/* 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, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [5, 5, -w^3 + w^2 + 5*w],\ [7, 7, -w^3 + 2*w^2 + 3*w - 3],\ [7, 7, w^3 - 2*w^2 - 3*w],\ [13, 13, -w^2 + w + 3],\ [13, 13, -w^2 + w + 2],\ [16, 2, 2],\ [23, 23, -w^2 + 3*w + 1],\ [23, 23, -2*w^3 + 3*w^2 + 9*w - 2],\ [25, 5, w^3 - 2*w^2 - 2*w + 2],\ [49, 7, w^3 - w^2 - 6*w - 1],\ [53, 53, 2*w^3 - 2*w^2 - 8*w - 3],\ [53, 53, 2*w^3 - 2*w^2 - 8*w - 1],\ [67, 67, 2*w^3 - 4*w^2 - 7*w + 2],\ [67, 67, -2*w^3 + 4*w^2 + 6*w - 1],\ [81, 3, -3],\ [83, 83, 2*w^3 - 3*w^2 - 6*w - 1],\ [83, 83, 3*w^3 - 4*w^2 - 12*w + 1],\ [103, 103, 3*w^3 - 4*w^2 - 12*w - 1],\ [103, 103, 2*w^3 - 3*w^2 - 6*w + 1],\ [107, 107, w^3 - w^2 - 3*w - 3],\ [107, 107, 2*w^3 - 2*w^2 - 9*w],\ [109, 109, -3*w^3 + 5*w^2 + 10*w - 5],\ [109, 109, -2*w^3 + 3*w^2 + 9*w + 1],\ [109, 109, -2*w^3 + 2*w^2 + 7*w],\ [109, 109, w^2 - 3*w - 4],\ [121, 11, 3*w^3 - 4*w^2 - 12*w],\ [121, 11, 2*w^2 - 3*w - 6],\ [139, 139, w^3 - 7*w - 1],\ [139, 139, -w^3 + 2*w^2 + 5*w - 4],\ [139, 139, 3*w^3 - 3*w^2 - 14*w - 2],\ [139, 139, -3*w^3 + 5*w^2 + 13*w - 2],\ [149, 149, -w^2 + 4*w + 1],\ [149, 149, -3*w^3 + 4*w^2 + 14*w - 1],\ [149, 149, 2*w^2 - 3*w - 5],\ [149, 149, -w^3 + 3*w^2 + 3*w - 4],\ [167, 167, -w^3 - w^2 + 7*w + 4],\ [167, 167, -4*w^3 + 6*w^2 + 15*w - 4],\ [169, 13, w^3 - w^2 - 6*w + 4],\ [173, 173, w^2 - 3*w - 5],\ [173, 173, 2*w^3 - 2*w^2 - 11*w],\ [179, 179, 2*w^3 - 2*w^2 - 9*w + 1],\ [179, 179, w^3 - w^2 - 3*w - 4],\ [179, 179, w^2 - 5],\ [179, 179, w^3 - 6*w - 1],\ [197, 197, w^3 + w^2 - 7*w - 6],\ [197, 197, 2*w^3 - 2*w^2 - 9*w + 2],\ [223, 223, -w^2 + 7],\ [223, 223, 2*w^2 - w - 4],\ [227, 227, 2*w^3 - 2*w^2 - 11*w - 2],\ [227, 227, w^3 - w^2 - 7*w - 1],\ [233, 233, w^3 - w^2 - 7*w],\ [233, 233, -2*w^3 + 2*w^2 + 11*w + 1],\ [257, 257, -2*w^3 + 4*w^2 + 8*w - 1],\ [257, 257, 2*w^2 - 4*w - 7],\ [277, 277, -4*w^3 + 6*w^2 + 15*w - 3],\ [277, 277, -3*w^3 + 5*w^2 + 9*w],\ [281, 281, -2*w^3 + 5*w^2 + 3*w - 5],\ [281, 281, -2*w^3 + 5*w^2 + 5*w - 7],\ [281, 281, 2*w^3 - 5*w^2 - 5*w + 4],\ [281, 281, -4*w^3 + 7*w^2 + 15*w - 4],\ [283, 283, 3*w^3 - 4*w^2 - 13*w - 3],\ [283, 283, -w^3 + 2*w^2 + w - 4],\ [313, 313, -2*w^3 + 4*w^2 + 5*w - 5],\ [313, 313, 3*w^3 - 