/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([1, -1, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([15, 15, -w^3 + 5*w + 1]) primes_array = [ [3, 3, w + 1],\ [5, 5, -w^3 + 5*w + 2],\ [5, 5, w - 1],\ [11, 11, -w^3 + 5*w],\ [13, 13, -w^3 + w^2 + 6*w - 3],\ [16, 2, 2],\ [19, 19, 2*w^3 - w^2 - 11*w + 2],\ [25, 5, -w^2 + w + 3],\ [27, 3, w^3 - w^2 - 5*w + 4],\ [31, 31, -w - 3],\ [37, 37, -w^3 - w^2 + 6*w + 4],\ [41, 41, w^3 + w^2 - 6*w - 5],\ [43, 43, w^3 - 7*w - 2],\ [47, 47, -2*w^3 + 11*w + 2],\ [61, 61, w^2 - 3],\ [67, 67, w^2 + w - 4],\ [79, 79, -4*w^3 + 2*w^2 + 22*w - 9],\ [97, 97, -3*w^3 + 2*w^2 + 19*w - 6],\ [97, 97, -w^3 + 6*w - 3],\ [97, 97, -3*w^3 + 16*w],\ [97, 97, -w^3 + w^2 + 4*w - 3],\ [101, 101, -2*w^3 + w^2 + 11*w - 6],\ [103, 103, w^3 - w^2 - 7*w],\ [109, 109, -w^3 - w^2 + 6*w + 2],\ [121, 11, 2*w^3 - 11*w + 1],\ [127, 127, -2*w^3 + w^2 + 13*w - 1],\ [127, 127, w^3 - w^2 - 6*w - 2],\ [157, 157, -2*w^3 + 2*w^2 + 10*w - 7],\ [157, 157, w - 4],\ [163, 163, 2*w^3 - w^2 - 13*w + 3],\ [173, 173, -2*w^3 + w^2 + 13*w - 6],\ [173, 173, 2*w^3 - 12*w + 1],\ [173, 173, 2*w^3 - 11*w + 2],\ [173, 173, w^2 + w - 5],\ [179, 179, -2*w^3 + 10*w + 3],\ [181, 181, -w^3 - w^2 + 7*w + 8],\ [191, 191, w^3 - 5*w - 5],\ [191, 191, 2*w^3 - w^2 - 9*w - 1],\ [193, 193, -3*w^3 + 3*w^2 + 19*w - 7],\ [197, 197, -w^3 + w^2 + 6*w - 7],\ [223, 223, w^3 + w^2 - 6*w + 1],\ [227, 227, 6*w^3 - 3*w^2 - 34*w + 9],\ [227, 227, w^3 + w^2 - 8*w],\ [233, 233, 2*w^3 - 13*w - 3],\ [233, 233, -w^3 + 2*w^2 + 4*w - 6],\ [239, 239, -2*w^3 + 10*w + 1],\ [239, 239, w^3 - 8*w - 2],\ [251, 251, -3*w^3 + w^2 + 18*w],\ [251, 251, -5*w^3 + 2*w^2 + 30*w - 11],\ [263, 263, -2*w^3 + w^2 + 10*w + 2],\ [269, 269, 3*w^3 - 18*w - 8],\ [271, 271, 2*w^3 + w^2 - 10*w - 2],\ [271, 271, -3*w^3 + 19*w],\ [277, 277, -w^3 + w^2 + 4*w - 8],\ [277, 277, 2*w^3 + w^2 - 14*w - 8],\ [281, 281, w^3 - 8*w],\ [307, 307, 3*w^3 - w^2 - 16*w + 2],\ [307, 307, 2*w^3 - w^2 - 9*w + 1],\ [307, 307, -2*w^3 + 2*w^2 + 13*w - 7],\ [307, 307, -w^3 + w^2 + 7*w - 6],\ [313, 313, -3*w^3 + w^2 + 15*w + 4],\ [317, 317, -4*w^3 + w^2 + 22*w - 3],\ [317, 317, 2*w^3 - w^2 - 10*w - 3],\ [337, 337, w^3 + w^2 - 4*w - 5],\ [349, 349, -2*w^3 - w^2 + 11*w + 3],\ [349, 349, -4*w^3 + 2*w^2 + 24*w - 11],\ [353, 353, 3*w^3 - w^2 - 17*w + 7],\ [353, 353, -3*w^3 - 2*w^2 + 17*w + 11],\ [367, 367, 2*w^3 - 11*w - 7],\ [373, 373, -w^3 + w^2 + 4*w - 5],\ [383, 383, -4*w^3 + 3*w^2 + 23*w - 9],\ [383, 383, 4*w^3 - w^2 - 22*w],\ [389, 389, w^2 - 2*w - 5],\ [401, 401, -3*w^3 + 2*w^2 + 16*w - 9],\ [401, 401, 2*w^3 - w^2 - 14*w - 1],\ [409, 409, 2*w^3 + 2*w^2 - 13*w - 9],\ [419, 419, -4*w^3 + 24*w + 7],\ [419, 419, 2*w^2 + 2*w - 7],\ [419, 419, -w^3 + 2*w^2 + 8*w - 6],\ [419, 419, 6*w^3 - 3*w^2 - 33*w + 13],\ [431, 431, 2*w^3 - w^2 - 12*w - 5],\ [439, 439, -3*w^3 + 2*w^2 + 18*w - 4],\ [439, 439, w^3 + 2*w^2 - 7*w - 3],\ [443, 443, 6*w^3 - 3*w^2 - 33*w + 7],\ [449, 449, -3*w^3 + w^2 + 14*w - 4],\ [457, 457, 2*w^3 - 9*w],\ [457, 457, 3*w^3 - w^2 - 16*w - 2],\ [479, 479, 3*w^3 - w^2 - 19*w + 4],\ [487, 487, -w^3 + w^2 + 4*w - 6],\ [491, 491, 3*w^3 + w^2 - 17*w - 5],\ [499, 499, 5*w^3 - 3*w^2 - 28*w + 8],\ [499, 499, 2*w^3 - w^2 - 12*w - 1],\ [503, 503, -2*w^3 - w^2 + 12*w + 3],\ [503, 503, w^3 - 3*w - 4],\ [509, 509, -3*w - 5],\ [509, 509, -4*w^3 + 3*w^2 + 25*w - 10],\ [523, 523, -2*w^3 + w^2 + 14*w - 6],\ [529, 23, 4*w^3 - 2*w^2 - 21*w + 5],\ [529, 23, -3*w^3 - w^2 + 15*w + 6],\ [557, 557, -2*w^3 + w^2 + 13*w + 2],\ [557, 557, 3*w^3 - 2*w^2 - 16*w + 3],\ [569, 569, 2*w^2 - w - 8],\ [587, 587, w^3 + 2*w^2 - 5*w - 16],\ [587, 587, -4*w^3 + w^2 + 25*w - 5],\ [599, 599, w^3 + 2*w^2 - 6*w - 5],\ [599, 599, -5*w^3 + 3*w^2 + 30*w - 15],\ [601, 601, w^3 - 6*w - 6],\ [607, 607, -w^3 + w^2 + 3*w - 4],\ [607, 607, -w^3 + 2*w - 3],\ [613, 613, 6*w^3 - 2*w^2 - 31*w + 11],\ [613, 613, w^3 + w^2 - 4*w - 6],\ [617, 617, 5*w^3 - w^2 - 30*w + 2],\ [617, 617, -2*w^3 + 2*w^2 + 9*w + 3],\ [619, 619, -w^3 + 2*w^2 + 6*w - 5],\ [619, 619, 2*w^3 - w^2 - 11*w - 3],\ [631, 631, -w^3 + 2*w^2 + 5*w - 5],\ [631, 631, -3*w^3 - w^2 + 18*w + 5],\ [647, 647, 6*w^3 - w^2 - 36*w + 2],\ [659, 659, 4*w^3 + w^2 - 24*w - 14],\ [661, 661, 2*w^2 - w - 4],\ [673, 673, 2*w^3 + w^2 - 14*w - 12],\ [677, 677, 3*w^3 + 2*w^2 - 16*w - 7],\ [677, 677, -3*w^3 + w^2 + 15*w + 1],\ [683, 683, 2*w^2 - 2*w - 9],\ [683, 683, 2*w^3 - 14*w + 1],\ [691, 691, 5*w^3 - 2*w^2 - 30*w + 6],\ [701, 701, -2*w^3 + w^2 + 12*w + 2],\ [701, 701, -5*w^3 + 2*w^2 + 26*w],\ [719, 719, 3*w^3 - 18*w + 1],\ [719, 719, 3*w^3 - w^2 - 19*w + 1],\ [727, 727, -3*w^2 - 2*w + 11],\ [727, 727, 2*w^3 + w^2 - 13*w - 2],\ [733, 733, 2*w^3 - 3*w^2 - 8*w + 5],\ [739, 739, -3*w^3 + 16*w - 6],\ [743, 743, -4*w^3 - w^2 + 25*w + 8],\ [751, 751, w^2 - 2*w - 7],\ [751, 751, -3*w^3 + 15*w + 5],\ [757, 757, 3*w^3 - 19*w - 2],\ [761, 761, 2*w^2 + w - 5],\ [769, 769, -2*w^3 + w^2 + 14*w - 9],\ [769, 769, 2*w^2 - w - 7],\ [773, 773, w^3 - w^2 - 5*w - 4],\ [773, 773, w^3 - 3*w - 6],\ [787, 787, 3*w^3 - 19*w - 3],\ [797, 797, 8*w^3 - 3*w^2 - 47*w + 10],\ [809, 809, 4*w^3 - 25*w - 1],\ [811, 811, w^3 - 9*w - 1],\ [821, 821, 6*w^3 - 3*w^2 - 35*w + 8],\ [823, 823, 2*w^3 + w^2 - 10*w + 1],\ [823, 823, -2*w^3 + 2*w^2 + 7*w - 4],\ [829, 829, -5*w^3 + w^2 + 27*w - 4],\ [829, 829, 5*w^3 - 2*w^2 - 26*w + 10],\ [839, 839, w^3 + 2*w^2 - 5*w - 7],\ [839, 839, 3*w^3 - w^2 - 15*w + 2],\ [841, 29, 3*w^3 + w^2 - 14*w - 2],\ [841, 29, 4*w^3 - 2*w^2 - 21*w + 10],\ [857, 857, -8*w^3 + 4*w^2 + 44*w - 11],\ [857, 857, -3*w^3 + 3*w^2 + 20*w - 9],\ [859, 859, w^3 + 2*w^2 - 6*w - 6],\ [877, 877, w^3 - 3*w^2 - 4*w + 14],\ [883, 883, 2*w^3 + w^2 - 10*w - 11],\ [887, 887, 3*w^3 + w^2 - 20*w - 2],\ [919, 919, 7*w^3 - 3*w^2 - 39*w + 9],\ [929, 929, 3*w^3 - 17*w + 2],\ [929, 929, -6*w^3 + 2*w^2 + 35*w - 10],\ [937, 937, w^3 + w^2 - 3*w - 5],\ [937, 937, -2*w^3 + 3*w^2 + 12*w - 12],\ [937, 937, -3*w^3 - 2*w^2 + 19*w + 8],\ [937, 937, 3*w^3 - 2*w^2 - 13*w + 3],\ [941, 941, w^3 + 2*w^2 - 6*w - 8],\ [941, 941, 5*w^3 - w^2 - 28*w - 3],\ [947, 947, -4*w^3 + 2*w^2 + 25*w - 6],\ [953, 953, w^3 - 7*w - 7],\ [953, 953, -3*w^3 + 2*w^2 + 18*w - 3],\ [967, 967, w^3 - 2*w^2 - 5*w - 3],\ [971, 971, w^2 + 2*w - 6],\ [977, 977, 2*w^3 - 11*w - 8],\ [997, 997, -3*w^3 + w^2 + 17*w + 2],\ [997, 997, -2*w^3 + 2*w^2 + 14*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 2*x^3 - 16*x^2 - 36*x - 12 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, 1, 1/2*e^3 - 8*e - 6, -4, -1/2*e^2 + e + 4, -1/2*e^3 - 1/2*e^2 + 8*e + 7, 1/2*e^3 + 1/2*e^2 - 9*e - 13, -1/2*e^3 - 1/2*e^2 + 7*e + 11, -1/2*e^3 - e^2 + 9*e + 10, -1/2*e^3 + 1/2*e^2 + 7*e - 1, 