/* 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([29, 16, -15, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25, 5, -4/23*w^3 + 6/23*w^2 + 40/23*w - 21/23]) primes_array = [ [4, 2, -2/23*w^3 + 3/23*w^2 - 3/23*w + 1/23],\ [9, 3, -2/23*w^3 + 3/23*w^2 + 43/23*w + 24/23],\ [9, 3, -2/23*w^3 + 3/23*w^2 + 43/23*w - 68/23],\ [19, 19, -2/23*w^3 + 3/23*w^2 - 3/23*w + 24/23],\ [19, 19, -6/23*w^3 + 9/23*w^2 + 83/23*w - 20/23],\ [19, 19, 6/23*w^3 - 9/23*w^2 - 83/23*w + 66/23],\ [19, 19, -2/23*w^3 + 3/23*w^2 - 3/23*w - 22/23],\ [25, 5, -4/23*w^3 + 6/23*w^2 + 40/23*w - 21/23],\ [29, 29, w],\ [29, 29, -4/23*w^3 + 6/23*w^2 + 63/23*w - 21/23],\ [29, 29, 4/23*w^3 - 6/23*w^2 - 63/23*w + 44/23],\ [29, 29, -w + 1],\ [31, 31, w + 2],\ [31, 31, -4/23*w^3 + 6/23*w^2 + 63/23*w + 25/23],\ [31, 31, 4/23*w^3 - 6/23*w^2 - 63/23*w + 90/23],\ [31, 31, -w + 3],\ [49, 7, -2/23*w^3 + 3/23*w^2 + 43/23*w - 22/23],\ [59, 59, -20/23*w^3 + 30/23*w^2 + 269/23*w - 128/23],\ [59, 59, -8/23*w^3 + 12/23*w^2 + 103/23*w - 88/23],\ [59, 59, -8/23*w^3 + 12/23*w^2 + 103/23*w - 19/23],\ [59, 59, -4/23*w^3 + 6/23*w^2 + 17/23*w - 44/23],\ [109, 109, -9/23*w^3 + 2/23*w^2 + 113/23*w + 62/23],\ [109, 109, 4/23*w^3 - 6/23*w^2 - 63/23*w - 71/23],\ [109, 109, 7/23*w^3 - 22/23*w^2 - 47/23*w + 100/23],\ [109, 109, 9/23*w^3 - 25/23*w^2 - 90/23*w + 168/23],\ [121, 11, -2/23*w^3 + 3/23*w^2 + 20/23*w - 91/23],\ [121, 11, 2/23*w^3 - 3/23*w^2 - 20/23*w - 70/23],\ [131, 131, -5/23*w^3 + 19/23*w^2 + 50/23*w - 101/23],\ [131, 131, -3/23*w^3 + 16/23*w^2 + 7/23*w - 125/23],\ [131, 131, 3/23*w^3 + 7/23*w^2 - 30/23*w - 105/23],\ [131, 131, 5/23*w^3 + 4/23*w^2 - 73/23*w - 37/23],\ [139, 139, -2/23*w^3 - 20/23*w^2 + 43/23*w + 323/23],\ [139, 139, 17/23*w^3 - 37/23*w^2 - 216/23*w + 302/23],\ [139, 139, 17/23*w^3 - 14/23*w^2 - 239/23*w - 66/23],\ [139, 139, -2/23*w^3 + 26/23*w^2 - 3/23*w - 344/23],\ [149, 149, -1/23*w^3 - 10/23*w^2 + 33/23*w - 34/23],\ [149, 149, 1/23*w^3 + 10/23*w^2 - 33/23*w - 196/23],\ [149, 149, -1/23*w^3 + 13/23*w^2 + 10/23*w - 218/23],\ [149, 149, 1/23*w^3 - 13/23*w^2 - 10/23*w - 12/23],\ [169, 