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Search: id:A001617
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| A001617 |
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Genus of modular group GAMMA_0 (n). Or, genus of modular curve X_0(n).
(Formerly M0188 N0069)
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+0
8
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| 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; listen)
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OFFSET
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1,22
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COMMENT
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Also the dimension of the space of cusp forms of weight two and level n. - Gene Smith (genewardsmith(AT)gmail.com), May 23 2006
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REFERENCES
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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.
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).
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..1000
S. R. Finch, Modular forms on SL_2(Z)
Index entries for sequences related to modular groups
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FORMULA
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a(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2.
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MAPLE
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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:
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:
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CROSSREFS
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Cf. A001615, A000089, A000086, A001616, A091401, A091403, A091404.
Sequence in context: A153241 A096830 A141647 this_sequence A143667 A084934 A125927
Adjacent sequences: A001614 A001615 A001616 this_sequence A001618 A001619 A001620
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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