/* 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([16, 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^3 - x^2 - 14*x + 16 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, e, 1, -1/2*e^2 - 1/2*e + 5, -1/2*e^2 + 1/2*e + 5, -e^2 - e + 10, e^2 + e - 12, e^2 - 4, 1/2*e^2 - 1/2*e + 1, -e - 2, -e - 2, -2, -1/2*e^2 - 3/2*e + 3, -1/2*e^2 - 3/2*e + 3, e^2 - e - 14, 4*e + 2, -1/2*e^2 + 5/2*e + 3, 1/2*e^2 - 3/2*e - 3, -e - 4, -2*e + 2, -1/2*e^2 - 5/2*e + 1, 1/2*e^2 + 5/2*e - 9, 2*e^2 - 18, -e + 4, -1/2*e^2 - 3/2*e + 1, 1/2*e^2 + 7/2*e - 9, -1/2*e^2 - 3/2*e + 9, 2*e^2 + 4*e - 26, 2*e^2 + 3*e - 18, e^2 - 2*e - 8, 3/2*e^2 + 5/2*e - 15, 1/2*e^2 - 5/2*e - 5, 1/2*e^2 - 5/2*e - 3, e^2 + 3*e - 8, -1/2*e^2 - 7/2*e + 7, -2*e^2 + 16, 2*e^2 + 2*e - 28, -2*e + 14, 3/2*e^2 - 3/2*e - 15, e, -2*e^2 + e + 16, e + 8, 2*e + 2, -4*e - 8, -e^2 + 4, -3/2*e^2 - 9/2*e + 23, e^2 - 4*e - 8, 4, e^2 + 5*e - 4, e^2 - 4*e - 2, -1/2*e^2 - 7/2*e + 21, -e^2 - 4*e + 14, -3/2*e^2 - 11/2*e + 13, 16, -3*e^2 - 2*e + 24, -7/2*e^2 - 13/2*e + 43, -3*e^2 + 2*e + 24, 3*e^2 + 5*e - 36, -2*e^2 - 2*e + 20, -e^2 + 3*e + 16, 1/2*e^2 + 7/2*e - 7, 3/2*e^2 - 1/2*e - 19, e^2 + 8*e - 14, 14, 1/2*e^2 - 1/2*e - 9, 3*e^2 - 18, -e^2 - 7*e + 4, 5/2*e^2 - 11/2*e - 33, -3*e^2 - 4*e + 34, 1/2*e^2 - 5/2*e - 13, -1/2*e^2 + 5/2*e - 7, -7/2*e^2 - 7/2*e + 33, -7/2*e^2 - 9/2*e + 39, 2*e + 26, -3/2*e^2 + 1/2*e + 11, e^2 + 6*e - 24, -e^2 - 4*e + 2, 2*e^2 + 2*e - 22, 2*e - 6, 2*e^2 - 6*e - 26, -1/2*e^2 + 9/2*e - 9, -8, 9/2*e^2 + 5/2*e - 35, -1/2*e^2 + 9/2*e + 21, e^2 - 16, 4*e - 4, e^2 - 5*e - 30, -e^2 - 6*e + 4, -3/2*e^2 - 21/2*e + 21, -4*e^2 + e + 32, e^2 - e - 24, 5/2*e^2 - 9/2*e - 15, -4*e^2 - 7*e + 56, 2*e^2 - 5*e - 4, -1/2*e^2 + 13/2*e + 19, 5/2*e^2 - 3/2*e - 41, 5/2*e^2 + 15/2*e - 17, e^2 + 2*e + 6, -2*e + 14, 3/2*e^2 + 15/2*e - 13, 9/2*e^2 - 9/2*e - 49, -e^2 + 2*e + 26, -3/2*e^2 + 9/2*e + 3, 3*e^2 + 7*e - 22, -4*e - 16, 1/2*e^2 + 11/2*e + 3, 3/2*e^2 - 7/2*e - 33, e^2 + 10*e - 8, -6*e^2 - 2*e + 42, -5*e^2 - e + 56, -3*e^2 + 3*e + 16, -2*e^2 + 4*e + 30, 3*e^2 + 7*e - 48, -3/2*e^2 + 1/2*e + 25, -8*e - 6, 5/2*e^2 + 3/2*e - 3, 3/2*e^2 + 19/2*e - 7, -3/2*e^2 - 3/2*e - 1, 4*e^2 + e - 46, 4*e^2 - 2*e - 14, 2*e^2 - 3*e + 8, -2*e^2 - e + 28, -3*e^2 - 2*e + 14, -5*e^2 - 4*e + 60, -6*e^2 + 4*e + 48, -e^2 + 7*e + 18, -e^2 - 11*e + 14, -3*e^2 + 2*e + 14, -e^2 + 11*e + 16, 7/2*e^2 + 25/2*e - 57, -9/2*e^2 - 9/2*e + 53, -3*e^2 - 3*e + 28, -4*e^2 - 4*e + 14, 5*e^2 - 5*e - 56, e^2 - 3*e - 34, -7/2*e^2 - 1/2*e + 33, 2*e^2 + 4*e + 6, 4*e^2 + 3*e - 50, -4*e - 32, -4*e^2 - 10*e + 56, -1/2*e^2 - 7/2*e + 31, 5*e^2 + 3*e - 62, 2*e^2 + 10*e - 38, -2*e^2 + 4*e + 36, -12, -3*e^2 - 10*e + 48, -7/2*e^2 + 23/2*e + 41, 3/2*e^2 + 9/2*e - 41, -3*e^2 + 3*e + 50, 24, 6*e^2 + 8*e - 54, 2*e^2 - 4*e - 6, -4*e^2 - 12*e + 38, -3*e^2 - 10*e + 58, 7/2*e^2 + 9/2*e - 49, 1/2*e^2 + 7/2*e + 15, -e^2 - 7*e - 2, e^2 + 3*e - 22, -4*e^2 - 3*e + 40, 2*e^2 - 26] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, w + 1])] = -1 AL_eigenvalues[ZF.ideal([8, 2, w^3 - w^2 - 4*w + 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]