13, 6/23*w^3 - 9/23*w^2 - 37/23*w + 20/23],\ [169, 13, -10/23*w^3 + 15/23*w^2 + 123/23*w - 64/23],\ [199, 199, -1/23*w^3 + 13/23*w^2 - 13/23*w - 195/23],\ [199, 199, 3/23*w^3 + 7/23*w^2 - 53/23*w + 33/23],\ [199, 199, 3/23*w^3 - 16/23*w^2 - 30/23*w + 10/23],\ [199, 199, 1/23*w^3 + 10/23*w^2 - 10/23*w - 196/23],\ [251, 251, 6/23*w^3 - 9/23*w^2 - 14/23*w + 43/23],\ [251, 251, 14/23*w^3 - 21/23*w^2 - 186/23*w + 62/23],\ [251, 251, 4/23*w^3 - 6/23*w^2 - 109/23*w + 228/23],\ [251, 251, 6/23*w^3 - 9/23*w^2 - 14/23*w - 26/23],\ [271, 271, -8/23*w^3 + 12/23*w^2 + 126/23*w + 27/23],\ [271, 271, 8/23*w^3 - 35/23*w^2 - 80/23*w + 226/23],\ [271, 271, 13/23*w^3 - 31/23*w^2 - 153/23*w + 189/23],\ [271, 271, 8/23*w^3 - 12/23*w^2 - 126/23*w + 157/23],\ [281, 281, -w^2 + 2*w + 9],\ [281, 281, 13/23*w^3 - 8/23*w^2 - 153/23*w + 28/23],\ [281, 281, -13/23*w^3 + 31/23*w^2 + 130/23*w - 120/23],\ [281, 281, 11/23*w^3 - 28/23*w^2 - 87/23*w + 190/23],\ [289, 17, -11/23*w^3 + 51/23*w^2 + 110/23*w - 581/23],\ [289, 17, -8/23*w^3 + 35/23*w^2 + 80/23*w - 410/23],\ [311, 311, 5/23*w^3 + 4/23*w^2 - 50/23*w - 106/23],\ [311, 311, -7/23*w^3 + 22/23*w^2 + 70/23*w - 100/23],\ [311, 311, 7/23*w^3 + 1/23*w^2 - 93/23*w - 15/23],\ [311, 311, -5/23*w^3 + 19/23*w^2 + 27/23*w - 147/23],\ [389, 389, -8/23*w^3 + 35/23*w^2 + 80/23*w - 295/23],\ [389, 389, -4/23*w^3 + 29/23*w^2 - 6/23*w - 136/23],\ [389, 389, -4/23*w^3 - 17/23*w^2 + 40/23*w + 117/23],\ [389, 389, 2/23*w^3 + 20/23*w^2 - 20/23*w - 185/23],\ [401, 401, 11/23*w^3 - 5/23*w^2 - 179/23*w - 17/23],\ [401, 401, 2/23*w^3 + 20/23*w^2 - 43/23*w - 93/23],\ [401, 401, 1/23*w^3 + 10/23*w^2 + 36/23*w - 104/23],\ [401, 401, 11/23*w^3 - 28/23*w^2 - 156/23*w + 190/23],\ [419, 419, -6/23*w^3 - 14/23*w^2 + 83/23*w + 72/23],\ [419, 419, 3/23*w^3 + 7/23*w^2 - 76/23*w + 79/23],\ [419, 419, -6/23*w^3 + 32/23*w^2 + 37/23*w - 296/23],\ [419, 419, -1/23*w^3 + 13/23*w^2 + 56/23*w - 172/23],\ [421, 421, -8/23*w^3 + 12/23*w^2 + 57/23*w - 65/23],\ [421, 421, -12/23*w^3 + 18/23*w^2 + 143/23*w - 109/23],\ [421, 421, 12/23*w^3 - 18/23*w^2 - 143/23*w + 40/23],\ [421, 421, 