0, -2*e + 2, 1/2*e^2 + e - 3, -1/2*e^3 - 1/2*e^2 + 9*e + 7, 1/2*e^3 + e^2 - 9*e - 14, -e^2 + e + 10, e^2 - 2*e - 8, -e^2 + 10, 3/2*e^3 + e^2 - 22*e - 16, 1/2*e^3 - 1/2*e^2 - 7*e - 7, 1/2*e^3 + e^2 - 11*e - 18, 1/2*e^3 + e^2 - 12*e - 16, -1/2*e^3 + 7*e - 2, 2*e - 8, 3/2*e^3 - 23*e - 14, -1/2*e^3 + 1/2*e^2 + 7*e - 1, e^2 - 4*e - 14, e^2 - 2, -1/2*e^3 - e^2 + 9*e + 20, -1/2*e^3 + 7*e + 18, 1/2*e^3 - 13*e - 6, 2*e^3 + 2*e^2 - 30*e - 36, 1/2*e^3 - e^2 - 3*e + 12, 1/2*e^3 + 3/2*e^2 - 11*e - 21, -3*e - 8, -3/2*e^3 + 21*e + 6, -1/2*e^3 + 3/2*e^2 + 6*e - 9, 3/2*e^3 + 1/2*e^2 - 24*e - 35, -1/2*e^3 + 7*e - 6, -1/2*e^3 + 9*e + 10, -5/2*e^3 - 5/2*e^2 + 38*e + 39, 2*e^2 - 2*e - 18, 3/2*e^3 - 25*e - 18, -3/2*e^3 - 1/2*e^2 + 23*e + 21, -1/2*e^3 + 3/2*e^2 + 9*e - 15, e^3 - 12*e - 6, -e^3 - 1/2*e^2 + 11*e + 15, -1/2*e^3 + e^2 + 7*e - 6, 1/2*e^3 - 8*e - 6, -3/2*e^3 - 1/2*e^2 + 23*e + 27, -1/2*e^3 + e^2 + 9*e - 14, 3/2*e^3 + 2*e^2 - 21*e - 34, e^3 + e^2 - 18*e - 14, e^3 + 2*e^2 - 13*e - 32, 5/2*e^3 + 2*e^2 - 39*e - 48, -1/2*e^3 + 3/2*e^2 + 9*e - 17, -e^2 - 6*e + 14, e^3 + 2*e^2 - 24*e - 38, -e^3 - 2*e^2 + 18*e + 32, -e^2 - 2*e + 10, -1/2*e^3 - 5/2*e^2 + 8*e + 15, 3/2*e^3 + 1/2*e^2 - 27*e - 15, -1/2*e^3 + 11*e - 2, e^3 - e^2 - 16*e - 8, -5/2*e^3 + 39*e + 34, -2*e^3 - e^2 + 34*e + 36, -2*e^3 - 7/2*e^2 + 35*e + 45, -3*e^3 - 2*e^2 + 48*e + 38, -e^3 - e^2 + 15*e + 34, -3/2*e^3 - 3/2*e^2 + 27*e + 27, -3*e^3 - 4*e^2 + 52*e + 60, -1/2*e^3 - 2*e^2 + 6*e + 18, 6*e - 12, 2*e^3 + 2*e^2 - 32*e - 42, 4*e^3 + 3*e^2 - 60*e - 62, 5/2*e^3 + 1/2*e^2 - 35*e - 27, -3*e^3 - 3*e^2 + 52*e + 54, 1/2*e^3 - 2*e^2 - 5*e + 12, -3/2*e^3 - 1/2*e^2 + 21*e + 3, -3/2*e^3 + 1/2*e^2 + 25*e + 21, -3/2*e^3 + 24*e + 14, e^3 + 3*e^2 - 18*e - 28, -1/2*e^3 + 5/2*e^2 + 2*e - 27, 3/2*e^3 + 3/2*e^2 - 17*e - 33, 1/2*e^3 + 1/2*e^2 - 9*e - 25, -3/2*e^3 - 3/2*e^2 + 20*e + 37, -3/2*e^3 - e^2 + 31*e + 24, -e^3 - 3/2*e^2 + 15*e + 29, -9/2*e^3 - 2*e^2 + 69*e + 66, 2*e^3 - 26*e - 20, -3*e^3 - e^2 + 50*e + 38, -4*e^2 + 36, e^2 - 4*e, 1/2*e^3 - 1/2*e^2 - 11*e + 9, 2*e^3 + 3*e^2 - 32*e - 42, 2*e^3 + e^2 - 32*e - 40, -3/2*e^3 + 1/2*e^2 + 26*e + 13, -3/2*e^3 + e^2 + 20*e + 8, -3*e^3 - 7/2*e^2 + 47*e + 45, -2*e^3 - e^2 + 24*e + 30, -5/2*e^3 - 3/2*e^2 + 38*e + 21, 2*e^3 + 2*e^2 - 40*e - 48, e^3 - e^2 - 22*e + 12, 3/2*e^3 - 5/2*e^2 - 27*e + 9, -e^3 - 4*e^2 + 16*e + 30, 3/2*e^3 + 4*e^2 - 21*e - 52, -5/2*e^2 - e + 11, 1/2*e^3 + 1/2*e^2 - 7*e - 11, 5/2*e^3 - 5/2*e^2 - 42*e - 5, -1/2*e^3 - 4*e^2 + 15*e + 44, -3/2*e^3 + 2*e^2 + 25*e - 18, 3/2*e^3 + 4*e^2 - 28*e - 42, -e^3 + 12*e + 16, -2*e^3 - 5/2*e^2 + 33*e + 59, 2*e^3 - 1/2*e^2 - 27*e - 13, -2*e^3 - 4*e^2 + 30*e + 62, 7/2*e^3 + 2*e^2 - 57*e - 42, -5/2*e^3 + e^2 + 37*e + 18, -3*e^3 - 3/2*e^2 + 45*e + 29, 3/2*e^2 + 3*e - 13, 4*e + 6, 2*e^3 + e^2 - 28*e - 42, 3/2*e^3 + 3*e^2 - 26*e - 60, -2*e^3 - 2*e^2 + 34*e + 30, 5/2*e^3 - 2*e^2 - 37*e - 2, -2*e^2 - 2*e + 48, 3/2*e^3 + 1/2*e^2 - 25*e - 21, 2*e^3 - 28*e, 9/2*e^3 + 5*e^2 - 73*e - 72, 7/2*e^3 + 9/2*e^2 - 55*e - 61, e^3 + 3*e^2 - 18*e - 32, 4*e^3 + 5*e^2 - 63*e - 74, e^3 - e^2 - 6*e + 8, -2*e - 30, -1/2*e^3 - 7/2*e^2 + 9*e + 49, 2*e^3 + 2*e^2 - 34*e - 52, -3/2*e^3 - 2*e^2 + 31*e + 28, -5/2*e^3 - 2*e^2 + 48*e + 30, 5*e^3 + 4*e^2 - 74*e - 82, -3*e^3 - 9/2*e^2 + 49*e + 59, -3*e^3 - 4*e^2 + 54*e + 54, -1/2*e^3 + 2*e^2 + 3*e - 30, -5/2*e^3 + 2*e^2 + 43*e + 14, -2*e^3 - 2*e^2 + 26*e + 36, -3/2*e^3 - 6*e^2 + 23*e + 60, -1/2*e^3 + 2*e^2 - e - 22, -3/2*e^3 + 3/2*e^2 + 27*e + 15, 9/2*e^3 + 3*e^2 - 69*e - 74, 5/2*e^3 + 1/2*e^2 - 38*e - 11, -3*e^2 + 8*e + 50, e^3 + 3/2*e^2 - 15*e - 1, e^3 + 1/2*e^2 - 11*e - 3, 3*e^3 + 3/2*e^2 - 51*e - 33, 3/2*e^3 + 4*e^2 - 26*e - 58, -e^3 + 3*e^2 + 11*e - 38, -e^3 + 1/2*e^2 + 11*e + 9, e^3 - 28*e - 12, -7/2*e^3 - 5/2*e^2 + 51*e + 43, -5/2*e^3 + 2*e^2 + 36*e - 10, -2*e^3 + 24*e - 16, -1/2*e^3 + e^2 + 13*e + 6, -1/2*e^3 + 1/2*e^2 + 7*e - 35, 3/2*e^3 - 2*e^2 - 13*e + 18, -3/2*e^3 + e^2 + 17*e - 18, -e^3 + 2*e^2 + 20*e - 16, -1/2*e^3 - e^2 + 18*e + 32, -e^3 - 4*e^2 + 10*e + 46, -3/2*e^3 + 5/2*e^2 + 21*e - 11, 3*e^3 + 2*e^2 - 44*e - 36, 1/2*e^3 - 3/2*e^2 - 6*e + 9, -e^3 + 2*e^2 + 18*e - 24, 1/2*e^3 - 3*e - 12, -e^3 + e^2 + 11*e + 6, 4*e^3 + 4*e^2 - 60*e - 76, 2*e^2 + 2*e + 6, -7/2*e^3 - 2*e^2 + 69*e + 54, e^3 + 9/2*e^2 - 9*e - 55, e^3 + 2*e^2 - 6*e - 34] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, w + 1])] = 1 AL_eigenvalues[ZF.ideal([5, 5, w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]