/* 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([19, 7, -10, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16,4,w - 1]) primes_array = [ [4, 2, -1/4*w^3 + 1/2*w + 3/4],\ [4, 2, 1/2*w^3 - 4*w - 1/2],\ [9, 3, 1/2*w^3 + w^2 - 3*w - 7/2],\ [11, 11, -1/4*w^3 + 1/2*w + 7/4],\ [11, 11, 1/2*w^3 - 4*w - 3/2],\ [19, 19, w],\ [19, 19, 1/4*w^3 - 5/2*w + 1/4],\ [25, 5, 1/2*w^3 - 3*w - 1/2],\ [29, 29, 3/4*w^3 + w^2 - 9/2*w - 25/4],\ [29, 29, 1/2*w^3 - w^2 - 3*w + 9/2],\ [41, 41, -w^3 + 7*w + 3],\ [41, 41, -3/4*w^3 + 11/2*w - 3/4],\ [41, 41, 1/4*w^3 + w^2 - 5/2*w - 11/4],\ [71, 71, 1/2*w^3 - 4*w + 5/2],\ [71, 71, 3/4*w^3 + w^2 - 11/2*w - 21/4],\ [79, 79, 1/4*w^3 - 2*w^2 + 1/2*w + 25/4],\ [79, 79, -3/4*w^3 + 13/2*w + 9/4],\ [89, 89, -7/4*w^3 - w^2 + 27/2*w + 41/4],\ [89, 89, -7/4*w^3 - w^2 + 27/2*w + 33/4],\ [101, 101, 1/4*w^3 + w^2 - 1/2*w - 27/4],\ [101, 101, -1/4*w^3 + w^2 + 5/2*w - 17/4],\ [109, 109, 1/2*w^3 + w^2 - 3*w - 3/2],\ [109, 109, -w^3 + 7*w + 1],\ [121, 11, 3/4*w^3 - 9/2*w - 5/4],\ [139, 139, 3/4*w^3 + 2*w^2 - 9/2*w - 45/4],\ [139, 139, 5/4*w^3 + w^2 - 17/2*w - 27/4],\ [139, 139, w^3 - w^2 - 6*w + 3],\ [139, 139, -1/4*w^3 + 2*w^2 + 3/2*w - 65/4],\ [149, 149, 3/4*w^3 + 2*w^2 - 13/2*w - 61/4],\ [149, 149, -1/4*w^3 - 2*w^2 + 7/2*w + 23/4],\ [151, 151, 5/4*w^3 + w^2 - 17/2*w - 31/4],\ [151, 151, w^3 + 2*w^2 - 7*w - 13],\ [179, 179, 3/4*w^3 + w^2 - 11/2*w - 17/4],\ [179, 179, -1/4*w^3 + w^2 + 1/2*w - 25/4],\ [181, 181, 1/4*w^3 + 2*w^2 - 3/2*w - 35/4],\ [181, 181, 1/4*w^3 + 2*w^2 - 3/2*w - 51/4],\ [191, 191, -1/4*w^3 - 2*w^2 + 3/2*w + 27/4],\ [191, 191, -5/4*w^3 + 19/2*w + 11/4],\ [199, 199, 2*w^2 - 13],\ [199, 199, 1/2*w^3 + 2*w^2 - 3*w - 17/2],\ [211, 211, 3/4*w^3 + 2*w^2 - 13/2*w - 41/4],\ [211, 211, -1/4*w^3 + 2*w^2 + 3/2*w - 49/4],\ [229, 229, -1/4*w^3 + 2*w^2 + 3/2*w - 45/4],\ [229, 229, -3/4*w^3 - 2*w^2 + 9/2*w + 41/4],\ [239, 239, 3/4*w^3 + w^2 - 9/2*w - 33/4],\ [239, 239, 1/4*w^3 + w^2 - 1/2*w - 31/4],\ [241, 241, -1/4*w^3 + 2*w^2 + 1/2*w - 41/4],\ [241, 241, -2*w^2 + w + 12],\ [241, 241, -w^3 - 2*w^2 + 7*w + 11],\ [241, 241, -3/4*w^3 - 2*w^2 + 11/2*w + 37/4],\ [251, 251, -2*w^2 + 3*w + 4],\ [251, 251, -1/4*w^3 - 1/2*w - 9/4],\ [251, 251, -3/2*w^3 - w^2 + 11*w + 13/2],\ [251, 251, -3/4*w^3 + 13/2*w - 11/4],\ [271, 271, -3/4*w^3 + w^2 + 13/2*w - 