8/23*w^3 - 12/23*w^2 - 57/23*w - 4/23],\ [439, 439, 7/23*w^3 - 22/23*w^2 - 93/23*w + 123/23],\ [439, 439, 1/23*w^3 + 10/23*w^2 + 13/23*w - 127/23],\ [439, 439, 5/23*w^3 + 4/23*w^2 - 73/23*w + 9/23],\ [439, 439, 7/23*w^3 + 1/23*w^2 - 116/23*w - 15/23],\ [449, 449, -11/23*w^3 + 5/23*w^2 + 133/23*w + 63/23],\ [449, 449, -9/23*w^3 + 2/23*w^2 + 90/23*w + 16/23],\ [449, 449, 9/23*w^3 - 25/23*w^2 - 67/23*w + 99/23],\ [449, 449, -11/23*w^3 + 28/23*w^2 + 110/23*w - 190/23],\ [479, 479, 10/23*w^3 - 15/23*w^2 - 123/23*w - 74/23],\ [479, 479, 6/23*w^3 - 9/23*w^2 - 37/23*w + 158/23],\ [479, 479, -6/23*w^3 + 9/23*w^2 + 37/23*w + 118/23],\ [479, 479, 10/23*w^3 - 15/23*w^2 - 123/23*w + 202/23],\ [529, 23, -4/23*w^3 + 29/23*w^2 + 40/23*w - 274/23],\ [529, 23, w^2 - 6],\ [541, 541, 6/23*w^3 + 14/23*w^2 - 83/23*w - 141/23],\ [541, 541, -6/23*w^3 + 32/23*w^2 + 37/23*w - 227/23],\ [541, 541, 6/23*w^3 + 14/23*w^2 - 83/23*w - 164/23],\ [541, 541, -6/23*w^3 + 32/23*w^2 + 37/23*w - 204/23],\ [569, 569, -5/23*w^3 - 4/23*w^2 + 50/23*w - 78/23],\ [569, 569, -7/23*w^3 + 22/23*w^2 + 70/23*w - 284/23],\ [569, 569, -5/23*w^3 + 19/23*w^2 + 50/23*w - 262/23],\ [569, 569, 5/23*w^3 - 19/23*w^2 - 27/23*w - 37/23],\ [619, 619, -2/23*w^3 - 43/23*w^2 + 66/23*w + 622/23],\ [619, 619, 17/23*w^3 - 37/23*w^2 - 216/23*w + 233/23],\ [619, 619, 17/23*w^3 - 14/23*w^2 - 239/23*w + 3/23],\ [619, 619, -2/23*w^3 + 49/23*w^2 - 26/23*w - 643/23],\ [641, 641, -1/23*w^3 + 13/23*w^2 - 59/23*w + 127/23],\ [641, 641, -14/23*w^3 + 44/23*w^2 + 117/23*w - 246/23],\ [641, 641, -11/23*w^3 + 5/23*w^2 + 179/23*w + 201/23],\ [641, 641, -1/23*w^3 - 10/23*w^2 - 36/23*w - 80/23],\ [691, 691, 5/23*w^3 - 65/23*w^2 - 4/23*w + 837/23],\ [691, 691, -2/23*w^3 + 26/23*w^2 + 20/23*w - 321/23],\ [691, 691, -2/23*w^3 - 20/23*w^2 + 66/23*w + 277/23],\ [691, 691, -5/23*w^3 - 50/23*w^2 + 119/23*w + 773/23],\ [701, 701, 2/23*w^3 + 20/23*w^2 - 66/23*w - 185/23],\ [701, 701, w^2 - 7],\ [701, 701, -2/23*w^3 + 26/23*w^2 + 20/23*w - 183/23],\ [701, 701, -2/23*w^3 + 26/23*w^2 + 20/23*w - 229/23],\ [709, 709, -18/23*w^3 + 27/23*w^2 + 226/23*w - 