/* 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([-5, -7, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25, 5, w^2 - 7]) primes_array = [ [5, 5, w],\ [8, 2, 2],\ [11, 11, -w^2 + 2*w + 4],\ [11, 11, -w + 2],\ [11, 11, w - 1],\ [13, 13, -w^2 + w + 4],\ [17, 17, -w^2 + w + 8],\ [17, 17, -w^2 + w + 3],\ [19, 19, -w^2 + 6],\ [23, 23, -w^2 + 3],\ [25, 5, w^2 - 7],\ [27, 3, -3],\ [31, 31, w^2 - 2*w - 8],\ [37, 37, w^2 - 2*w - 6],\ [41, 41, -w - 4],\ [41, 41, w^2 - 2*w - 7],\ [47, 47, 2*w^2 - 3*w - 7],\ [53, 53, -w^2 + w + 9],\ [61, 61, 3*w^2 - 2*w - 17],\ [67, 67, 2*w - 1],\ [73, 73, 2*w^2 - 2*w - 9],\ [89, 89, 2*w^2 - 3*w - 11],\ [97, 97, 2*w^2 - w - 9],\ [97, 97, w^2 - 3*w - 6],\ [97, 97, 2*w - 3],\ [101, 101, 3*w^2 - 4*w - 13],\ [107, 107, 2*w^2 - 3*w - 14],\ [113, 113, 2*w^2 - 3*w - 12],\ [127, 127, -w^2 + w - 1],\ [127, 127, 2*w^2 - 2*w - 7],\ [127, 127, w^2 + 3*w - 1],\ [131, 131, w^2 - 3*w - 8],\ [137, 137, 2*w^2 - w - 8],\ [139, 139, 4*w^2 - 5*w - 19],\ [149, 149, w^2 + w - 7],\ [151, 151, w^2 - 4*w - 6],\ [157, 157, w^2 + w - 9],\ [163, 163, w^2 + w - 8],\ [169, 13, 2*w^2 - w - 7],\ [173, 173, w^2 - 5*w - 3],\ [179, 179, -w - 6],\ [191, 191, w^2 + 2*w - 4],\ [197, 197, 3*w - 1],\ [229, 229, 3*w^2 - 4*w - 18],\ [233, 233, -w^2 + w - 2],\ [233, 233, -3*w^2 + 4*w + 11],\ [233, 233, 3*w^2 - w - 14],\ [239, 239, 3*w^2 - 5*w - 14],\ [241, 241, w^2 - 4*w - 11],\ [257, 257, -2*w^2 + 2*w + 17],\ [263, 263, 3*w^2 - 3*w - 23],\ [269, 269, w^2 - 2*w - 12],\ [277, 277, w^2 - 4*w - 9],\ [281, 281, 4*w^2 - 2*w - 21],\ [283, 283, 3*w^2 - 3*w - 13],\ [293, 293, 3*w^2 - 4*w - 9],\ [331, 331, w^2 + 3*w - 3],\ [337, 337, w^2 + 2*w - 6],\ [343, 7, -7],\ [361, 19, w^2 - 5*w - 1],\ [367, 367, w^2 - 5*w - 8],\ [373, 373, w^2 + 2*w - 16],\ [373, 373, 2*w^2 + w - 11],\ [373, 373, 3*w^2 - 2*w - 13],\ [379, 379, -w^2 + w - 3],\ [383, 383, 2*w^2 - 6*w - 9],\ [401, 401, w^2 + 2*w - 7],\ [419, 419, 3*w^2 - 6*w - 13],\ [421, 421, 4*w^2 - 5*w - 26],\ [431, 431, 4*w - 1],\ [439, 439, w^2 - 5*w - 9],\ [439, 439, w^2 - 2*w - 13],\ [439, 439, 4*w^2 - 7*w - 17],\ [443, 443, 3*w^2 - 3*w - 8],\ [443, 443, w^2 - 13],\ [443, 443, 3*w^2 - w - 12],\ [449, 449, 4*w^2 - 6*w - 13],\ [457, 457, w^2 + 2*w - 11],\ [461, 461, -w - 8],\ [461, 461, -w^2 + 5*w - 1],\ [461, 461, 2*w^2 + w - 12],\ [463, 463, 3*w^2 + w - 9],\ [487, 487, 5*w^2 - 3*w - 27],\ [499, 499, 4*w^2 - w - 24],\ [499, 499, w^2 - 5*w - 11],\ [499, 499, 5*w^2 - 4*w - 27],\ [503, 503, 3*w^2 - 5*w - 22],\ [503, 503, 3*w^2 - 5*w - 17],\ [503, 503, 5*w^2 - 8*w - 22],\ [509, 509, -w^2 - 4],\ [529, 23, 4*w^2 - 5*w - 16],\ [541, 541, 2*w^2 + w - 17],\ [547, 547, -w^2 + 2*w - 4],\ [547, 547, 3*w^2 - 6*w - 14],\ [547, 547, 2*w^2 - 5*w - 17],\ [557, 557, 5*w^2 - 2*w - 31],\ [563, 563, 3*w^2 - 5*w - 18],\ [571, 571, -w^2 + w - 4],\ [571, 571, w^2 - 6*w - 2],\ [571, 571, 3*w^2 + w - 8],\ [577, 577, 2*w^2 - w - 19],\ [587, 587, 2*w^2 - 5*w - 13],\ [593, 593, 2*w^2 + w - 16],\ [599, 599, 3*w^2 - 19],\ [599, 599, 3*w^2 - 5*w - 19],\ [599, 599, 4*w^2 - w - 18],\ [607, 607, 4*w^2 - 2*w - 19],\ [617, 617, 4*w^2 - 3*w - 19],\ [619, 619, 4*w^2 - 7*w - 18],\ [643, 643, 3*w^2 + w - 7],\ [643, 643, 2*w^2 - 2*w - 19],\ [643, 643, 5*w^2 - 8*w - 16],\ [653, 653, 3*w^2 - w - 9],\ [653, 653, 2*w^2 - 6*w - 11],\ [653, 653, 3*w^2 - w - 7],\ [659, 659, w^2 - 2*w - 14],\ [661, 661, 5*w^2 - 6*w - 31],\ [661, 661, w^2 - 14],\ [661, 661, w - 9],\ [673, 673, -2*w^2 + 4*w - 1],\ [673, 673, 4*w^2 - w - 26],\ [673, 673, 6*w^2 - 5*w - 33],\ [677, 677, 3*w^2 - 8],\ [719, 719, 6*w^2 - 7*w - 29],\ [719, 719, 4*w^2 - 6*w - 29],\ [719, 719, 4*w^2 - 5*w - 14],\ [727, 727, 4*w^2 - 4*w - 31],\ [743, 743, w^2 - w - 14],\ [757, 757, 5*w^2 - 8*w - 23],\ [761, 761, w^2 - 6*w - 11],\ [761, 761, 4*w - 7],\ [761, 761, 4*w^2 - 3*w - 18],\ [773, 773, -5*w^2 + 10*w + 18],\ [787, 787, 5*w^2 - 7*w - 19],\ [797, 797, 4*w^2 - 5*w - 12],\ [797, 797, 3*w^2 + w - 16],\ [797, 797, 3*w^2 - 6*w - 16],\ [811, 811, w^2 + 3*w - 7],\ [821, 821, 4*w^2 - 6*w - 23],\ [829, 829, 2*w^2 + 2*w - 11],\ [829, 829, 2*w^2 - 7*w - 11],\ [829, 829, 5*w^2 - 5*w - 24],\ [853, 853, 2*w^2 - 6*w - 19],\ [857, 857, w^2 + 4*w - 4],\ [877, 877, -w^2 - w - 6],\ [881, 881, -4*w^2 + 4*w + 9],\ [883, 883, 2*w^2 - 6*w - 13],\ [907, 907, 3*w^2 - 6*w - 17],\ [911, 911, w^2 + 3*w - 8],\ [919, 919, 4*w^2 - 8*w - 17],\ [937, 937, 4*w^2 - 6*w - 27],\ [941, 941, 4*w^2 - 