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A001617 Genus of modular group Gamma_0(n). Or, genus of modular curve X_0(n).
(Formerly M0188 N0069)
20
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
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, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Steven R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
Ralf Hemmecke, Peter Paule, and Silviu Radu, Construction of Modular Function Bases for Gamma_0(121) related to p*(11*n + 6), (2019).
Nicolas Allen Smoot, Computer algebra with the fifth operation: applications of modular functions to partition congruences, Ph. D. Thesis, Johannes Kepler Univ., Linz (Austria 2022), 33.
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
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: # Gene Ward Smith, May 23 2006
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-François 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
Sequence in context: A141647 A366386 A358729 * A333628 A342707 A282947
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved

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Last modified June 17 15:57 EDT 2024. Contains 373463 sequences. (Running on oeis4.)