5*w^2 - 11*w],\ [347, 347, -w^3 + 2*w^2 + w - 5],\ [347, 347, 3*w^3 - 4*w^2 - 13*w - 4],\ [353, 353, 3*w^3 - 5*w^2 - 12*w + 1],\ [353, 353, -w^3 + 3*w^2 - 5],\ [361, 19, 3*w^3 - 3*w^2 - 13*w - 9],\ [361, 19, 2*w^3 - 2*w^2 - 7*w + 4],\ [373, 373, -3*w^3 + 4*w^2 + 11*w + 1],\ [373, 373, -3*w^3 + 4*w^2 + 11*w],\ [383, 383, 3*w^3 - 5*w^2 - 13*w + 1],\ [383, 383, 2*w^2 - 5*w - 6],\ [397, 397, 4*w^3 - 5*w^2 - 19*w - 1],\ [397, 397, 2*w^3 - 2*w^2 - 9*w - 7],\ [431, 431, 2*w^3 - 2*w^2 - 7*w - 5],\ [431, 431, 3*w^3 - 6*w^2 - 11*w + 3],\ [431, 431, 3*w^3 - 3*w^2 - 13*w],\ [431, 431, w^3 - 4*w^2 + w + 8],\ [457, 457, 4*w^3 - 7*w^2 - 13*w],\ [457, 457, -3*w^2 + 4*w + 5],\ [463, 463, -3*w^3 + 5*w^2 + 11*w + 1],\ [463, 463, 2*w^3 - 4*w^2 - 5*w + 6],\ [487, 487, -w^3 + 4*w^2 - 7],\ [487, 487, -2*w^3 + 5*w^2 + 6*w - 5],\ [499, 499, -6*w^3 + 8*w^2 + 25*w - 1],\ [499, 499, -w^3 + 4*w^2 - 5],\ [499, 499, -2*w^3 + 5*w^2 + 6*w - 7],\ [499, 499, 3*w^3 - 5*w^2 - 7*w],\ [509, 509, -4*w^3 + 8*w^2 + 15*w - 8],\ [509, 509, 3*w^3 - 4*w^2 - 10*w - 2],\ [509, 509, -3*w^3 + 3*w^2 + 11*w + 6],\ [509, 509, 4*w^3 - 5*w^2 - 16*w],\ [521, 521, -4*w^3 + 5*w^2 + 15*w - 2],\ [521, 521, w^3 - w^2 - 8*w + 1],\ [521, 521, -4*w^3 + 7*w^2 + 12*w - 6],\ [521, 521, 3*w^3 - 3*w^2 - 16*w - 1],\ [523, 523, 3*w^3 - w^2 - 17*w - 8],\ [523, 523, -5*w^3 + 5*w^2 + 23*w + 4],\ [529, 23, 3*w^3 - 6*w^2 - 10*w + 2],\ [547, 547, -2*w^3 + 5*w^2 + 6*w - 6],\ [547, 547, -w^3 + 4*w^2 - 6],\ [557, 557, -3*w^3 + 6*w^2 + 9*w - 10],\ [557, 557, 3*w^3 - 6*w^2 - 9*w - 1],\ [571, 571, 4*w^3 - 4*w^2 - 17*w - 6],\ [571, 571, -2*w^3 + 6*w^2 + 3*w - 10],\ [571, 571, 4*w^3 - 5*w^2 - 18*w + 3],\ [571, 571, -3*w^3 + 7*w^2 + 9*w - 5],\ [587, 587, 3*w^3 - 3*w^2 - 14*w],\ [587, 587, w^3 - w^2 - 2*w - 4],\ [593, 593, -w^3 + 3*w^2 + 2*w - 10],\ [593, 593, -5*w^3 + 7*w^2 + 19*w - 4],\ [613, 613, 3*w^3 - 5*w^2 - 12*w - 4],\ [613, 613, -w^3 + 3*w^2 - 10],\ [631, 631, 4*w^3 - 8*w^2 - 13*w + 4],\ [631, 631, 3*w^3 - 4*w^2 - 15*w + 2],\ [631, 631, w^3 - 9*w - 1],\ [631, 631, 3*w^3 - 7*w^2 - 7*w + 9],\ [643, 643, 3*w^3 - 3*w^2 - 16*w - 2],\ [643, 643, w^3 - w^2 - 8*w],\ [647, 647, -w^3 + 4*w^2 + 2*w - 4],\ [647, 647, -3*w^2 + 4*w + 10],\ [673, 673, w^3 + 2*w^2 - 9*w - 8],\ [673, 673, -w^3 + 