/* 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([49, 7, w^3 - w^2 - 6*w - 1]) 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^4 + 6*x^3 - 6*x^2 - 80*x - 100 K. = NumberField(heckePol) hecke_eigenvalues_array = [1/5*e^3 + 1/5*e^2 - 16/5*e - 5, e, -1/5*e^3 - 1/5*e^2 + 11/5*e, 2/5*e^3 + 7/5*e^2 - 27/5*e - 15, -3/5*e^3 - 8/5*e^2 + 43/5*e + 21, 1/5*e^3 + 1/5*e^2 - 16/5*e - 3, -3/5*e^3 - 8/5*e^2 + 38/5*e + 14, 3/5*e^3 + 8/5*e^2 - 38/5*e - 22, 2/5*e^3 + 2/5*e^2 - 32/5*e - 9, 1, -e^3 - 3*e^2 + 12*e + 35, 7/5*e^3 + 17/5*e^2 - 92/5*e - 37, -3/5*e^3 - 8/5*e^2 + 53/5*e + 22, e^2 - e - 14, -4/5*e^3 - 4/5*e^2 + 64/5*e + 15, e^3 + 2*e^2 - 14*e - 26, -1/5*e^3 - 6/5*e^2 + 6/5*e + 10, 1/5*e^3 + 6/5*e^2 - 31/5*e - 22, -e^2 + 3*e + 14, 2/5*e^3 + 7/5*e^2 - 27/5*e - 12, -3/5*e^3 - 8/5*e^2 + 43/5*e + 24, -1/5*e^3 - 6/5*e^2 + 21/5*e + 21, -e^3 - 4*e^2 + 14*e + 39, 2/5*e^3 + 7/5*e^2 - 37/5*e - 15, 9/5*e^3 + 24/5*e^2 - 134/5*e - 69, 7/5*e^3 + 17/5*e^2 - 92/5*e - 39, -e^3 - 3*e^2 + 12*e + 33, -8/5*e^3 - 23/5*e^2 + 103/5*e + 44, 9/5*e^3 + 24/5*e^2 - 119/5*e - 64, -2*e^3 - 6*e^2 + 29*e + 68, 9/5*e^3 + 29/5*e^2 - 129/5*e - 76, -6/5*e^3 - 26/5*e^2 + 81/5*e + 51, 13/5*e^3 + 33/5*e^2 - 193/5*e - 93, -8/5*e^3 - 23/5*e^2 + 128/5*e + 57, 4/5*e^3 + 19/5*e^2 - 64/5*e - 51, -11/5*e^3 - 36/5*e^2 + 146/5*e + 84, 3*e^3 + 8*e^2 - 42*e - 96, -4/5*e^3 - 4/5*e^2 + 64/5*e + 7, -6/5*e^3 - 16/5*e^2 + 96/5*e + 34, 2/5*e^3 + 12/5*e^2 - 32/5*e - 38, -7/5*e^3 - 12/5*e^2 + 117/5*e + 38, -4/5*e^3 + 1/5*e^2 + 59/5*e + 2, -4*e - 4, 4/5*e^3 + 4/5*e^2 - 44/5*e - 4, 11/5*e^3 + 31/5*e^2 - 156/5*e - 66, -9/5*e^3 - 29/5*e^2 + 124/5*e + 78, -9/5*e^3 - 24/5*e^2 + 144/5*e + 68, 3/5*e^3 + 18/5*e^2 - 48/5*e - 40, 4/5*e^3 + 14/5*e^2 - 49/5*e - 30, -7/5*e^3 - 17/5*e^2 + 97/5*e + 42, 14/5*e^3 + 44/5*e^2 - 209/5*e - 111, -13/5*e^3 - 43/5*e^2 + 193/5*e + 105, -2*e^3 - 2*e^2 + 32*e + 27, -2*e^3 - 2*e^2 + 32*e + 27, -e^3 - 3*e^2 + 12*e + 34, 