/* 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, -5, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([23, 23, -w^2 + 2]) primes_array = [ [2, 2, w + 1],\ [5, 5, w^3 - 4*w],\ [8, 2, w^3 - w^2 - 4*w + 3],\ [11, 11, w^3 - 5*w + 1],\ [17, 17, -w^3 - w^2 + 5*w + 4],\ [23, 23, -w^2 + 2],\ [29, 29, -w^2 - w + 3],\ [31, 31, -w^3 + 6*w + 2],\ [43, 43, w^3 - 6*w],\ [47, 47, 2*w^3 - 9*w],\ [47, 47, -2*w^3 + w^2 + 8*w - 2],\ [53, 53, -4*w^3 + 2*w^2 + 18*w - 5],\ [59, 59, w^2 - w - 5],\ [59, 59, 3*w^3 - 15*w - 5],\ [71, 71, w^3 - 3*w - 3],\ [81, 3, -3],\ [83, 83, -2*w^3 + 8*w + 3],\ [89, 89, -2*w^3 + 11*w - 2],\ [101, 101, 2*w^3 - w^2 - 10*w],\ [101, 101, 2*w^3 - 2*w^2 - 10*w + 3],\ [107, 107, 3*w^3 - 15*w - 1],\ [107, 107, 4*w^3 - 2*w^2 - 18*w + 3],\ [109, 109, 2*w^3 - 10*w + 3],\ [109, 109, 3*w^3 - 2*w^2 - 13*w + 3],\ [113, 113, 3*w^3 - w^2 - 15*w + 2],\ [113, 113, 3*w^3 - 13*w - 5],\ [121, 11, 3*w^3 - 13*w - 1],\ [125, 5, -2*w^3 + w^2 + 7*w + 1],\ [127, 127, 2*w^3 - 8*w - 1],\ [127, 127, -w^3 + 2*w^2 + 3*w - 3],\ [137, 137, 2*w^3 - 10*w + 1],\ [137, 137, 2*w^3 - 11*w],\ [139, 139, 2*w^3 - 9*w - 6],\ [157, 157, -4*w^3 - w^2 + 19*w + 7],\ [163, 163, 5*w^3 - 2*w^2 - 25*w + 7],\ [167, 167, w^2 + w - 5],\ [167, 167, -w^3 + 2*w^2 + 4*w - 6],\ [169, 13, -w^3 + w^2 + 3*w - 4],\ [169, 13, w^2 - 2*w - 4],\ [173, 173, -2*w^3 + w^2 + 11*w - 7],\ [173, 173, w - 4],\ [181, 181, w + 4],\ [181, 181, -2*w - 5],\ [191, 191, -w^3 + 2*w^2 + 6*w - 6],\ [191, 191, 2*w^3 - w^2 - 11*w + 1],\ [193, 193, -3*w^3 + 16*w],\ [227, 227, 6*w^3 - 3*w^2 - 27*w + 5],\ [227, 227, 2*w^3 - 11*w - 2],\ [229, 229, -2*w^3 + w^2 + 10*w - 6],\ [233, 233, 4*w^3 - 3*w^2 - 17*w + 5],\ [233, 233, 2*w^3 + w^2 - 7*w - 5],\ [233, 233, w^3 - 7*w - 1],\ [233, 233, 2*w^3 - 7*w],\ [239, 239, 2*w^2 - 2*w - 7],\ [239, 239, -4*w^3 + w^2 + 18*w],\ [241, 241, w^3 - 5*w - 5],\ [263, 263, -4*w^3 + w^2 + 20*w - 4],\ [269, 269, w^3 + 2*w^2 - 6*w - 6],\ [271, 271, 2*w^3 + w^2 - 12*w - 8],\ [277, 277, w^3 + w^2 - 7*w - 6],\ [283, 283, 6*w^3 - 3*w^2 - 27*w + 7],\ [283, 283, -3*w^3 + 3*w^2 + 13*w - 8],\ [293, 293, -2*w^3 + w^2 + 9*w + 3],\ [293, 293, 4*w^3 - w^2 - 19*w + 3],\ [313, 313, 2*w^2 - w - 4],\ [313, 313, -2*w^3 + 2*w^2 + 8*w - 1],\ [317, 317, 3*w^3 + w^2 - 13*w - 10],\ [331, 331, w^3 + 2*w^2 - 4*w - 6],\ [349, 349, -w^3 + 3*w^2 + w - 4],\ [353, 353, 3*w^3 - 14*w],\ [353, 353, -2*w^3 + w^2 + 7*w + 5],\ [361, 19, -5*w^3 + w^2 + 23*w - 2],\ [361, 19, -2*w^3 + 7*w + 6],\ [373, 373, -w^3 + 4*w - 4],\ [379, 379, -w^3 - 2*w^2 + 5*w + 7],\ [383, 383, w^2 - 2*w - 6],\ [389, 389, -4*w^3 + 2*w^2 + 16*w - 5],\ [397, 397, 5*w^3 - 2*w^2 - 24*w + 2],\ [397, 397, 3*w^3 + 2*w^2 - 16*w - 14],\ [401, 401, 2*w^2 - 5],\ [419, 419, -w^3 - w^2 + 9*w + 6],\ [439, 439, 3*w^3 - w^2 - 15*w - 2],\ [449, 449, 4*w^3 + w^2 - 17*w - 5],\ [449, 449, 4*w^3 - w^2 - 17*w - 5],\ [467, 467, -3*w^3 + 16*w - 2],\ [467, 467, 2*w^3 + w^2 - 10*w - 2],\ [479, 479, 4*w^3 - 19*w - 8],\ [491, 491, 3*w - 4],\ [491, 491, -w^3 + 2*w^2 + 5*w - 11],\ [509, 509, -7*w^3 + 3*w^2 + 33*w - 10],\ [509, 509, 4*w^3 - w^2 - 21*w - 3],\ [523, 523, -w^3 - 2*w^2 + 6*w + 2],\ [541, 541, -5*w^3 + 2*w^2 + 26*w - 8],\ [541, 541, 2*w^3 + w^2 - 9*w - 1],\ [571, 571, -4*w^3 + 3*w^2 + 20*w - 14],\ [571, 571, -3*w^3 - w^2 + 15*w + 4],\ [593, 593, w^3 - 4*w - 6],\ [613, 613, 4*w^3 - 2*w^2 - 19*w + 2],\ [617, 617, -w^3 + 3*w - 5],\ [617, 617, -w^3 + 2*w^2 + 3*w - 9],\ [617, 617, 8*w^3 - 3*w^2 - 38*w + 8],\ [617, 617, w^3 - 8*w],\ [619, 619, 6*w^3 - 3*w^2 - 26*w + 4],\ [631, 631, 3*w^3 + w^2 - 17*w - 8],\ [631, 631, w^2 - 3*w - 5],\ [641, 641, -3*w^3 + 2*w^2 + 16*w - 10],\ [643, 643, -2*w^3 + 3*w^2 + 7*w - 9],\ [647, 647, w^3 - w^2 - 3*w - 4],\ [653, 653, 2*w^3 + w^2 - 11*w - 1],\ [653, 653, w^3 + 2*w^2 - 7*w - 9],\ [653, 653, -4*w^3 - w^2 + 17*w + 3],\ [653, 653, 3*w^3 - 2*w^2 - 13*w + 1],\ [661, 661, 2*w^3 - 3*w^2 - 8*w + 8],\ [673, 673, 3*w^3 - 12*w - 2],\ [673, 673, 2*w^3 - w^2 - 12*w],\ [683, 683, -w^3 + 2*w^2 + 2*w - 6],\ [691, 691, -5*w^3 + 22*w + 8],\ [691, 691, 2*w^3 - 12*w - 3],\ [719, 719, 6*w^3 + w^2 - 28*w - 8],\ [733, 733, -3*w^2 + 3*w + 7],\ [733, 733, 3*w^3 - 14*w + 2],\ [733, 733, -w^3 + w^2 + 7*w - 8],\ [733, 733, 2*w^3 - 6*w - 5],\ [751, 751, 3*w^3 + 3*w^2 - 13*w - 12],\ [751, 751, 5*w^3 - 2*w^2 - 25*w + 5],\ [751, 751, -4*w^3 + 3*w^2 + 20*w - 12],\ [751, 751, 2*w^3 + w^2 - 9*w + 1],\ [757, 757, -3*w^3 + w^2 + 11*w + 2],\ [773, 773, w^3 - w^2 - 7*w + 6],\ [787, 787, -9*w^3 + 4*w^2 + 43*w - 13],\ [787, 787, w^2 + 2*w - 6],\ [797, 797, -w^3 + 4*w^2 - 2*w - 6],\ [797, 797, 2*w^3 - w^2 - 9*w + 7],\ [821, 821, -6*w^3 + 27*w + 2],\ [823, 823, 8*w^3 - 4*w^2 - 37*w + 8],\ [827, 827, -3*w^3 + 3*w^2 + 15*w - 4],\ [829, 829, w^3 + w^2 - 3*w - 8],\ [839, 839, 2*w^3 - 3*w^2 - 8*w + 6],\ [839, 839, 8*w^3 - 2*w^2 - 38*w + 3],\ [859, 859, 4*w^3 - 2*w^2 - 18*w + 1],\ [859, 859, -4*w^3 + 4*w^2 + 17*w - 10],\ [863, 863, -4*w^3 + w^2 + 19*w + 3],\ [877, 877, -4*w^3 + w^2 + 18*w + 4],\ [883, 883, -6*w^3 + 3*w^2 + 31*w - 13],\ [883, 883, 4*w^3 - w^2 - 17*w - 1],\ [887, 887, 3*w^3 - 4*w^2 - 11*w + 11],\ [907, 907, 5*w^3 - 4*w^2 - 21*w + 11],\ [907, 907, -2*w^3 + 2*w^2 + 8*w - 11],\ [911, 911, 2*w^3 - 2*w^2 - 11*w],\ [919, 919, 4*w^3 - 17*w - 2],\ [937, 937, 3*w^3 - 2*w^2 - 15*w + 1],\ [937, 937, 3*w^3 - 17*w + 1],\ [941, 941, 4*w^3 - w^2 - 16*w - 4],\ [953, 953, -2*w^3 + 2*w^2 + 13*w - 6],\ [971, 971, -3*w^3 + 11*w + 9],\ [977, 977, 8*w^3 - 4*w^2 - 39*w + 16],\ [983, 983, 4*w^3 - w^2 - 21*w + 3],\ [983, 983, 2*w^3 - 3*w^2 - 9*w + 7],\ [997, 997, w^3 - w^2 - w - 6],\ [997, 997, w^3 - 9*w - 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 3*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1, -3, -2*e - 3, 2*e - 1, 1, 3, -4*e - 5, -3, 2*e + 6, 6*e + 14, 6*e + 13, -6*e - 2, -8, 8*e + 12, -4*e - 14, -12*e - 16, -2*e, -10*e - 16, -6*e - 16, -2*e + 14, 4, 10*e + 20, -2*e - 10, 12*e + 20, 10*e + 6, -10*e - 14, -2*e - 6, -2*e - 7, 4*e - 6, 4*e + 17, 2*e + 9, 10*e + 16, -2*e - 21, 8*e + 15, 10*e + 8, 10*e + 12, 10*e + 12, -8*e - 12, 2*e - 7, 16*e + 20, 2*e + 23, -10*e - 25, -6*e + 8, -6*e - 18, -16*e - 26, -12*e - 12, 4*e, 16*e + 25, 4*e + 6, 4*e, -14*e - 23, -8*e - 26, 8*e + 17, -16*e - 18, 14*e + 28, 20*e + 26, -8*e - 29, -2*e - 28, -16*e - 11, -8*e - 24, 18*e + 29, -14, -10*e - 19, -18*e - 38, -8*e - 31, -16*e - 17, -16*e - 29, -12*e - 28, -8*e - 36, 4*e - 18, 26*e + 41, 10*e + 18, 32*e + 50, -10*e - 12, 2*e - 24, 10*e + 12, 14*e + 2, -8*e - 27, 14*e + 8, -12*e - 9, -12*e - 28, 14*e + 39, -6*e - 34, 14*e + 46, 4*e + 32, -16*e - 48, -8*e - 12, -4*e, -26*e - 43, 20*e + 36, -4*e + 22, -10*e + 10, 4*e + 6, 10*e + 6, -16*e - 32, 32*e + 52, 31, 10*e + 18, -28*e - 30, 6*e + 2, -24*e - 50, 10*e + 9, -10*e + 8, 4*e + 23, 6*e + 14, -16*e - 56, 18*e + 32, 8*e - 18, 4*e - 30, -2*e + 15, -12*e + 2, -6*e - 32, 28*e + 40, -34*e - 40, 36*e + 48, 26*e + 30, -34*e - 59, 12*e + 24, -16*e - 20, -8*e - 53, -18*e - 51, 6*e + 3, -26*e - 42, -8*e + 20, -12*e - 40, -14*e - 24, 10*e - 10, -24, -8*e + 3, -6*e + 18, 12*e + 52, -2*e - 10, -8*e + 1, -8*e - 23, -8, 26*e + 48, 18*e + 28, 12*e - 2, -34*e - 33, 28*e + 55, 10*e + 39, -18*e - 28, -10*e - 13, -42*e - 61, 20*e + 20, -2*e - 5, 26*e + 25, 12*e + 62, -4*e - 44, -28*e - 33, 28*e + 42, -34*e - 34, -8*e - 34, 12*e + 24, -2*e - 18, -26*e - 53, 6*e + 9, -36*e - 62, 2*e + 2] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([23, 23, -w^2 + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]