/* 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([8, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([20, 20, w^2 - 4]) primes_array = [ [2, 2, -w + 2],\ [4, 2, -w^2 - w + 5],\ [5, 5, -w + 3],\ [7, 7, -w^2 - 2*w + 3],\ [11, 11, -w^2 + 5],\ [13, 13, w + 1],\ [23, 23, w^2 - 3],\ [25, 5, w^2 + 2*w - 1],\ [27, 3, -3],\ [29, 29, -3*w + 7],\ [43, 43, -3*w^2 - 2*w + 17],\ [49, 7, -2*w^2 + w + 11],\ [67, 67, -2*w^2 - 4*w + 3],\ [71, 71, -2*w^2 + w + 9],\ [73, 73, w^2 + 2*w - 7],\ [73, 73, -2*w^2 - 2*w + 11],\ [73, 73, w - 5],\ [89, 89, 2*w^2 + w - 9],\ [89, 89, -w^2 - 2*w + 9],\ [89, 89, -2*w - 1],\ [101, 101, 4*w^2 + 4*w - 19],\ [103, 103, 2*w + 3],\ [107, 107, w + 5],\ [109, 109, 3*w^2 + 2*w - 15],\ [113, 113, 2*w - 7],\ [121, 11, 2*w^2 + w - 7],\ [127, 127, 2*w^2 - 9],\ [127, 127, -4*w + 11],\ [127, 127, -2*w^2 + 2*w + 9],\ [131, 131, 3*w^2 + 4*w - 13],\ [137, 137, -w^2 + 2*w - 3],\ [149, 149, 2*w^2 - 6*w + 1],\ [151, 151, 3*w - 1],\ [157, 157, -3*w^2 - 6*w + 7],\ [157, 157, -2*w^2 + 3*w + 3],\ [157, 157, 2*w^2 + 2*w - 15],\ [163, 163, 3*w^2 + 2*w - 19],\ [163, 163, 4*w^2 + 5*w - 17],\ [163, 163, -5*w^2 - 6*w + 19],\ [167, 167, w^2 - 11],\ [167, 167, -3*w - 7],\ [167, 167, -w^2 - 2*w + 13],\ [169, 13, w^2 - 2*w - 5],\ [181, 181, -2*w - 7],\ [181, 181, 2*w^2 - 7],\ [181, 181, 2*w^2 - 2*w - 5],\ [193, 193, 2*w^2 + 2*w - 3],\ [197, 197, -4*w^2 - 7*w + 9],\ [197, 197, 2*w^2 - 2*w - 19],\ [197, 197, 2*w^2 + 3*w - 11],\ [199, 199, -6*w^2 + 37],\ [227, 227, 2*w^2 - 3*w - 7],\ [233, 233, w^2 + 4*w - 15],\ [251, 251, -4*w^2 + 23],\ [257, 257, w^2 - 2*w - 7],\ [269, 269, -4*w^2 - w + 23],\ [271, 271, 2*w^2 - w - 19],\ [277, 277, -3*w^2 - 2*w + 9],\ [281, 281, w^2 - 2*w - 9],\ [283, 283, 6*w^2 + 5*w - 31],\ [293, 293, 2*w^2 - 2*w - 21],\ [311, 311, -w^2 - 4*w - 1],\ [317, 317, 3*w^2 + 2*w - 11],\ [317, 317, 2*w^2 + 3*w - 13],\ [317, 317, 2*w^2 - 2*w - 11],\ [331, 331, 4*w^2 - w - 21],\ [331, 331, 3*w + 5],\ [331, 331, 4*w^2 + 2*w - 21],\ [337, 337, -2*w^2 + 4*w + 5],\ [349, 349, w^2 - 4*w - 1],\ [353, 353, -w^2 - 4*w - 5],\ [367, 367, 6*w^2 + 7*w - 23],\ [379, 379, 2*w - 9],\ [389, 389, 2*w^2 + 3*w - 15],\ [389, 389, w^2 + 4*w - 13],\ [389, 389, 2*w^2 + 4*w - 1],\ [397, 397, 4*w - 1],\ [409, 409, 2*w^2 + w - 19],\ [409, 409, -w^2 + 6*w - 11],\ [409, 409, 2*w^2 + 4*w - 21],\ [419, 419, 4*w^2 + 4*w - 21],\ [421, 421, -w^2 - 3],\ [421, 421, -4*w^2 - 6*w + 17],\ [421, 421, 4*w^2 + 3*w - 31],\ [439, 439, 5*w^2 - 29],\ [443, 443, w^2 - 13],\ [449, 449, 4*w^2 - w - 27],\ [457, 457, 2*w^2 - 19],\ [467, 467, 4*w + 9],\ [467, 467, 3*w^2 - 2*w - 17],\ [467, 467, 2*w^2 - 2*w - 13],\ [479, 479, -w^2 + 4*w + 19],\ [487, 487, 4*w^2 + 5*w - 19],\ [491, 491, 3*w - 11],\ [503, 503, 4*w^2 + 3*w - 17],\ [523, 523, -2*w^2 - 5*w - 1],\ [529, 23, -4*w^2 - w + 21],\ [547, 547, 5*w^2 + 4*w - 23],\ [557, 557, 4*w^2 - 2*w - 33],\ [563, 563, 2*w^2 + 2*w - 1],\ [563, 563, 3*w^2 - 8*w - 1],\ [563, 563, 3*w^2 + 4*w - 5],\ [571, 571, 5*w^2 + 10*w - 11],\ [577, 577, -4*w + 13],\ [587, 587, 3*w^2 + 4*w - 17],\ [593, 593, 3*w^2 - 11],\ [593, 593, 4*w^2 + 2*w - 19],\ [593, 593, w - 9],\ [601, 601, -4*w^2 - 10*w + 3],\ [607, 607, 3*w^2 + 4*w - 27],\ [607, 607, -4*w^2 - 2*w + 33],\ [607, 607, 2*w^2 - 7*w + 9],\ [617, 617, -2*w^2 + 9*w - 5],\ [619, 619, -4*w - 1],\ [631, 631, -8*w^2 - 10*w + 29],\ [641, 641, w^2 - 4*w - 3],\ [653, 653, 3*w^2 - 2*w - 9],\ [659, 659, 2*w^2 + 4*w - 19],\ [661, 661, 6*w^2 + 6*w - 25],\ [673, 673, 6*w^2 + 7*w - 27],\ [683, 683, 5*w^2 + 2*w - 27],\ [701, 701, 7*w - 13],\ [709, 709, 6*w^2 + 8*w - 25],\ [719, 719, -2*w^2 + 4*w + 7],\ [719, 719, -7*w^2 - 10*w + 27],\ [719, 719, 4*w^2 - w - 19],\ [727, 727, 3*w^2 + 8*w + 1],\ [733, 733, 3*w^2 - 2*w - 7],\ [733, 733, 3*w^2 - 2*w - 29],\ [739, 739, -w - 9],\ [751, 751, -w^2 - 4*w + 19],\ [757, 757, -4*w - 7],\ [761, 761, -5*w^2 - 4*w + 29],\ [787, 787, -2*w^2 - 6*w + 3],\ [809, 809, -w^2 + 4*w - 9],\ [809, 809, -4*w^2 + 3*w + 15],\ [809, 809, 2*w^2 - 9*w + 13],\ [821, 821, -4*w^2 - 9*w + 7],\ [821, 821, 5*w - 1],\ [821, 821, 8*w^2 + 11*w - 31],\ [827, 827, 8*w^2 + 8*w - 35],\ [829, 829, 2*w^2 - 5*w - 27],\ [839, 839, -w^2 - 6*w - 7],\ [841, 29, 3*w^2 + 4*w - 31],\ [863, 863, 4*w^2 - 19],\ [883, 883, -5*w^2 + 27],\ [887, 887, 5*w^2 + 4*w - 21],\ [907, 907, -6*w^2 + 2*w + 31],\ [907, 907, 2*w^2 - 5*w - 29],\ [907, 907, 2*w^2 + 5*w - 15],\ [911, 911, w^2 - 15],\ [919, 919, 8*w - 17],\ [941, 941, -4*w^2 + 5*w + 15],\ [947, 947, 4*w^2 + 2*w - 13],\ [947, 947, 3*w^2 - 6*w - 7],\ [947, 947, -5*w^2 - 4*w + 15]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 2*x^3 - 12*x^2 - 20*x + 17 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -1, 1/2*e^3 - 1/2*e^2 - 9/2*e + 9/2, -1/2*e^3 + 1/2*e^2 + 11/2*e - 3/2, e^2 - 5, -1/2*e^3 - 1/2*e^2 + 7/2*e + 7/2, -1/2*e^3 - 1/2*e^2 + 9/2*e + 13/2, -1/2*e^3 - 1/2*e^2 + 7/2*e + 3/2, 1/2*e^3 - 1/2*e^2 - 11/2*e + 15/2, e^3 - 11*e - 2, -1/2*e^3 - 3/2*e^2 + 11/2*e + 17/2, -e^3 + 9*e + 4, e^2 + 2*e - 5, e^3 - e^2 - 7*e + 9, 1/2*e^3 + 3/2*e^2 - 7/2*e - 17/2, 3/2*e^3 - 1/2*e^2 - 33/2*e - 1/2, -e^3 - e^2 + 7*e + 7, -1/2*e^3 - 1/2*e^2 + 13/2*e + 25/2, -e^3 - 2*e^2 + 11*e + 12, 2*e^2 - 2*e - 10, e^2 - 17, -e^3 + 2*e^2 + 7*e - 20, -e^3 + 2*e^2 + 11*e - 14, 1/2*e^3 - 5/2*e^2 - 13/2*e + 29/2, -2*e, 1/2*e^3 - 3/2*e^2 - 3/2*e + 29/2, 4*e + 6, e^3 + e^2 - 9*e - 7, -1/2*e^3 + 1/2*e^2 + 17/2*e - 9/2, 1/2*e^3 - 1/2*e^2 - 17/2*e + 21/2, -3*e^2 + 21, 3/2*e^3 + 1/2*e^2 - 23/2*e - 5/2, -2*e^3 + 18*e, -e^3 - e^2 + 5*e + 7, e^3 - 11*e + 10, -3*e^2 + 17, -1/2*e^3 - 1/2*e^2 + 9/2*e + 1/2, e^3 + 2*e^2 - 7*e - 12, -e^2 - 2*e + 11, -e^3 - 2*e^2 + 11*e + 12, 5/2*e^3 - 1/2*e^2 - 47/2*e + 7/2, -e^3 + e^2 + 9*e - 9, e^3 - 9*e - 6, 2*e^3 + e^2 - 18*e - 3, 1/2*e^3 - 3/2*e^2 - 1/2*e + 31/2, e^3 - 2*e^2 - 15*e + 14, e^3 + e^2 - 7*e - 1, 3/2*e^3 + 1/2*e^2 - 31/2*e - 17/2, -5/2*e^3 - 3/2*e^2 + 45/2*e + 7/2, -e^3 + 7*e + 8, -2*e^2 - 4*e + 22, -e^3 + 9*e + 20, -2*e^3 - 2*e^2 + 20*e + 8, -5/2*e^3 + 5/2*e^2 + 43/2*e - 51/2, 5/2*e^3 - 3/2*e^2 - 51/2*e + 17/2, e^3 + 2*e^2 - 13*e - 12, -2*e^3 + 3*e^2 + 18*e - 27, 1/2*e^3 + 5/2*e^2 - 11/2*e + 9/2, 2*e^2 - 4*e - 22, e^3 - 9*e - 2, 6*e, e^3 - 2*e^2 - 13*e + 8, e^3 + 2*e^2 - 11*e - 6, -2*e^3 + 2*e^2 + 16*e - 30, -3/2*e^3 + 9/2*e^2 + 35/2*e - 81/2, 2*e^3 - 3*e^2 - 26*e + 17, -2*e^3 + 3*e^2 + 18*e - 27, e^3 + 2*e^2 - 13*e - 12, e^3 - 6*e^2 - 15*e + 42, -e^3 - 3*e^2 + 3*e + 35, -1/2*e^3 + 3/2*e^2 + 21/2*e - 15/2, 1/2*e^3 - 7/2*e^2 - 3/2*e + 45/2, 2*e^3 - e^2 - 14*e + 7, -e^3 + 15*e + 20, -e^3 + 2*e^2 + 7*e - 8, 9/2*e^3 - 1/2*e^2 - 87/2*e - 1/2, 7/2*e^3 + 1/2*e^2 - 65/2*e - 19/2, e^3 + 2*e^2 - 7*e - 6, 4*e^3 - 3*e^2 - 32*e + 37, 1/2*e^3 - 5/2*e^2 - 19/2*e + 35/2, 1/2*e^3 - 3/2*e^2 + 9/2*e + 29/2, 4*e^3 + 3*e^2 - 36*e - 9, -3*e^3 + 3*e^2 + 33*e - 1, -5/2*e^3 - 1/2*e^2 + 39/2*e - 9/2, -3/2*e^3 - 5/2*e^2 + 11/2*e + 37/2, -2*e^3 + 2*e^2 + 16*e - 30, 2*e^3 - 3*e^2 - 24*e + 25, -3*e^2 + 27, -e^3 - 4*e^2 + e + 28, e^2 + 2*e - 5, e^3 - 4*e^2 - 11*e + 18, 7/2*e^3 - 1/2*e^2 - 81/2*e - 21/2, 5*e^2 - 2*e - 31, -3/2*e^3 - 3/2*e^2 + 15/2*e + 39/2, -2*e^3 - 4*e^2 + 26*e + 24, -2*e^3 + 6*e^2 + 18*e - 48, e^3 - 9*e - 12, -e^3 + 9*e + 20, -2*e^3 + 5*e^2 + 16*e - 51, -e^3 - e^2 + 7*e + 1, 2*e^3 + 6*e^2 - 18*e - 34, e^3 + 2*e^2 - 7*e - 36, 2*e^2 + 2*e - 22, 2*e^3 - e^2 - 14*e + 1, -4*e^3 - e^2 + 40*e + 7, 2*e^3 - e^2 - 26*e + 7, -3*e^3 - 3*e^2 + 33*e + 27, 4*e^3 - 5*e^2 - 42*e + 35, -2*e^2 + 6*e + 40, 6*e + 14, -2*e^2 - 6*e + 4, -1/2*e^3 - 9/2*e^2 - 9/2*e + 59/2, 2*e^2 - 10*e - 16, -1/2*e^3 + 1/2*e^2 + 13/2*e - 45/2, -3*e^3 + 4*e^2 + 35*e - 14, -2*e^3 + 5*e^2 + 22*e - 21, 1/2*e^3 - 3/2*e^2 - 21/2*e + 31/2, -2*e^3 - 2*e^2 + 24*e + 14, 3*e^3 - 27*e + 14, -36, -e^3 + 3*e^2 + 15*e - 7, -3*e^3 + 21*e + 8, 5/2*e^3 - 15/2*e^2 - 51/2*e + 113/2, -2*e^3 + 3*e^2 + 18*e - 17, 3*e^3 - 27*e + 12, -3*e^3 - e^2 + 35*e + 17, 5/2*e^3 - 1/2*e^2 - 75/2*e - 17/2, -3*e^3 + 2*e^2 + 29*e - 22, -1/2*e^3 + 3/2*e^2 + 15/2*e - 21/2, 2*e^3 - 2*e^2 - 24*e + 24, -e^3 + 6*e^2 + 21*e - 44, -1/2*e^3 - 11/2*e^2 + 11/2*e + 33/2, -2*e^3 + 6*e^2 + 24*e - 30, 3*e^3 + e^2 - 25*e + 7, -e^3 + 3*e^2 + 9*e - 37, -3/2*e^3 + 7/2*e^2 + 35/2*e - 107/2, -1/2*e^3 - 5/2*e^2 - 1/2*e + 51/2, -4*e^3 + e^2 + 38*e - 9, e^3 + e^2 - 13*e + 11, 2*e^2 - 2*e - 16, -3*e^3 + 6*e^2 + 27*e - 64, 5*e^3 - e^2 - 53*e - 23, -9/2*e^3 + 3/2*e^2 + 75/2*e - 17/2, -7/2*e^3 + 1/2*e^2 + 71/2*e - 33/2, 5*e^3 + 4*e^2 - 39*e - 30, 2*e^3 - 5*e^2 - 28*e + 33, -2*e^3 + 9*e^2 + 28*e - 71, 9/2*e^3 - 3/2*e^2 - 97/2*e + 3/2, -2*e^3 + 3*e^2 + 22*e + 1, 6*e^2 - 30, e^3 - 6*e^2 - 3*e + 36, 3*e^3 - 21*e + 24, -3*e^3 + 3*e^2 + 27*e - 15, -13/2*e^3 - 1/2*e^2 + 121/2*e + 37/2, -3*e^3 + 6*e^2 + 27*e - 54] 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 + 2])] = -1 AL_eigenvalues[ZF.ideal([5, 5, -w + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]