A159046 Dimension of the space of newforms of weight 2 on the subgroup Gamma_1(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3, 5, 4, 12, 5, 12, 6, 13, 8, 22, 7, 26, 13, 19, 11, 25, 13, 40, 14, 29, 19, 51, 13, 57, 25, 39, 21, 70, 23, 69, 24, 55, 37, 92, 22, 79, 42, 71, 34, 117, 34, 126, 39, 87, 61, 117, 31, 155, 68, 109, 45, 176, 55, 187, 56, 119, 87

Offset

1,13

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

G. Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331.

Formula

a(n) = A029937(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.

Dirichlet convolution of A007247 and A029937. - Michael Somos, May 10 2015

Example

a(p) = A029937(p) = (p-5)*(p-7)/24 for any prime p>3.
G.f. = x^11 + 2*x^13 + x^14 + x^15 + 2*x^16 + 5*x^17 + 2*x^18 + 7*x^19 + ...

Mathematica

a[ n_] := If[ n < 1, 0, Sum[ DivisorSum[ n/j, MoebiusMu[#] MoebiusMu[n/j/#] &] If[ j < 5, 0, 1 + DivisorSum[ j, #^2 MoebiusMu[ j/#] / 24 - EulerPhi [#] EulerPhi[j/#] / 4 &]], {j, Divisors@n}]]; (* Michael Somos, May 10 2015 *)

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv(n, j, sumdiv(n/j, k, moebius(k) * moebius(n/j/k)) * if( j<5, 0, 1 + sumdiv(j, k, k^2 * moebius(j/k) / 24 - eulerphi(k) * eulerphi(j/k) / 4))))}; /* Michael Somos, May 10 2015 */

Crossrefs

Cf. A007427, A029937, A029938, A127788.

Sequence in context: A309522 A336878 A305313 * A029937 A368926 A289772

Adjacent sequences: A159043 A159044 A159045 * A159047 A159048 A159049

Keyword

nonn

Author

Steven Finch, Apr 03 2009

Status

approved