2*w - 17],\ [947, 947, 3*w^2 - 8*w - 13],\ [961, 31, 6*w^2 - 7*w - 39],\ [967, 967, 5*w - 2],\ [971, 971, 5*w^2 - 9*w - 21],\ [977, 977, 3*w^2 + w - 17],\ [983, 983, 2*w - 11],\ [991, 991, 6*w^2 - 7*w - 38],\ [991, 991, w^2 + 3*w - 9],\ [991, 991, w^2 + 3*w - 13],\ [997, 997, 4*w^2 - w - 16],\ [997, 997, 5*w - 12],\ [997, 997, 3*w^2 - 2*w - 26]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 19*x^2 + 9*x + 34 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/3*e^2 + 11/3, 1/3*e^2 + e - 11/3, -2/15*e^3 - 2/15*e^2 + 16/15*e + 13/15, -1/15*e^3 + 4/15*e^2 + 23/15*e - 26/15, 1/15*e^3 - 4/15*e^2 - 23/15*e + 86/15, -e + 1, -1/15*e^3 - 11/15*e^2 + 8/15*e + 94/15, 1/5*e^3 + 8/15*e^2 - 18/5*e - 22/15, 1/5*e^3 + 1/5*e^2 - 18/5*e + 26/5, 1, e - 3, 1/5*e^3 - 2/15*e^2 - 18/5*e - 32/15, -1/5*e^3 - 13/15*e^2 + 13/5*e + 62/15, -1/3*e^3 - 1/3*e^2 + 17/3*e - 4/3, 1/3*e^2 + 7/3, -4/15*e^3 + 2/5*e^2 + 62/15*e - 13/5, -4/15*e^3 - 4/15*e^2 + 47/15*e + 86/15, -1/3*e^2 - e - 4/3, 7/15*e^3 + 7/15*e^2 - 116/15*e + 52/15, 1/3*e^3 + 1/3*e^2 - 8/3*e + 10/3, 8/15*e^3 + 8/15*e^2 - 79/15*e - 22/15, -2/3*e^2 - 2*e + 10/3, 2/3*e^3 + 1/3*e^2 - 34/3*e + 4/3, -1/5*e^3 + 17/15*e^2 + 33/5*e - 193/15, -7/15*e^3 - 7/15*e^2 + 146/15*e + 83/15, -8/15*e^3 + 2/15*e^2 + 109/15*e - 148/15, 14/15*e^3 + 4/15*e^2 - 187/15*e + 64/15, 7/15*e^3 + 17/15*e^2 - 101/15*e - 13/15, -1/3*e^3 - 4/3*e^2 + 8/3*e + 23/3, -4/15*e^3 - 3/5*e^2 + 77/15*e + 77/5, -2/5*e^3 - 2/5*e^2 + 21/5*e + 43/5, -3/5*e^3 + 1/15*e^2 + 44/5*e + 91/15, -1/15*e^3 + 8/5*e^2 - 7/15*e - 102/5, 1/5*e^3 - 4/5*e^2 - 8/5*e + 61/5, -1/5*e^3 + 32/15*e^2 + 13/5*e - 358/15, 1/3*e^3 - 14/3*e + 8, 4/15*e^3 - 11/15*e^2 - 92/15*e + 94/15, 1/3*e^3 + 2*e^2 - 11/3*e - 12, 1/3*e^3 + 1/3*e^2 - 17/3*e - 11/3, -8/15*e^3 - 28/15*e^2 + 124/15*e + 227/15, -1/5*e^3 + 2/15*e^2 + 43/5*e - 13/15, 6/5*e^3 + 28/15*e^2 - 78/5*e - 152/15, -11/15*e^3 + 4/15*e^2 + 163/15*e - 161/15, -1/3*e^2 - 6*e + 8/3, -1/5*e^3 + 22/15*e^2 + 33/5*e - 248/15, -4/5*e^3 + 1/5*e^2 + 57/5*e - 59/5, -1/15*e^3 - 11/15*e^2 - 67/15*e + 34/15, -1/15*e^3 + 4/15*e^2 + 8/15*e - 281/15, 4/15*e^3 + 19/15*e^2 - 2/15*e - 266/15, 2/3*e^3 + e^2 - 28/3*e + 11, -4/5*e^3 - 32/15*e^2 + 57/5*e + 