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A029937 Genus of modular curve X_1(n). 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3, 5, 6, 12, 5, 12, 10, 13, 10, 22, 9, 26, 17, 21, 21, 25, 17, 40, 28, 33, 25, 51, 25, 57, 36, 41, 45, 70, 37, 69, 48, 65, 55, 92, 52, 81, 61, 85, 78, 117, 57, 126, 91, 97 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,13

COMMENTS

Also the dimension of the space of cusp forms of weight two on Gamma1(n). [Steven Finch, Apr 03 2009]

REFERENCES

F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 161.

A. V. Sutherland, Notes on torsion subgroups of elliptic curves over number fields, 2012, http://math.mit.edu/~drew/MazursTheoremSubsequentResults.pdf. - From N. J. A. Sloane, Feb 02 2013

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]

Chang Heon Kim, Ja Kyung Koo, On the genus of some modular curves of level N, Bull Austral. Math. Soc. 54 (1996) 291-297.

A. V. Sutherland, Torsion subgroups of elliptic curves over number fields, 2012. - From N. J. A. Sloane, Feb 03 2013

FORMULA

a(n) = 1+A115000(n)-A029935(n)/4, n>4. [Kim and Koo, Theorem 1]

MAPLE

with(numtheory); A029937 := proc(n) local i, j; j := 1+(1/24)*phi(n)*A001615(n); for i in divisors(n) do j := j-(1/4)*phi(i)*phi(n/i) od; j; end;

MATHEMATICA

a[n_ /; n<5] = 0; a[n_] := 1+Sum[d^2*MoebiusMu[n/d]/24 - EulerPhi[d]*EulerPhi[n/d]/4, {d, Divisors[n]}]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Jan 13 2014 *)

PROG

(PARI)

A029935(n) = {

  my(f = factor(n), fsz = matsize(f)[1],

     g = prod(k=1, fsz, f[k, 1]),

     h = prod(k=1, fsz, sqr(f[k, 1]-1)*f[k, 2] + sqr(f[k, 1])-1));

  return(h*n\sqr(g));

};

a(n) = {

  if (n < 5, return(0));

  my(f = factor(n), fsz = matsize(f)[1],

     g = prod(k=1, fsz, f[k, 1]),

     h = prod(k=1, fsz, sqr(f[k, 1]) - 1));

  return(1 + sqr(n\g)*h/24 - A029935(n)/4);

};

vector(63, n, a(n))  \\ Gheorghe Coserea, Oct 23 2016

CROSSREFS

Cf. A001617, A029938. [Steven Finch, Apr 03 2009]

Sequence in context: A309522 A305313 A159046 * A289772 A283615 A216396

Adjacent sequences:  A029934 A029935 A029936 * A029938 A029939 A029940

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified December 12 12:30 EST 2019. Contains 329958 sequences. (Running on oeis4.)