4*w^2 + 3*w - 7],\ [683, 683, 3*w^3 - 3*w^2 - 11*w - 2],\ [683, 683, 4*w^3 - 4*w^2 - 17*w - 5],\ [709, 709, -5*w^3 + 7*w^2 + 21*w + 1],\ [709, 709, -3*w^3 + 2*w^2 + 16*w + 4],\ [709, 709, 4*w^3 - 4*w^2 - 19*w - 2],\ [709, 709, -2*w^3 + 4*w^2 + 3*w - 4],\ [787, 787, 2*w^2 - w - 9],\ [787, 787, w^3 + w^2 - 7*w - 2],\ [811, 811, 4*w^3 - 7*w^2 - 11*w],\ [811, 811, 3*w^3 - 4*w^2 - 9*w - 1],\ [811, 811, -6*w^3 + 9*w^2 + 23*w - 5],\ [811, 811, 5*w^3 - 6*w^2 - 21*w - 2],\ [821, 821, -3*w^3 + 6*w^2 + 11*w - 2],\ [821, 821, 2*w^3 - 13*w - 5],\ [821, 821, -w^3 + 4*w^2 - w - 9],\ [821, 821, -w^3 + 3*w^2 + 5*w - 6],\ [841, 29, 2*w^3 - 2*w^2 - 12*w - 1],\ [857, 857, w^3 - 4*w^2 + 3*w + 6],\ [857, 857, -5*w^3 + 8*w^2 + 21*w - 3],\ [863, 863, -w^3 + 4*w^2 + w - 7],\ [863, 863, -w^3 + 4*w^2 + w - 6],\ [877, 877, -w^3 - w^2 + 8*w + 2],\ [877, 877, -w^3 + 3*w^2 + 4*w - 8],\ [883, 883, 4*w^3 - 7*w^2 - 14*w + 1],\ [883, 883, 3*w^3 - 6*w^2 - 8*w + 7],\ [929, 929, 3*w^3 - 5*w^2 - 12*w - 3],\ [929, 929, -w^3 + 3*w^2 - 9],\ [929, 929, 3*w - 5],\ [929, 929, 3*w^3 - 3*w^2 - 15*w + 2],\ [937, 937, w^3 - 2*w^2 - 6*w + 6],\ [937, 937, 2*w^3 - w^2 - 12*w],\ [941, 941, -w^3 + 4*w^2 + 2*w - 7],\ [941, 941, -4*w^3 + 6*w^2 + 13*w - 3],\ [941, 941, -5*w^3 + 7*w^2 + 19*w + 2],\ [941, 941, 3*w^2 - 4*w - 7],\ [953, 953, 3*w^3 - 5*w^2 - 12*w - 2],\ [953, 953, -w^3 + 3*w^2 - 8],\ [961, 31, 4*w^3 - 6*w^2 - 13*w - 2],\ [961, 31, 5*w^3 - 7*w^2 - 19*w + 3],\ [977, 977, -4*w^3 + 7*w^2 + 19*w - 6],\ [977, 977, -3*w^3 + 7*w^2 + 10*w - 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - 7*x - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -e^2 + e + 4, -e^2 + e + 4, -e^2 + 2*e + 4, -e^2 + 2*e + 4, -1, e^2 - e - 6, e^2 - e - 6, 9, -e^2 - 2*e + 13, e^2 + 4*e - 6, e^2 + 4*e - 6, -3*e^2 + 2*e + 11, -3*e^2 + 2*e + 11, e^2 - 4*e + 2, 2*e^2 - 5*e - 9, 2*e^2 - 5*e - 9, 2*e^2 - 2, 2*e^2 - 2, 4*e^2 - 5*e - 23, 4*e^2 - 5*e - 23, 2*e^2 - 3*e - 12, 2*e^2 + 3*e - 14, 2*e^2 - 3*e - 12, 2*e^2 + 3*e - 14, 3*e^2 - 3*e - 19, 3*e^2 - 3*e - 19, -2*e^2 - 3*e + 9, -2*e^2 - 3*e + 9, 2*e^2 - e - 11, 2*e^2 - e - 11, e^2 + 4*e - 12, e^2 + 4*e - 12, e^2 - 2*e - 10, e^2 - 2*e - 10, -4*e^2 - 2*e + 18, -4*e^2 - 2*e + 18, 2*e^2 - 8*e - 7, 6*e + 2, 6*e + 2, 5*e^2 - 4*e - 27, 