152/23],\ [709, 709, -33/23*w^3 + 38/23*w^2 + 445/23*w - 41/23],\ [709, 709, -33/23*w^3 + 61/23*w^2 + 422/23*w - 409/23],\ [709, 709, 18/23*w^3 - 27/23*w^2 - 226/23*w + 83/23],\ [719, 719, -4/23*w^3 + 29/23*w^2 + 40/23*w - 136/23],\ [719, 719, 12/23*w^3 - 18/23*w^2 - 166/23*w + 155/23],\ [719, 719, w^2 - 12],\ [719, 719, 4/23*w^3 + 17/23*w^2 - 86/23*w - 71/23],\ [809, 809, -10/23*w^3 + 15/23*w^2 + 123/23*w - 248/23],\ [809, 809, 7/23*w^3 - 22/23*w^2 - 93/23*w + 307/23],\ [809, 809, 7/23*w^3 + 1/23*w^2 - 116/23*w - 199/23],\ [809, 809, 10/23*w^3 - 15/23*w^2 - 123/23*w - 120/23],\ [811, 811, -7/23*w^3 - 1/23*w^2 + 93/23*w - 31/23],\ [811, 811, -7/23*w^3 + 22/23*w^2 + 93/23*w - 100/23],\ [811, 811, 7/23*w^3 + 1/23*w^2 - 116/23*w + 8/23],\ [811, 811, 1/23*w^3 + 10/23*w^2 + 13/23*w - 150/23],\ [821, 821, -4/23*w^3 + 29/23*w^2 + 17/23*w - 182/23],\ [821, 821, 4/23*w^3 + 17/23*w^2 - 63/23*w - 186/23],\ [821, 821, -4/23*w^3 + 29/23*w^2 + 17/23*w - 228/23],\ [821, 821, 4/23*w^3 + 17/23*w^2 - 63/23*w - 140/23],\ [839, 839, -16/23*w^3 + 70/23*w^2 + 160/23*w - 751/23],\ [839, 839, 22/23*w^3 - 33/23*w^2 - 289/23*w + 196/23],\ [839, 839, -22/23*w^3 + 33/23*w^2 + 289/23*w - 104/23],\ [839, 839, -16/23*w^3 - 22/23*w^2 + 252/23*w + 537/23],\ [859, 859, -7/23*w^3 + 22/23*w^2 + 47/23*w - 192/23],\ [859, 859, -9/23*w^3 + 2/23*w^2 + 113/23*w - 30/23],\ [859, 859, -9/23*w^3 + 25/23*w^2 + 90/23*w - 76/23],\ [859, 859, 7/23*w^3 + 1/23*w^2 - 70/23*w - 130/23],\ [971, 971, -10/23*w^3 + 15/23*w^2 + 146/23*w + 97/23],\ [971, 971, -14/23*w^3 - 2/23*w^2 + 186/23*w + 99/23],\ [971, 971, 14/23*w^3 - 44/23*w^2 - 140/23*w + 269/23],\ [971, 971, -10/23*w^3 + 15/23*w^2 + 146/23*w - 248/23]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 76*x^2 + 1152 K. = NumberField(heckePol) hecke_eigenvalues_array = [-4, 1/4*e^2 - 8, 1/4*e^2 - 8, e, e, -e, -e, -1, 1/4*e^2 - 10, 1/4*e^2 - 10, 1/4*e^2 - 10, 1/4*e^2 - 10, -1/24*e^3 + 13/6*e, 1/24*e^3 - 13/6*e, -1/24*e^3 + 13/6*e, 1/24*e^3 - 13/6*e, -1/2*e^2 + 24, -1/12*e^3 + 10/3*e, 1/12*e^3 - 10/3*e, -1/12*e^3 + 10/3*e, 1/12*e^3 - 10/3*e, 1/4*e^2 - 12, 