroup Gamma_0(n).
%D H. Cohen, Number Theory. Vol. II. Analytic and Modern Tools. Springer, 2007, pp. 496-497.
%D Toshitsune Miyake, Modular Forms, Springer-Verlag, 1989. See Table A.
%H Alois P. Heinz, <a href="/A127788/b127788.txt">Table of n, a(n) for n = 1..10000</a>
%H E. Halberstadt and A. Kraus, <a href="https://www.researchgate.net/publication/243111985_Courbes_de_Fermat_resultats_et_problemes">Courbes de Fermat: résultats et problèmes</a>, J. Reine Angew. Math. 548 (2002) 167-234. [_Steven Finch_, Mar 27 2009}
%H Xian-Jin Li, <a href="http://arXiv.org/abs/math/0403148">An arithmetic formula for certain coefficients of the Euler product of Hecke polynomials</a>, arXiv:math/0403148 [math.NT], 2004; J. Number Theory 113 (2005) 175-200. See Formula (5.8).
%H G. Martin, <a href="http://dx.doi.org/10.1016/j.jnt.2004.10.009">Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N)</a>, J. Numb. Theory 112 (2005) 298-331. [_Steven Finch_, Mar 27 2009]
%F a(n) = A001617(n) - sum a(m)*d(n/m), where the summation is over all divisors 1 < m < n of n and d is the divisor function.
%e a(p) = A001617(p) for any prime p.
%e G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^23 + x^24 + ...
%p seq( g0star(2,N),N=1..80); # using the source in A063195 - _R. J. Mathar_, Jul 15 2015
%t A001617[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@n}] - Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4]; a[n_ /; n < 10] = 0; a[n_] := a[n] = A001617[n] - Sum[a[m]*DivisorSigma[0, n/m], {m, Divisors[n][[2 ;; -2]]}]; Table[a[n], {n, 1, 102}] (* _Jean-François Alcover_, Sep 07 2015, A001617 code due to _Michael Somos_ *)
%o (PARI) {a(n) = my(v = [1, 3, 4, 6], A, p, e); if( n<1, 0, A = factor(n); for( k=1, matsize(A)[1], [p, e] = A[k,]; v[1] *= if( e==1, p-1, e==2, p^2-p-1, p^(e-3) * (p+1) * (p-1)^2); v[2] *= if( p==2, (e==3) - (e<3), e==1, kronecker(-4, p) - 1, e==2, -kronecker(-4, p)); v[3] *= if( p==3, (e==3) - (e<3), e==1, kronecker(-3, p) - 1, e==2, -kronecker(-3, p)); v[4] *= if( e%2, 0, e==2, p-2, p^(e/2-2) * (p-1)^2)); moebius(n) + (v[1] - v[2] - v[3] - v[4]) / 12 )}; /* _Michael Somos_, Jun 06 2015 */
%Y Cf. A001617, A116563.
%K nonn
%O 1,23
%A _Steven Finch_, Apr 04 2007
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