/* 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^3 + 2*x^2 - 8*x - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, 1, 1/2*e^2 - 4, 1/2*e^2 + e, -1/2*e^2 - e + 3, -e^2 - 2*e + 4, -e^2 - e + 4, 1/2*e^2 + 2*e - 8, e^2 - 4, -3*e - 4, -1/2*e^2 - e - 4, 2*e + 4, -2*e - 8, e^2 - 3*e - 12, -e^2, 1/2*e^2 + 2*e - 8, -e^2 - 4*e + 6, -2*e^2 - 2*e + 2, -e, -2*e^2 - 3*e + 8, 2*e^2 + 2*e - 10, -1/2*e^2 + 2*e + 4, -2*e^2 - 2*e + 14, -e^2 + 2*e + 14, e^2 - 12, 5/2*e^2 + 4*e - 12, -1/2*e^2 + 3*e + 8, -2*e - 10, -e^2 - 8, 2*e^2 - 4*e - 26, -4*e^2 - 6*e + 22, -2*e^2 - 4*e + 14, e^2 + 2*e - 6, 5/2*e^2 + 4*e - 12, -e^2 + e + 16, 2*e^2 + 6*e, -3*e^2 + 20, -1/2*e^2 + e + 12, 2*e^2 + 2*e - 18, 3*e^2 + 12*e - 20, -2*e^2 + 4*e + 24, -4*e - 4, 6*e + 10, -e^2 - 3*e - 8, -1/2*e^2 - 2*e, e^2 - 4, e^2 - 4*e, -3*e^2 - 4*e + 24, -1/2*e^2 - 4*e + 16, 3*e + 8, e^2 + 2*e - 12, 8*e + 8, 7/2*e^2 + 7*e - 24, -5*e^2 - 5*e + 28, 2*e^2 + 2, 3/2*e^2 - 8*e - 20, -e^2 - 12, 3*e^2 - 4*e - 32, -2*e^2 - 2*e + 4, 1/2*e^2 + 7*e - 4, 9/2*e^2 + 3*e - 20, -7*e + 4, -2*e^2 - 6*e + 10, -e^2 - 6*e - 14, 2*e^2 + 10*e - 26, -e^2 - 12*e + 6, 1/2*e^2 - 5*e - 12, -4*e^2 - 2*e + 32, -5*e^2 - 5*e + 36, 5/2*e^2 + 10*e - 24, 2*e^2 + 6*e - 4, 3*e^2 + 5*e, -9/2*e^2 - e + 40, -4*e^2 + 26, -5/2*e^2 - 3*e, -7/2*e^2 - 4*e + 20, 5*e^2 + 8*e - 24, 4*e^2 - 36, 9/2*e^2 + 4*e - 12, -5/2*e^2 - 8*e, 5/2*e^2 + 12*e - 16, -e^2 + 8*e + 4, 2*e^2 + 4*e - 24, e^2 - 9*e - 20, -e^2 + 5*e + 24, -7/2*e^2 - e, -e^2 - 4, 4*e^2 + 8*e - 12, 3*e^2 + 10*e - 24, -e^2 - 2*e - 8, 7*e^2 + 6*e - 40, 4*e + 8, -3*e^2 + 8*e + 36, -9*e + 8, 2*e^2 + 6*e + 6, e^2 + 8*e + 16, -1/2*e^2 + 5*e + 12, 4*e^2 + 5*e - 20, 9/2*e^2 - 3*e - 36, -1/2*e^2 + 11*e + 20, -1/2*e^2 - 7*e + 4, 6*e^2 + 4*e - 20, 3*e^2 + 4*e - 16, 1/2*e^2 - 2*e - 32, e^2 - 2*e - 20, -6*e^2 - 8*e + 42, 2*e^2 - 2*e - 16, 7/2*e^2 + 2*e - 20, -9/2*e^2 - 7*e, 4*e + 14, -4*e^2 - 10*e + 26, 4*e^2 + 17*e - 36, -4*e^2 - 4*e + 20, -5*e^2 - 6*e + 44, e^2 + 6*e - 8, 8*e^2 + 4*e - 48, 2*e^2 + 14*e - 16, -3*e^2 + 2*e, -9/2*e^2 - 5*e + 16, -13/2*e^2 - 11*e + 40, -4*e^2 + 8*e + 38, 9/2*e^2 + 11*e - 20, 5/2*e^2 + 2*e - 8, -2*e^2 - 4*e + 4, 5*e^2 + 2*e - 48, 5/2*e^2 - 9*e - 40, 6*e^2 + 17*e - 36, -8, 3*e^2 - 12, 3/2*e^2 + 4*e - 4, 2*e^2 - 8*e - 16, 5*e^2 - 5*e - 60, 3*e^2 + 12*e - 24, -7*e^2 - 4*e + 36, 5/2*e^2 - 2*e + 4, 2*e - 12, -3*e^2 + 2*e - 6, 3*e^2 + 13*e - 12, 4*e^2 - 8*e - 46, -11/2*e^2 + e + 32, 6*e^2 + 4*e - 42, -2*e^2 - 6*e + 30, 2*e^2 + 10*e - 4, -2*e^2 - 4*e + 14, 5*e^2 + 12*e - 26, -6*e - 4, 9*e^2 + 11*e - 40, 3*e^2 + 6*e - 28, -5*e^2 - 10*e + 24, -9/2*e^2 - 5*e + 4, 5/2*e^2 - 3*e - 28, -3*e^2 - 4, e^2 - 44, -e^2 + 3*e - 8, e^2 + e + 12, 9/2*e^2 - 7*e - 48, 2*e^2 + 4*e - 22, 11/2*e^2 + 2*e - 32, -e^2 - e + 36, 52, e^2 - 8*e - 28, 11/2*e^2 + 14*e - 20, 8*e^2 - 6*e - 70, 6*e^2 + 6*e - 30, -8*e^2 - 4*e + 58, -7*e^2 - 9*e + 40, -2*e^2 - 12*e + 2, -e^2 - e + 28, -5/2*e^2 - 9*e + 4, -3/2*e^2 - 7*e + 32, -4*e^2 + 52, -4*e^2 - 8*e + 10, 5*e^2 - e - 48, -4*e^2 - 8*e + 24, -3*e^2 + 2*e, -2*e^2 - 6*e + 26, -9/2*e^2 + 5*e + 52, 8*e + 6] 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]