1/4*e^2 - 12, 1/4*e^2 - 12, 1/4*e^2 - 12, -1/4*e^2 + 14, -1/4*e^2 + 14, 1/24*e^3 - 19/6*e, -1/24*e^3 + 19/6*e, 1/24*e^3 - 19/6*e, -1/24*e^3 + 19/6*e, -2*e, 2*e, -2*e, 2*e, 1/2*e^2 - 20, 1/2*e^2 - 20, 1/2*e^2 - 20, 1/2*e^2 - 20, 1/4*e^2 - 2, 1/4*e^2 - 2, e, e, -e, -e, -5/24*e^3 + 59/6*e, 5/24*e^3 - 59/6*e, -5/24*e^3 + 59/6*e, 5/24*e^3 - 59/6*e, 1/24*e^3 - 7/6*e, 1/24*e^3 - 7/6*e, -1/24*e^3 + 7/6*e, -1/24*e^3 + 7/6*e, 3/4*e^2 - 36, 3/4*e^2 - 36, 3/4*e^2 - 36, 3/4*e^2 - 36, 1/4*e^2 + 6, 1/4*e^2 + 6, 1/24*e^3 - 37/6*e, 1/24*e^3 - 37/6*e, -1/24*e^3 + 37/6*e, -1/24*e^3 + 37/6*e, -3/4*e^2 + 36, -3/4*e^2 + 36, -3/4*e^2 + 36, -3/4*e^2 + 36, -1/4*e^2 - 2, -1/4*e^2 - 2, -1/4*e^2 - 2, -1/4*e^2 - 2, -3*e, 3*e, -3*e, 3*e, -5/4*e^2 + 40, -5/4*e^2 + 40, -5/4*e^2 + 40, -5/4*e^2 + 40, -1/12*e^3 + 13/3*e, -1/12*e^3 + 13/3*e, 1/12*e^3 - 13/3*e, 1/12*e^3 - 13/3*e, 5/4*e^2 - 56, 5/4*e^2 - 56, 5/4*e^2 - 56, 5/4*e^2 - 56, -1/3*e^3 + 40/3*e, 1/3*e^3 - 40/3*e, -1/3*e^3 + 40/3*e, 1/3*e^3 - 40/3*e, -1/2*e^2 - 8, -1/2*e^2 - 8, -1/4*e^2 - 10, -1/4*e^2 - 10, -1/4*e^2 - 10, -1/4*e^2 - 10, -3/2*e^2 + 48, -3/2*e^2 + 48, -3/2*e^2 + 48, -3/2*e^2 + 48, 1/12*e^3 - 13/3*e, 1/12*e^3 - 13/3*e, -1/12*e^3 + 13/3*e, -1/12*e^3 + 13/3*e, -3/2*e^2 + 36, -3/2*e^2 + 36, -3/2*e^2 + 36, -3/2*e^2 + 36, 1/8*e^3 - 11/2*e, -1/8*e^3 + 11/2*e, 1/8*e^3 - 11/2*e, -1/8*e^3 + 11/2*e, -5/4*e^2 + 56, -5/4*e^2 + 56, -5/4*e^2 + 56, -5/4*e^2 + 56, -3/4*e^2 + 2, -3/4*e^2 + 2, -3/4*e^2 + 2, -3/4*e^2 + 2, -1/12*e^3 + 1/3*e, 1/12*e^3 - 1/3*e, -1/12*e^3 + 1/3*e, 1/12*e^3 - 1/3*e, -3/4*e^2 + 30, -3/4*e^2 + 30, -3/4*e^2 + 30, -3/4*e^2 + 30, -11/24*e^3 + 113/6*e, 11/24*e^3 - 113/6*e, -11/24*e^3 + 113/6*e, 11/24*e^3 - 113/6*e, 3/4*e^2, 3/4*e^2, 3/4*e^2, 3/4*e^2, -1/12*e^3 + 1/3*e, -1/12*e^3 + 1/3*e, 1/12*e^3 - 1/3*e, 1/12*e^3 - 1/3*e, -5/12*e^3 + 53/3*e, -5/12*e^3 + 53/3*e, 5/12*e^3 - 53/3*e, 5/12*e^3 - 53/3*e, 7/24*e^3 - 79/6*e, -7/24*e^3 + 79/6*e, 7/24*e^3 - 79/6*e, -7/24*e^3 + 79/6*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([25,5,-4/23*w^3+6/23*w^2+40/23*w-21/23])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]