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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

a(n) seems to be A001615(2n)/3. - Enrique Pérez Herrero, Jan 31 2014

REFERENCES

K. 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.

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.

MAGMA Calculator.

Scanned copy of Cohen-Oesterle.

Wikipedia, Cusp Form

FORMULA

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

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 *)

PROG

(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]];

CROSSREFS

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

Sequence in context: A288529 A288772 A053196 * A186690 A002131 A230641

Adjacent sequences:  A159631 A159632 A159633 * A159635 A159636 A159637

KEYWORD

nonn,look

AUTHOR

Steven Finch, Apr 17 2009

STATUS

approved

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Last modified November 21 22:32 EST 2017. Contains 295054 sequences.