223/15, -1/3*e^2 + 4*e - 1/3, -8/5*e^3 - 29/15*e^2 + 109/5*e + 106/15, -6/5*e^3 - 11/5*e^2 + 63/5*e + 94/5, -6/5*e^3 - 38/15*e^2 + 78/5*e + 247/15, -1/5*e^3 - 1/5*e^2 + 13/5*e + 44/5, -7/5*e^3 - 26/15*e^2 + 86/5*e + 94/15, 11/15*e^3 + 11/15*e^2 - 283/15*e - 139/15, 4/5*e^3 - 1/5*e^2 - 67/5*e + 14/5, 8/15*e^3 + 43/15*e^2 - 109/15*e - 212/15, -1/3*e^3 + 1/3*e^2 + 23/3*e + 4/3, 11/15*e^3 + 31/15*e^2 - 178/15*e + 106/15, 2/3*e^3 - 5/3*e^2 - 40/3*e + 43/3, 16/15*e^3 + 2/5*e^2 - 308/15*e - 38/5, 14/15*e^3 - 26/15*e^2 - 232/15*e + 319/15, -2/15*e^3 - 2/15*e^2 - 89/15*e + 28/15, 1/15*e^3 - 14/15*e^2 + 7/15*e + 121/15, -2/15*e^3 + 1/5*e^2 + 31/15*e - 64/5, 1/15*e^3 - 34/15*e^2 - 68/15*e + 416/15, -3/5*e^3 - 3/5*e^2 + 54/5*e - 58/5, -1/15*e^3 + 8/5*e^2 + 83/15*e - 142/5, -28/15*e^3 - 8/15*e^2 + 419/15*e + 52/15, -1/5*e^3 - 23/15*e^2 + 23/5*e + 277/15, 13/15*e^3 + 43/15*e^2 - 149/15*e - 197/15, -1/5*e^3 - 8/15*e^2 + 8/5*e + 247/15, -1/5*e^3 - 16/5*e^2 - 17/5*e + 189/5, 1/3*e^3 + 4/3*e^2 - 5/3*e + 4/3, 2/15*e^3 - 13/15*e^2 - 106/15*e + 47/15, 8/15*e^3 + 16/5*e^2 - 169/15*e - 169/5, -7/15*e^3 + 8/15*e^2 + 86/15*e + 173/15, 4/3*e^3 - 2/3*e^2 - 65/3*e + 37/3, 9/5*e^3 + 22/15*e^2 - 117/5*e + 127/15, 22/15*e^3 - 13/15*e^2 - 311/15*e + 167/15, -22/15*e^3 - 17/15*e^2 + 296/15*e + 43/15, -4/15*e^3 + 16/15*e^2 + 122/15*e - 224/15, 1/5*e^3 + 1/5*e^2 - 23/5*e - 14/5, -4/3*e^3 - 10/3*e^2 + 65/3*e + 62/3, 1/15*e^3 + 11/15*e^2 - 128/15*e - 154/15, -14/15*e^3 + 1/15*e^2 + 202/15*e - 209/15, -13/15*e^3 - 1/5*e^2 + 209/15*e - 56/5, -2/3*e^3 + 22/3*e - 1, 1/15*e^3 - 3/5*e^2 - 98/15*e + 37/5, -2/3*e^3 - 4/3*e^2 + 16/3*e + 50/3, -4/15*e^3 + 11/15*e^2 + 92/15*e - 64/15, -4/3*e^2 + 26/3, 7/15*e^3 - 43/15*e^2 - 26/15*e + 737/15, -5/3*e^3 - 2/3*e^2 + 94/3*e - 8/3, -8/5*e^3 - 3/5*e^2 + 124/5*e - 8/5, 2/15*e^3 + 9/5*e^2 + 44/15*e - 46/5, 1/3*e^3 - 4/3*e^2 + 1/3*e + 68/3, 3/5*e^3 + 3/5*e^2 - 79/5*e + 28/5, 1/3*e^2 - 2*e - 26/3, 13/15*e^3 - 4/5*e^2 - 239/15*e - 4/5, 22/15*e^3 + 17/15*e^2 - 356/15*e - 73/15, -4/5*e^3 - 4/5*e^2 + 62/5*e - 54/5, -7/15*e^3 - 4/5*e^2 + 161/15*e + 91/5, -1/15*e^3 + 29/15*e^2 - 37/15*e - 541/15, -2/3*e^2 - 9*e + 31/3, 4/3*e^2 - 5*e - 119/3, 8/15*e^3 - 2/15*e^2 - 139/15*e - 