%I A001617 M0188 N0069
%S A001617 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,
%T A001617 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,
%U A001617 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
%N A001617 Genus of modular group GAMMA_0 (n). Or, genus of modular curve X_0(n).
%C A001617 Also the dimension of the space of cusp forms of weight two and level 
               n. - Gene Smith (genewardsmith(AT)gmail.com), May 23 2006
%D A001617 Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups 
               of linear fractional transformations. J. Res. Nat. Bur. Standards 
               Sect. B 67B 1963 61-68.
%D A001617 B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, 
               p. 103.
%D A001617 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A001617 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A001617 N. J. A. Sloane, <a href="b001617.txt">Table of n, a(n) for n = 1..1000</
               a>
%H A001617 S. R. Finch, <A HREF="http://algo.inria.fr/bsolve/">Modular forms on 
               SL_2(Z)</A>
%H A001617 <a href="Sindx_Gre.html#groups_modular">Index entries for sequences related 
               to modular groups</a>
%F A001617 a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2.
%p A001617 Maple program from Gene Smith, May 23 2006: 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:
%p A001617 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:
%p A001617 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:
%p A001617 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;
%p A001617 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:
%Y A001617 Cf. A001615, A000089, A000086, A001616, A091401, A091403, A091404.
%Y A001617 Sequence in context: A153241 A096830 A141647 this_sequence A143667 A084934 
               A125927
%Y A001617 Adjacent sequences: A001614 A001615 A001616 this_sequence A001618 A001619 
               A001620
%K A001617 nonn,easy,nice
%O A001617 1,22
%A A001617 N. J. A. Sloane (njas(AT)research.att.com).