7/5*e^3 + 17/5*e^2 - 92/5*e - 38, -12/5*e^3 - 27/5*e^2 + 167/5*e + 51, 3*e^3 + 8*e^2 - 44*e - 115, -9/5*e^3 - 34/5*e^2 + 124/5*e + 65, e^3 + 4*e^2 - 11*e - 57, 13/5*e^3 + 43/5*e^2 - 193/5*e - 108, -14/5*e^3 - 44/5*e^2 + 209/5*e + 108, 3*e^3 + 8*e^2 - 46*e - 111, -7/5*e^3 - 32/5*e^2 + 102/5*e + 69, 9/5*e^3 + 24/5*e^2 - 99/5*e - 50, -12/5*e^3 - 27/5*e^2 + 147/5*e + 58, -12/5*e^3 - 32/5*e^2 + 162/5*e + 86, 2*e^3 + 6*e^2 - 26*e - 58, 8/5*e^3 + 13/5*e^2 - 148/5*e - 42, 8/5*e^3 + 3/5*e^2 - 108/5*e - 6, -9/5*e^3 - 24/5*e^2 + 154/5*e + 63, 1/5*e^3 + 16/5*e^2 - 26/5*e - 45, 9/5*e^3 + 24/5*e^2 - 139/5*e - 60, -4/5*e^3 - 19/5*e^2 + 59/5*e + 48, 19/5*e^3 + 44/5*e^2 - 259/5*e - 107, -2*e^3 - 7*e^2 + 23*e + 73, 16/5*e^3 + 46/5*e^2 - 236/5*e - 110, -11/5*e^3 - 31/5*e^2 + 161/5*e + 76, -12/5*e^3 - 42/5*e^2 + 172/5*e + 106, 8/5*e^3 + 28/5*e^2 - 113/5*e - 68, 3*e^3 + 9*e^2 - 40*e - 90, -17/5*e^3 - 47/5*e^2 + 232/5*e + 126, -4/5*e^3 - 14/5*e^2 + 84/5*e + 26, 2*e^2 - 4*e - 46, -9/5*e^3 - 24/5*e^2 + 134/5*e + 38, e^3 + 4*e^2 - 14*e - 70, -e^3 - 6*e^2 + 16*e + 66, 4/5*e^3 + 4/5*e^2 - 84/5*e - 18, 8/5*e^3 + 8/5*e^2 - 108/5*e - 18, 3*e^3 + 8*e^2 - 48*e - 114, 2/5*e^3 + 22/5*e^2 - 62/5*e - 71, 3/5*e^3 + 8/5*e^2 - 53/5*e - 31, -8/5*e^3 - 28/5*e^2 + 158/5*e + 73, -e^2 + e + 5, -28/5*e^3 - 78/5*e^2 + 363/5*e + 181, -9/5*e^3 - 9/5*e^2 + 134/5*e + 1, 29/5*e^3 + 79/5*e^2 - 379/5*e - 179, -7/5*e^3 - 7/5*e^2 + 122/5*e + 1, -11/5*e^3 - 21/5*e^2 + 161/5*e + 34, 2*e^2 + 3*e - 38, 3/5*e^3 + 3/5*e^2 - 48/5*e + 2, 3/5*e^3 + 13/5*e^2 - 33/5*e - 16, -8/5*e^3 - 18/5*e^2 + 113/5*e + 56, -12/5*e^3 - 37/5*e^2 + 152/5*e + 74, 16/5*e^3 + 41/5*e^2 - 216/5*e - 106, e^3 - 2*e^2 - 14*e + 20, -2/5*e^3 - 12/5*e^2 - 3/5*e + 14, 3*e^3 + 6*e^2 - 50*e - 88, 13/5*e^3 + 23/5*e^2 - 173/5*e - 58, 8/5*e^3 + 8/5*e^2 - 143/5*e - 20, 11/5*e^3 + 11/5*e^2 - 161/5*e - 20, 17/5*e^3 + 37/5*e^2 - 217/5*e - 67, -2*e^3 - 6*e^2 + 21*e + 77, 2/5*e^3 + 32/5*e^2 - 17/5*e - 61, -5*e^3 - 11*e^2 + 77*e + 155, -16/5*e^3 - 26/5*e^2 + 266/5*e + 76, -3/5*e^3 - 3/5*e^2 + 13/5*e - 6, 4/5*e^3 + 4/5*e^2 - 29/5*e - 6, -2*e^3 + 30*e + 4, 17/5*e^3 + 52/5*e^2 - 232/5*e - 130, -19/5*e^3 - 54/5*e^2 + 264/5*e + 122, e^3 + 3*e^2 - 17*e - 40, -2/5*e^3 - 12/5*e^2 + 37/5*e + 32, 13/5*e^3 + 28/5*e^2 - 228/5*e - 87, e^3 - 2*e^2 - 12*e + 21, 28/5*e^3 + 73/5*e^2 - 383/5*e - 166, -21/5*e^3 - 66/5*e^2 + 271/5*e + 158, -3*e^3 - 11*e^2 + 45*e + 143, 7/5*e^3 + 37/5*e^2 - 117/5*e - 97, -16/5*e^3 - 46/5*e^2 + 261/5*e + 119, 4*e^3 + 12*e^2 - 61*e - 145, -4/5*e^3 - 24/5*e^2 + 9/5*e + 58, 23/5*e^3 + 43/5*e^2 - 313/5*e - 86, -19/5*e^3 - 64/5*e^2 + 284/5*e + 150, -5*e^3 - 16*e^2 + 69*e + 192, 21/5*e^3 + 66/5*e^2 - 316/5*e - 174, 6*e^3 + 17*e^2 - 85*e - 204, 26/5*e^3 + 81/5*e^2 - 341/5*e - 179, 3/5*e^3 + 3/5*e^2 - 48/5*e - 17, -33/5*e^3 - 88/5*e^2 + 453/5*e + 217, 3/5*e^3 + 3/5*e^2 - 48/5*e - 17, 6/5*e^3 + 6/5*e^2 - 96/5*e - 18, 24/5*e^3 + 79/5*e^2 - 344/5*e - 185, -28/5*e^3 - 83/5*e^2 + 408/5*e + 211, 22/5*e^3 + 72/5*e^2 - 287/5*e - 170, -31/5*e^3 - 81/5*e^2 + 431/5*e + 190, -5*e^3 - 9*e^2 + 68*e + 87, 3/5*e^3 + 23/5*e^2 + 12/5*e - 57, 12/5*e^3 + 32/5*e^2 - 142/5*e - 94, -14/5*e^3 - 34/5*e^2 + 174/5*e + 50, 14/5*e^3 + 49/5*e^2 - 184/5*e - 122, -22/5*e^3 - 57/5*e^2 + 312/5*e + 130, 6/5*e^3 + 11/5*e^2 - 116/5*e - 22, 6/5*e^3 + 1/5*e^2 - 76/5*e + 14, -8/5*e^3 - 18/5*e^2 + 148/5*e + 66, -4/5*e^3 + 6/5*e^2 + 44/5*e - 6, -3/5*e^3 - 13/5*e^2 + 68/5*e + 27, 19/5*e^3 + 69/5*e^2 - 289/5*e - 153, -24/5*e^3 - 74/5*e^2 + 369/5*e + 207, 1/5*e^3 + 11/5*e^2 - 36/5*e - 45, -e^3 - 6*e^2 + 17*e + 63, 14/5*e^3 + 39/5*e^2 - 229/5*e - 117, 6/5*e^3 + 11/5*e^2 - 46/5*e - 1, -8/5*e^3 - 13/5*e^2 + 78/5*e + 35, -4/5*e^3 - 9/5*e^2 + 4/5*e + 7, 12/5*e^3 + 17/5*e^2 - 132/5*e - 29] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([49, 7, w^3 - w^2 - 6*w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]