23/4],\ [271, 271, w^2 - 10],\ [271, 271, 1/2*w^3 + w^2 - w - 11/2],\ [271, 271, 1/4*w^3 + w^2 - 3/2*w - 3/4],\ [281, 281, -5/4*w^3 - 2*w^2 + 11/2*w + 39/4],\ [281, 281, 3/2*w^3 - 11*w - 7/2],\ [311, 311, w^2 - 2*w - 6],\ [311, 311, -1/4*w^3 + w^2 + 1/2*w - 29/4],\ [331, 331, 2*w^2 - 11],\ [331, 331, 1/2*w^3 + 2*w^2 - 3*w - 21/2],\ [361, 19, w^3 - 6*w - 2],\ [389, 389, 1/4*w^3 + w^2 - 1/2*w - 35/4],\ [389, 389, 3/4*w^3 + 2*w^2 - 9/2*w - 53/4],\ [389, 389, 1/4*w^3 - 2*w^2 - 3/2*w + 33/4],\ [389, 389, w^3 + w^2 - 6*w - 9],\ [401, 401, 3/4*w^3 + w^2 - 7/2*w - 29/4],\ [401, 401, -3/4*w^3 + w^2 + 11/2*w - 15/4],\ [401, 401, 3/4*w^3 + 2*w^2 - 13/2*w - 33/4],\ [401, 401, 1/4*w^3 + 2*w^2 - 7/2*w - 51/4],\ [409, 409, w^3 - 5*w - 1],\ [409, 409, -7/4*w^3 - w^2 + 15/2*w + 21/4],\ [409, 409, 5/4*w^3 - 17/2*w - 3/4],\ [409, 409, 9/4*w^3 - w^2 - 33/2*w + 25/4],\ [419, 419, 3/4*w^3 + w^2 - 11/2*w - 9/4],\ [419, 419, -3/4*w^3 - 2*w^2 + 11/2*w + 21/4],\ [419, 419, -1/4*w^3 - 4*w^2 + 7/2*w + 47/4],\ [419, 419, -1/4*w^3 + w^2 + 1/2*w - 33/4],\ [421, 421, -1/2*w^3 - 4*w^2 + 4*w + 59/2],\ [421, 421, -7/4*w^3 + 25/2*w + 1/4],\ [431, 431, -5/4*w^3 - w^2 + 19/2*w + 23/4],\ [431, 431, 1/2*w^3 + 2*w^2 - 5*w - 13/2],\ [431, 431, -1/2*w^3 + w^2 + w - 9/2],\ [431, 431, -1/2*w^3 - 2*w^2 + 5*w + 29/2],\ [439, 439, -3/2*w^3 + 10*w - 3/2],\ [439, 439, 5/4*w^3 - 13/2*w + 5/4],\ [449, 449, -3/4*w^3 + 13/2*w - 15/4],\ [449, 449, 1/4*w^3 + 1/2*w + 21/4],\ [449, 449, 5/4*w^3 + 2*w^2 - 19/2*w - 43/4],\ [449, 449, 5/4*w^3 + 2*w^2 - 17/2*w - 59/4],\ [461, 461, -5/4*w^3 + 19/2*w + 15/4],\ [461, 461, -1/4*w^3 - 3*w^2 + 5/2*w + 31/4],\ [461, 461, 3/2*w^3 + w^2 - 9*w - 21/2],\ [461, 461, 3/4*w^3 - 5/2*w - 17/4],\ [479, 479, 3/2*w^3 + 2*w^2 - 10*w - 29/2],\ [479, 479, 3/4*w^3 - 2*w^2 - 7/2*w + 27/4],\ [491, 491, -5/4*w^3 + w^2 + 7/2*w + 7/4],\ [491, 491, 1/4*w^3 + w^2 - 1/2*w - 39/4],\ [491, 491, -5/2*w^3 - w^2 + 19*w + 23/2],\ [491, 491, -1/4*w^3 + w^2 + 5/2*w - 5/4],\ [499, 499, -9/4*w^3 + w^2 + 35/2*w - 21/4],\ [499, 499, -1/4*w^3 - 4*w^2 + 9/2*w + 47/4],\ [509, 509, w^3 + w^2 - 4*w - 7],\ [509, 509, -5/4*w^3 + w^2 + 19/2*w - 17/4],\ [521, 521, -3/4*w^3 + w^2 + 7/2*w - 27/4],\ [521, 521, 5/4*w^3 + w^2 - 17/2*w - 15/4],\ [529, 23, -9/4*w^3 + w^2 + 33/2*w - 21/4],\ [529, 23, 7/4*w^3 + w^2 - 15/2*w - 25/4],\ [541, 541, -7/4*w^3 + 13/2*w + 13/4],\ [541, 541, 1/4*w^3 + 2*w^2 - 5/2*w - 39/4],\ [541, 541, 1/2*w^3 + 2*w^2 - 4*w - 23/2],\ [541, 541, 3*w^2 - 2*w - 10],\ [571, 571, -5/4*w^3 + w^2 + 9/2*w - 5/4],\ [571, 571, 1/2*w^3 + 4*w^2 - 5*w - 25/2],\ [601, 601, -3/4*w^3 - 2*w^2 + 9/2*w + 17/4],\ [601, 601, -1/4*w^3 + 2*w^2 + 3/2*w - 69/4],\ [641, 641, 5/4*w^3 + 2*w^2 - 15/2*w - 51/4],\ [641, 641, -3/4*w^3 + 2*w^2 + 9/2*w - 35/4],\ [659, 659, 5/4*w^3 + 2*w^2 - 17/2*w - 35/4],\ [659, 659, -1/2*w^3 + 2*w^2 + 2*w - 25/2],\ [661, 661, -11/4*w^3 - w^2 + 41/2*w + 41/4],\ [661, 661, 11/4*w^3 + 5*w^2 - 43/2*w - 157/4],\ [661, 661, 1/4*w^3 - 5*w^2 + 7/2*w + 53/4],\ [661, 661, 3/2*w^3 - w^2 - 5*w - 1/2],\ [691, 691, 5/4*w^3 + w^2 - 7/2*w - 31/4],\ [691, 691, -2*w^3 + w^2 + 16*w - 4],\ [709, 709, 3/4*w^3 + 3*w^2 - 11/2*w - 45/4],\ [709, 709, 1/4*w^3 + 3*w^2 - 5/2*w - 83/4],\ [719, 719, 1/4*w^3 + 2*w^2 - 1/2*w - 55/4],\ [719, 719, 3/4*w^3 - w^2 - 9/2*w - 1/4],\ [751, 751, 2*w^3 + 2*w^2 - 13*w - 12],\ [751, 751, 3/4*w^3 + 3*w^2 - 9/2*w - 73/4],\ [809, 809, 5/4*w^3 + 2*w^2 - 11/2*w - 43/4],\ [809, 809, 1/2*w^3 + 2*w^2 - 3*w - 33/2],\ [821, 821, -5/4*w^3 + 19/2*w - 9/4],\ [821, 821, 3/4*w^3 + 2*w^2 - 13/2*w - 29/4],\ [821, 821, -1/4*w^3 - 3/2*w + 31/4],\ [821, 821, 1/4*w^3 + 2*w^2 - 9/2*w - 15/4],\ [839, 839, -5/4*w^3 - w^2 + 13/2*w + 35/4],\ [839, 839, 5/4*w^3 - w^2 - 17/2*w + 9/4],\ [841, 29, -5/4*w^3 + 15/2*w - 1/4],\ [859, 859, 1/4*w^3 + w^2 - 7/2*w - 51/4],\ [859, 859, -3/2*w^3 - 3*w^2 + 11*w + 41/2],\ [859, 859, 2*w^3 + 2*w^2 - 12*w - 9],\ [859, 859, 5/4*w^3 + 3*w^2 - 15/2*w - 63/4],\ [919, 919, 5/4*w^3 - 2*w^2 - 11/2*w + 29/4],\ [919, 919, 9/4*w^3 + 2*w^2 - 31/2*w - 55/4],\ [929, 929, -3/2*w^3 - 3*w^2 + 11*w + 49/2],\ [929, 929, -5/4*w^3 - w^2 + 9/2*w + 31/4],\ [961, 31, 5/4*w^3 - 15/2*w - 7/4],\ [961, 31, 1/2*w^3 - 3*w - 13/2],\ [971, 971, 1/4*w^3 + 4*w^2 - 11/2*w - 43/4],\ [971, 971, 5/4*w^3 - w^2 - 5/2*w - 11/4],\ [991, 991, 5/4*w^3 - w^2 - 15/2*w + 17/4],\ [991, 991, 3/2*w^3 + w^2 - 9*w - 13/2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 4*x^3 - 13*x^2 + 64*x - 44 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 0, 0, 0, 0, 2*e^3 - e^2 - 30*e + 29, -3/2*e^3 + 43/2*e - 10, e^3 - e^2 - 13*e + 19, 0, 0, 0, 0, 0, 0, 0, -7/2*e^3 + 103/2*e - 26, e^3 - e^2 - 17*e + 