A159634 Coefficient for dimensions of spaces of modular & cusp forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.

1, 2, 4, 4, 6, 8, 8, 8, 12, 12, 12, 16, 14, 16, 24, 16, 18, 24, 20, 24, 32, 24, 24, 32, 30, 28, 36, 32, 30, 48, 32, 32, 48, 36, 48, 48, 38, 40, 56, 48, 42, 64, 44, 48, 72, 48, 48, 64, 56, 60, 72, 56, 54, 72, 72, 64, 80, 60, 60, 96, 62, 64, 96, 64, 84, 96, 68, 72, 96, 96

Offset: 1

Author

Steven Finch, Apr 17 2009

Keywords

• nonn
• look
• mult

Comments

Denote dim{M_k(Gamma_0(N))} by m(k,N) and dim{S_k(Gamma_0(N))} by s(k,N).

We have

m(7/2,N)+s(5/2,N) = m(5/2,N)+s(7/2,N) =

(m(11/2,N)+s(9/2,N))/2 = (m(9/2,N)+s(11/2,N))/2 =

(m(15/2,N)+s(13/2,N))/3 = (m(13/2,N)+s(15/2,N))/3 = ...

(m((4j+3)/2,N)+s((4j+1)/2,N))/j = (m((4j+1)/2,N)+s((4j+3)/2,N))/j = ...

where N is any positive multiple of 4 and j>=1.

Multiplicative because A001615 is multiplicative and a(1) = A001615(2)/3 = 1. - Andrew Howroyd, Aug 08 2018

References

• Ken Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and q-series. American Mathematical Society, 2004, (p. 16, theorem 1.56).

Links

• Peter Luschny, Table of n, a(n) for n = 1..1000
• H. Cohen and J. Oesterle, Dimensions des espaces de formes modulaires, Modular Functions of One Variable. VI, Proc. 1976 Bonn conf., Lect. Notes in Math. 627, Springer-Verlag, 1977, pp. 69-78; Scanned copy.
• S. R. Finch, Primitive Cusp Forms, April 27, 2009. [Cached copy, with permission of the author]
• Peter Humphries, Answer to: "A conjecture related to the Cohen-Oesterlé dimension formula", MathOverflow, 2014.
• Jon Maiga, Computer-generated formulas for A159634, Sequence Machine.
• Wikipedia, Cusp Form.

Crossrefs

Cf. A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A000010, A000082, A001221, A001615, A008683, A059956, A080512, A159630.

Programs

Magma

[[4*n,(Dimension(HalfIntegralWeightForms(4*n,7/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2))))/2] : n in [1..70]]; [[4*n,(Dimension(HalfIntegralWeightForms(4*n,5/2))+ Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,7/2))))/2] : n in [1..70]];

Mathematica

(* per Enrique Pérez Herrero's conjecture proved by P. Humphries, see link *)
dedekindPsi[n_Integer]:=n Apply[Times,1+1/Map[First,FactorInteger[n]]];
1/3 dedekindPsi /@ (2 Range[70]) (* Wouter Meeussen, Apr 06 2014 *)

PARI

a(n) = 2*n*sumdiv( 2*n, d, moebius(d)^2 / d)/3; \\ Andrew Howroyd, Aug 08 2018

Formula

a(n) = A159636(n) + A159630(n). - Enrique Pérez Herrero, Apr 15 2014

a(n) = A001615(2*n)/3. - Enrique Pérez Herrero, Jan 31 2014

From Peter Bala, Mar 19 2019: (Start)

a(n)= n*Product_{p|n, p odd prime} (1 + 1/p).

a(n) = Sum_{d|n, d odd} mu(d)^2*n/d, where mu(n) = A008683(n) is the Möbius function.

If n = m*2^k , where 2^k is the largest power of 2 dividing n, then

a(n) = (2^k)*a(m) = 2^k * Sum_{d^2|m} mu(d)*sigma(m/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also

a(n) = 2^k * Sum_{d|m} 2^omega(d)*phi(m/d), where omega(n) = A001221(n) is the number of different primes dividing n and phi(n) = A000010 is the Euler totient function.

O.g.f.: Sum_{n >= 1} mu(2*n-1)^2*x^(2^n-1)/(1 - x^(2*n-1))^2. (End)

a(n) = A000082(n)/A080512(n). [obvious by prime products, discovered by Sequence Machine]. - R. J. Mathar, Jun 24 2021

From Amiram Eldar, Nov 17 2022: (Start)

Multiplicative with a(2^e) = 2^e, and a(p^e) = (p+1)*p^(e-1) for p > 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 6/Pi^2 = 0.607927... (A059956). (End)