92/15, 4/3*e^3 + 1/3*e^2 - 59/3*e + 28/3, 4/3*e^3 - 1/3*e^2 - 56/3*e - 34/3, 8/15*e^3 - 22/15*e^2 - 139/15*e - 37/15, 19/15*e^3 - 7/5*e^2 - 272/15*e + 48/5, 3/5*e^3 - 7/5*e^2 - 54/5*e + 168/5, -16/15*e^3 - 17/5*e^2 + 263/15*e + 98/5, 2/15*e^3 - 38/15*e^2 - 46/15*e + 262/15, -13/15*e^3 - 23/15*e^2 + 164/15*e + 382/15, 1/3*e^3 - 20/3*e - 38, 2/5*e^3 + 7/5*e^2 - 46/5*e - 53/5, 4/3*e^3 + 5/3*e^2 - 68/3*e - 22/3, 13/15*e^3 + 23/15*e^2 - 194/15*e - 22/15, -2/15*e^3 - 17/15*e^2 - 29/15*e - 167/15, 6/5*e^3 + 1/5*e^2 - 88/5*e + 41/5, 17/15*e^3 + 17/15*e^2 - 226/15*e - 163/15, -28/15*e^3 - 6/5*e^2 + 434/15*e + 44/5, -13/5*e^3 - 44/15*e^2 + 184/5*e + 16/15, -2/5*e^3 - 17/5*e^2 + 26/5*e + 203/5, -19/15*e^3 - 3/5*e^2 + 167/15*e - 68/5, 4/5*e^3 + 32/15*e^2 - 47/5*e - 223/15, 4/5*e^3 + 2/15*e^2 - 52/5*e + 212/15, 4/3*e^3 + 10/3*e^2 - 47/3*e - 35/3, -32/15*e^3 - 2/15*e^2 + 526/15*e - 212/15, 9/5*e^3 + 62/15*e^2 - 97/5*e - 628/15, -19/15*e^3 + 1/15*e^2 + 212/15*e - 179/15, 1/15*e^3 - 3/5*e^2 - 113/15*e + 32/5, -2/5*e^3 - 16/15*e^2 + 11/5*e - 136/15, -16/15*e^3 - 22/5*e^2 + 218/15*e + 98/5, -8/15*e^3 + 2/15*e^2 + 154/15*e + 92/15, -14/15*e^3 - 18/5*e^2 + 52/15*e + 222/5, 29/15*e^3 + 18/5*e^2 - 307/15*e - 112/5, 1/15*e^3 - 4/15*e^2 + 82/15*e - 214/15, -8/15*e^3 - 8/15*e^2 - 11/15*e + 52/15, 11/15*e^3 - 4/15*e^2 - 103/15*e + 71/15, e^3 + 3*e^2 - 7*e - 32, -19/15*e^3 + 16/15*e^2 + 167/15*e - 374/15, 1/5*e^3 + 23/15*e^2 + 17/5*e - 232/15, -6/5*e^3 + 9/5*e^2 + 98/5*e - 71/5, 1/15*e^3 + 31/15*e^2 - 68/15*e - 584/15, -19/15*e^3 - 14/15*e^2 + 332/15*e - 14/15, 2/5*e^3 - 8/5*e^2 - 31/5*e + 107/5, 22/15*e^3 - 8/15*e^2 - 401/15*e + 202/15, -3/5*e^3 + 7/5*e^2 + 39/5*e - 78/5, -19/15*e^3 + 1/15*e^2 + 272/15*e - 344/15, 23/15*e^3 - 7/15*e^2 - 274/15*e + 188/15, 8/15*e^3 + 28/15*e^2 - 154/15*e - 377/15, -2*e^3 - 8/3*e^2 + 21*e + 70/3, -7/5*e^3 - 12/5*e^2 + 106/5*e + 148/5, -9/5*e^3 - 17/15*e^2 + 147/5*e - 182/15, 2/15*e^3 - 13/15*e^2 - 31/15*e + 242/15, -1/3*e^3 + 5*e^2 + 29/3*e - 40, 4/15*e^3 - 16/15*e^2 - 107/15*e - 91/15, -34/15*e^3 + 26/15*e^2 + 647/15*e - 184/15] 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, w^2 - 7])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]