5*e^2 - 4*e - 27, -3*e^2 - 2*e + 11, -3*e^2 - 2*e + 11, -4*e^2 - 4*e + 26, -4*e^2 - 4*e + 26, 4*e - 4, 4*e - 4, 6*e^2 - 6*e - 20, 6*e^2 - 6*e - 20, -2*e^2 + 2*e + 5, -2*e^2 + 2*e + 5, -3*e^2 + 7*e + 35, -3*e^2 + 7*e + 35, -2*e^2 + 6*e + 8, -2*e^2 + 6*e + 8, -4*e^2 + 4*e + 11, -7, -7, -4*e^2 + 4*e + 11, -5*e^2 + 9*e + 36, -5*e^2 + 9*e + 36, -4*e^2 + 11*e + 18, -4*e^2 + 11*e + 18, e^2 + 6*e - 13, e^2 + 6*e - 13, e^2 - 5, e^2 - 5, -9*e^2 + 8*e + 43, -9*e^2 + 8*e + 43, 5*e + 4, 5*e + 4, -3*e^2 - 3*e + 34, -3*e^2 - 3*e + 34, 2*e^2 - 7*e - 6, 2*e^2 - 7*e - 6, -11*e^2 + 7*e + 38, 7*e^2 - e - 18, -11*e^2 + 7*e + 38, 7*e^2 - e - 18, 7*e^2 - 8*e - 35, 7*e^2 - 8*e - 35, -e^2 - 5*e + 20, -e^2 - 5*e + 20, -7*e^2 + 5*e + 30, -7*e^2 + 5*e + 30, -9*e - 7, -6*e^2 + 13*e + 43, -6*e^2 + 13*e + 43, -9*e - 7, 3*e^2 - 14*e - 24, -e^2 - 4*e - 8, 3*e^2 - 14*e - 24, -e^2 - 4*e - 8, -7*e^2 + 9*e + 23, 8*e^2 - 7*e - 30, -7*e^2 + 9*e + 23, 8*e^2 - 7*e - 30, 9*e^2 - 12*e - 51, 9*e^2 - 12*e - 51, -8*e^2 + 14*e + 54, e^2 - 3*e + 20, e^2 - 3*e + 20, 10*e^2 - 8*e - 26, 10*e^2 - 8*e - 26, -e^2 + 10*e + 21, -8*e + 8, -e^2 + 10*e + 21, -8*e + 8, 2*e^2 - 5*e - 15, 2*e^2 - 5*e - 15, 5*e^2 - 13*e - 35, 5*e^2 - 13*e - 35, -10*e^2 + 3*e + 38, -10*e^2 + 3*e + 38, -4*e^2 + 6*e + 26, 4*e^2 - 14*e - 6, 4*e^2 - 14*e - 6, -4*e^2 + 6*e + 26, -5*e + 11, -5*e + 11, -5*e^2 - 5*e + 42, -5*e^2 - 5*e + 42, -9*e^2 + 20*e + 62, -9*e^2 + 20*e + 62, -e^2 - 12*e + 27, -e^2 - 12*e + 27, 3*e^2 + 4*e, -7*e^2 + 10*e + 18, -7*e^2 + 10*e + 18, 3*e^2 + 4*e, 7*e^2 - 8*e - 45, 7*e^2 - 8*e - 45, 3*e^2 + e - 16, -8*e^2 + 7*e + 21, 3*e^2 + e - 16, -8*e^2 + 7*e + 21, 5*e^2 - 12*e - 34, -3*e^2 + 14*e - 4, 5*e^2 - 12*e - 34, -3*e^2 + 14*e - 4, 11*e^2 - 4*e - 23, -7*e^2 + e + 33, -7*e^2 + e + 33, 4*e^2 - 10*e - 2, 4*e^2 - 10*e - 2, 11*e^2 - 18*e - 56, 11*e^2 - 18*e - 56, -10*e^2 + 5*e + 41, -10*e^2 + 5*e + 41, -10*e^2 + 4*e + 42, -10*e^2 + 4*e + 42, 3*e^2 + 23, 3*e^2 + 23, 3*e^2 + 4*e - 27, 3*e^2 + 4*e - 27, 4*e^2 - e - 12, -8*e^2 - e + 46, -8*e^2 - e + 46, 4*e^2 - e - 12, -2*e^2 - 12*e - 7, -2*e^2 - 12*e - 7, -5*e^2 - 7*e + 11, -5*e^2 - 7*e + 11, 13*e - 18, 13*e - 18] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([16, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]