17, 0, 0, 0, 0, -3*e^3 + 41*e - 16, -e^3 + 19*e - 8, -7/2*e^3 + 4*e^2 + 99/2*e - 66, -2*e^3 + 4*e^2 + 30*e - 48, 6*e^3 - 4*e^2 - 90*e + 100, -13/2*e^3 + 4*e^2 + 197/2*e - 106, -2*e^3 + 3*e^2 + 30*e - 39, 0, 0, -2*e^3 + 4*e^2 + 30*e - 48, 6*e^3 - 4*e^2 - 90*e + 100, 0, 0, -5/2*e^3 + 65/2*e - 2, -3/2*e^3 + 2*e^2 + 47/2*e - 48, 0, 0, -5/2*e^3 + 69/2*e - 22, -5*e^3 + e^2 + 69*e - 33, 5*e^3 - 5*e^2 - 77*e + 97, 9/2*e^3 - 2*e^2 - 137/2*e + 68, 3/2*e^3 - 39/2*e + 14, -15/2*e^3 + 6*e^2 + 219/2*e - 124, 0, 0, -5/2*e^3 + 65/2*e - 2, 5/2*e^3 + 2*e^2 - 73/2*e + 8, -3/2*e^3 + 4*e^2 + 39/2*e - 50, -5/2*e^3 + 2*e^2 + 73/2*e - 60, 0, 0, 0, 0, 3*e^3 - 3*e^2 - 51*e + 59, 3*e^2 + 4*e - 27, -3/2*e^3 + 51/2*e - 10, -3/2*e^3 - 2*e^2 + 43/2*e - 4, 0, 0, 0, 0, 4*e^3 - 4*e^2 - 52*e + 76, -8*e^3 + 4*e^2 + 112*e - 88, 3*e^3 - 45*e + 8, 0, 0, 0, 0, 0, 0, 0, 0, 15/2*e^3 - 6*e^2 - 219/2*e + 116, -e^3 - 3*e^2 + 13*e + 13, 21/2*e^3 - 6*e^2 - 297/2*e + 128, 3*e^3 - e^2 - 47*e + 15, 0, 0, 0, 0, -2*e^3 - e^2 + 22*e + 11, 5*e^3 - e^2 - 73*e + 27, 0, 0, 0, 0, 9*e^3 - 7*e^2 - 137*e + 159, 1/2*e^3 - 4*e^2 - 33/2*e + 50, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3*e^3 + 3*e^2 + 51*e - 79, -9/2*e^3 + 6*e^2 + 153/2*e - 112, 0, 0, 0, 0, -8*e^3 + 9*e^2 + 116*e - 143, -10*e^3 + 7*e^2 + 142*e - 137, 3/2*e^3 + 2*e^2 - 63/2*e - 20, -e^3 - 3*e^2 + 21*e + 29, 10*e^3 - 5*e^2 - 150*e + 147, -11/2*e^3 + 159/2*e - 46, 9/2*e^3 - 6*e^2 - 137/2*e + 80, 5*e^3 - 5*e^2 - 85*e + 105, 21/2*e^3 - 2*e^2 - 297/2*e + 76, 7/2*e^3 - 2*e^2 - 83/2*e + 48, 0, 0, 0, 0, -12*e^3 + 5*e^2 + 168*e - 119, 17/2*e^3 - 261/2*e + 66, -7/2*e^3 - 2*e^2 + 107/2*e - 28, 5*e^3 - 3*e^2 - 81*e + 65, 9/2*e^3 - 6*e^2 - 121/2*e + 96, 11*e^3 - e^2 - 163*e + 93, -5*e^3 + 8*e^2 + 83*e - 128, 7*e^3 - 8*e^2 - 113*e + 152, 0, 0, 2*e^3 - 4*e^2 - 22*e + 60, -10*e^3 + 4*e^2 + 142*e - 104, 0, 0, 0, 0, 0, 0, 0, 0, -15*e^3 + 9*e^2 + 219*e - 211, 14*e^3 - 4*e^2 - 210*e + 164, 3*e^3 - 9*e^2 - 51*e + 101, 9/2*e^3 - 6*e^2 - 153/2*e + 104, 6*e^3 + 4*e^2 - 90*e + 16, -3/2*e^3 + 6*e^2 + 27/2*e - 76, -6*e^3 - 3*e^2 + 90*e - 13, 0, 0, -9*e^3 + 135*e - 88, 9*e^3 - 135*e + 56, 0, 0, 14*e^3 - 4*e^2 - 210*e + 164, 6*e^3 + 4*e^2 - 90*e + 16] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4,2,-1/2*w^3 + 4*w + 1/2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]