

A159633


Dimension of Eisenstein subspace of the space of modular forms of weight k/2, level 4*n and trivial character, where k>=5 is odd.


1



2, 3, 4, 6, 4, 6, 4, 8, 8, 6, 4, 12, 4, 6, 8, 12, 4, 12, 4, 12, 8, 6, 4, 16, 12, 6, 12, 12, 4, 12, 4, 16, 8, 6, 8, 24, 4, 6, 8, 16, 4, 12, 4, 12, 16, 6, 4, 24, 16, 18, 8, 12, 4, 18, 8, 16, 8, 6, 4, 24, 4, 6, 16, 24, 8, 12, 4, 12, 8, 12, 4, 32, 4, 6, 24, 12, 8, 12, 4, 24, 24, 6, 4, 24, 8, 6, 8, 16, 4
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OFFSET

1,1


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(3/2,N)s(3/2,N)+m(1/2,N)s(1/2,N) =
m(5/2,N)s(5/2,N) = m(7/2,N)s(7/2,N) =
m(9/2,N)s(9/2,N) = m(11/2,N)s(11/2,N) = ...
m(k/2,N)s(k/2,N) = ...
where N is any positive multiple of 4 and k>=5 is odd.
a(n) = A159635(n)  A159636(n).  Steven Finch, Apr 22 2009
Conjecture: a(n) = 2*chi(n)  if(mod(n+2,4)=0, chi(n)/2, 0) with chi(n) = Sum(dn; phi(gcd(d,n/d)); checked up to n=1024.  Wouter Meeussen, Apr 02 2014


REFERENCES

K. Ono, The Web of Modularity: Arithmetic of Coefficients of Modular Forms and qseries. American Mathematical Society, 2004 (p. 16, theorem 1.56).


LINKS

Table of n, a(n) for n=1..89.
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, SpringerVerlag, 1977, pp. 6978.
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 CohenOesterlĂ© dimension formula", MathOverflow, 2014.
MAGMA Calculator.


MATHEMATICA

(* see link, conjecture proved by P. Humphries *)
chi[n_Integer]:=Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
2 chi[#]  If[Mod[# + 2, 4] == 0, chi[#]/2, 0] & /@ Range[89]
(* Wouter Meeussen, Apr 06 2014 *)


PROG

(MAGMA) [[4*n, Dimension(HalfIntegralWeightForms(4*n, 5/2))Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 5/2)))] : n in [1..100]]; [[4*n, Dimension(HalfIntegralWeightForms(4*n, 7/2))Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 7/2)))] : n in [1..100]]; [[4*n, Dimension(HalfIntegralWeightForms(4*n, 3/2))Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 3/2)))+Dimension(HalfIntegralWeightForms(4*n, 1/2))Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n, 1/2)))] : n in [1..100]];


CROSSREFS

Cf. A159630, A159631, A159632, A159634, A159635, A159636.  Steven Finch, Apr 22 2009
Cf. A001616.
Sequence in context: A074103 A051785 A144825 * A049044 A102284 A273098
Adjacent sequences: A159630 A159631 A159632 * A159634 A159635 A159636


KEYWORD

nonn


AUTHOR

Steven Finch, Apr 17 2009


STATUS

approved



