A159634 -id:A159634 - cp's OEIS frontend

A159636 Dimension of space of cusp forms of weight 5/2, level 4*n and trivial character.

0, 0, 1, 0, 3, 3, 4, 2, 6, 6, 7, 6, 9, 9, 14, 6, 12, 12, 13, 12, 20, 15, 16, 16, 18, 18, 21, 18, 21, 30, 22, 16, 32, 24, 32, 24, 27, 27, 38, 28, 30, 42, 31, 30, 48, 33, 34, 36, 36, 36, 50, 36, 39, 45, 50, 40, 56, 42, 43, 60

Offset: 1

Steven Finch, Apr 17 2009

nonn

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

Cf. A159630, A159634.

[[4*n,Dimension(CuspidalSubspace(HalfIntegralWeightForms(4*n,5/2)))] : n in [1..75]];

dedekindPsi[n_Integer] := n*Times @@ (1 + 1/First /@ FactorInteger[n]);
\[Chi][n_Integer] := Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
r[(p_)?PrimeQ, n_Integer] := -1+ Last[Flatten[Cases[FactorInteger[p*n], {p, _}]]];
\[Alpha][n_Integer] := Block[{rn}, rn = r[2, n]; If[EvenQ[rn], 3*2^(rn/2 - 1), 2^(rn/2 + 1/2)]];
\[Beta][n_Integer] := Block[{rn}, rn = r[2, n]; Which[rn >= 4, \[Alpha][n], rn === 3, 3, rn === 2 && Or @@ OddQ[(r[#1, n] & ) /@ Select[First /@ FactorInteger[n], Mod[#1, 4] === 3 & ]], 2, True, 3/2]];
s[5/2, n_Integer] := (1/8)* dedekindPsi[n] - \[Beta][n]*(\[Chi][n]/2/\[Alpha][n]);
s[5/2, #] & /@ Range[4, 240, 4] (* Wouter Meeussen, cf. Finch reference, Mar 31 2014 *)

A159630 Dimension of space of modular forms for Gamma_0(4*n) of weight 3/2 and trivial character.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 4, 6, 6, 6, 5, 10, 5, 7, 10, 10, 6, 12, 7, 12, 12, 9, 8, 16, 12, 10, 15, 14, 9, 18, 10, 16, 16, 12, 16, 24, 11, 13, 18, 20, 12, 22, 13, 18, 24, 15, 14, 28, 20, 24, 22, 20, 15, 27, 22, 24, 24, 18, 17, 36, 17, 19, 32, 28, 24, 30, 19, 24, 28, 30, 20, 40, 20, 22, 42

Offset: 1

Steven Finch, Apr 17 2009

nonn

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.

Cf. A159634, A159636.

[Dimension(HalfIntegralWeightForms(4*n,3/2)) : n in [1..80]];

a := func< n | Dimension( ModularForms( Gamma0(4*n), 3/2))>; /* Michael Somos, Jul 27 2014 */

A031359 Bisection of A001615.

Original entry on oeis.org

1, 4, 6, 8, 12, 12, 14, 24, 18, 20, 32, 24, 30, 36, 30, 32, 48, 48, 38, 56, 42, 44, 72, 48, 56, 72, 54, 72, 80, 60, 62, 96, 84, 68, 96, 72, 74, 120, 96, 80, 108, 84, 108, 120, 90, 112, 128, 120, 98, 144, 102, 104, 192, 108, 110, 152, 114, 144, 168, 144, 132, 168

Offset: 1

N. J. A. Sloane

Number of coincidence site lattices of index 2n-1 in lattice Z^3.

			G.f. = x + 4*x^2 + 6*x^3 + 8*x^4 + 12*x^5 + 12*x^6 + 14*x^7 + 24*x^8 + ...
G.f. = q + 4*q^3 + 6*q^5 + 8*q^7 + 12*q^9 + 12*q^11 + 14*q^13 + 24*q^15 + ...

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Mathematics of Long-Range Aperiodic Order, Kluwer, 1997, pp. 9-44.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael Baake, Solution of coincidence problem in dimensions d<=4, arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Peter A. B. Pleasants, Algebraic solution of the coincidence problem in two and three dimensions, Zeitschrift für Naturforschung A 50.8 (1995): 711-717. See page 715, the Dirichlet g.f. following Eq. (18).

Cf. A000010, A000203, A001615, A008683, A159634.

a031359 = a001615 . (subtract 1) . (* 2)
-- Reinhard Zumkeller, Jun 03 2013

A001615 := n -> mul((op(1,i)+1)*op(1,i)^(op(2,i)-1),i=op(2,numtheory[ifactors](n)));
A031359 := n -> A001615(2*n-1); # Peter Luschny, Oct 23 2010

a[n_] := (2n-1)*Sum[ MoebiusMu[d]^2/d, {d, Divisors[2n-1]}]; Table[a[n], {n, 1, 62}] (* Jean-François Alcover, Jan 18 2012, after Michael Somos *)
a[ n_] := If[ n < 1, 0, With[{m = 2 n - 1}, m Sum[ MoebiusMu[ d]^2 / d, {d, Divisors[m]}]]] (* Michael Somos, Nov 22 2013 *)

{a(n) = my(m); if( n<1, 0, m = 2*n - 1; m * sumdiv( m, d, moebius(d)^2 / d))} /* Michael Somos, Nov 22 2013 */

{a(n) = my(m); if( n<1, 0, m = 2*n - 1; direuler( p=2, m, (1 + X) / (1 - p*X))[ m])} /* Michael Somos, Nov 22 2013 */

{a(n) = my(A, p, e); if( n<1, 0, n = 2*n - 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, p^(e-1) * (p + 1)))))} /* Michael Somos, Nov 22 2013 */

a(n) = b(2*n - 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^(e-1) * (p+1) if p > 2. - Michael Somos, Nov 22 2013
Dirichlet series: Product (1+p^(-s))/(1-p^(1-s)); p != 2.
a(n) = A001615(2*n - 1).
From Peter Bala, Mar 19 2019: (Start)
a(n) = (2*n - 1)*Product_{p|(2*n-1), p prime} (1 + 1/p).
a(n) = Sum_{ d|(2*n-1) } mu(d)^2*(2*n-1)/d, where mu(n) = A008683(n) is the Möbius function.
a(n) = Sum_{ d^2|(2*n-1) } mu(d)*sigma((2*n-1)/d^2), where sigma(n) = A000203(n) is the sum of the divisors of n, and also
a(n) = Sum_{ d|(2*n-1) } 2^omega(d)*phi((2*n-1)/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^n*(1 + x^(2*n-1))/(1 - x^(2*n-1))^2.
Bisection of A159634. (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 12/Pi^2 = 1.215854... . - Amiram Eldar, Nov 24 2022

Better description from Vladeta Jovovic, Jan 25 2002

More terms from Sascha Kurz, Mar 24 2002

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.

Original entry on oeis.org

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

Offset: 1

Steven Finch, Apr 17 2009

nonn

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) = A001616(n) = Sum_{d|n} phi(gcd(d,n/d)); checked up to n=1024. - Wouter Meeussen, Apr 02 2014

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

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.

Cf. A159630, A159631, A159632, A159634, A159635, A159636. - Steven Finch, Apr 22 2009

Cf. A001616.

[[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]];

(* 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 *)

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A159636 Dimension of space of cusp forms of weight 5/2, level 4*n and trivial character.

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A159630 Dimension of space of modular forms for Gamma_0(4*n) of weight 3/2 and trivial character.

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A031359 Bisection of A001615.

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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.

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