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A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).
(Formerly M0188 N0069)
13
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 0, 2, 1, 2, 2, 3, 2, 1, 3, 3, 3, 1, 2, 4, 3, 3, 3, 5, 3, 4, 3, 5, 4, 3, 1, 2, 5, 5, 4, 4, 5, 5, 5, 6, 5, 7, 4, 7, 5, 3, 5, 9, 5, 7, 7, 9, 6, 5, 5, 8, 5, 8, 7, 11, 6, 7, 4, 9, 7, 11, 7, 10, 9, 9, 7, 11, 7, 10, 9, 11, 9, 9, 7, 7, 9, 7, 8, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,22

COMMENTS

Also the dimension of the space of cusp forms of weight two and level n. - Gene Ward Smith, May 23 2006

REFERENCES

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

N. J. A. Sloane and Gheorghe Coserea, Table of n, a(n) for n = 1..50000, first 1000 terms from N. J. A. Sloane

J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Harriet Fell, Morris Newman, Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.

S. R. Finch, Modular forms on SL_2(Z)

Index entries for sequences related to modular groups

FORMULA

a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2.

From Gheorghe Coserea, May 20 2016: (Start)

limsup a(n) / (n*log(log(n))) = exp(Euler)/(2*Pi^2), where Euler is A001620.

a(n) >= (n-5*sqrt(n)-8)/12, with equality iff n = p^2 for prime p=1 (mod 12) (see A068228).

a(n) < n * exp(Euler)/(2*Pi^2) * (log(log(n)) + 2/log(log(n))) for n>=3 (see Csirik link).

(End)

EXAMPLE

G.f. = x^11 + x^14 + x^15 + x^17 + x^19 + x^20 + x^21 + 2*x^22 + 2*x^23 + ...

MAPLE

From Gene Ward Smith, May 23 2006: (Start)

nu2 := proc (n) # number of elliptic points of order two (A000089) local i, s; if modp(n, 4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1, i))) fi od; s end:

nu3 := proc (n) # number of elliptic points of order three (A000086) local d, s; if modp(n, 9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3, d))) fi od; s end:

nupara := proc (n) # number of parabolic cusps (A001616) local b, d; b := 0; for d to n do if modp(n, d) = 0 then b := b+eval(phi(gcd(d, n/d))) fi od; b end:

A001615 := proc(n) local i, j; j := n; for i in divisors(n) do if isprime(i) then j := j*(1+1/i); fi; od; j; end;

genx := proc (n) # genus of X0(n) (A001617) #1+1/12*psi(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end: 1+1/12*A001615(n)-1/4*nu2(n)-1/3*nu3(n)-1/2*nupara(n) end:

(End)

MATHEMATICA

nu2[n_] := Module[{i, s}, If[Mod[n, 4] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[i] && i > 2, s = s*(1 + JacobiSymbol[-1, i])], {i, Divisors[n]}]; s];

nu3[n_] := Module[{d, s}, If[Mod[n, 9] == 0, Return[0]]; s = 1; Do[ If[ PrimeQ[d], s = s*(1 + JacobiSymbol[-3, d])], {d, Divisors[n]}]; s];

nupara[n_] := Module[{b, d}, b = 0; For[d = 1, d <= n, d++, If[ Mod[n, d] == 0, b = b + EulerPhi[ GCD[d, n/d]]]]; b];

A001615[n_] := Module[{i, j}, j = n; Do[ If[ PrimeQ[i], j = j*(1 + 1/i)], {i, Divisors[n]}]; j];

genx[n_] := 1 + (1/12)*A001615[n] - (1/4)*nu2[n] - (1/3)*nu3[n] - (1/2)*nupara[n];

A001617 = Table[ genx[n], {n, 1, 102}] (* Jean-Fran├žois Alcover, Jan 04 2012, after Gene Ward Smith's Maple program *)

a[ n_] := If[ n < 1, 0, 1 + Sum[ MoebiusMu[ d]^2 n/d / 12 - EulerPhi[ GCD[ d, n/d]] / 2, {d, Divisors @n}] - Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 - Count[ (#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4]; (* Michael Somos, May 08 2015 *)

PROG

(MAGMA) a := func< n | n lt 1 select 0 else Dimension( CuspForms( Gamma0(n), 2))>; /* Michael Somos, May 08 2015 */

(PARI)

A000089(n) = {

  if (n%4 == 0 || n%4 == 3, return(0));

  if (n%2 == 0, n \= 2);

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

  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));

};

A000086(n) = {

  if (n%9 == 0 || n%3 == 2, return(0));

  if (n%3 == 0, n \= 3);

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

  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));

};

A001615(n) = {

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

     g = prod(k=1, fsz, (f[k, 1]+1)),

     h = prod(k=1, fsz, f[k, 1]));

  return((n*g)\h);

};

A001616(n) = {

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

  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));

};

a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;

vector(102, n, a(n))  \\ Gheorghe Coserea, May 20 2016

CROSSREFS

Cf. A001615, A000089, A000086, A001616, A054728, A091401, A091403, A091404.

Sequence in context: A257109 A096830 A141647 * A143667 A246785 A084934

Adjacent sequences:  A001614 A001615 A001616 * A001618 A001619 A001620

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 1 12:57 EDT 2016. Contains 274320 sequences.