/* 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, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([38, 38, -13*w - 57]) primes_array = [ [2, 2, -3*w - 13],\ [3, 3, w + 4],\ [3, 3, w - 4],\ [5, 5, 2*w + 9],\ [5, 5, -2*w + 9],\ [17, 17, w + 6],\ [17, 17, -w + 6],\ [19, 19, w],\ [31, 31, 20*w + 87],\ [31, 31, 7*w + 30],\ [49, 7, -7],\ [59, 59, 6*w + 25],\ [59, 59, -6*w + 25],\ [61, 61, -9*w - 40],\ [61, 61, 9*w - 40],\ [67, 67, 2*w - 3],\ [67, 67, -2*w - 3],\ [71, 71, 3*w + 10],\ [71, 71, 3*w - 10],\ [73, 73, 27*w + 118],\ [73, 73, -12*w - 53],\ [79, 79, 4*w - 15],\ [79, 79, -4*w - 15],\ [101, 101, 5*w + 24],\ [101, 101, 5*w - 24],\ [103, 103, 29*w + 126],\ [103, 103, 16*w + 69],\ [107, 107, 3*w - 8],\ [107, 107, -3*w - 8],\ [121, 11, -11],\ [127, 127, -8*w - 33],\ [127, 127, -8*w + 33],\ [137, 137, -4*w + 21],\ [137, 137, 4*w + 21],\ [149, 149, 2*w - 15],\ [149, 149, -2*w - 15],\ [151, 151, -5*w - 18],\ [151, 151, 5*w - 18],\ [157, 157, 6*w + 29],\ [157, 157, 6*w - 29],\ [167, 167, 3*w - 2],\ [167, 167, -3*w - 2],\ [169, 13, -13],\ [179, 179, -15*w - 64],\ [179, 179, -15*w + 64],\ [197, 197, -19*w - 84],\ [197, 197, 46*w + 201],\ [211, 211, 38*w + 165],\ [211, 211, 25*w + 108],\ [223, 223, -4*w - 9],\ [223, 223, 4*w - 9],\ [227, 227, 57*w + 248],\ [227, 227, 18*w + 77],\ [229, 229, -3*w - 20],\ [229, 229, 3*w - 20],\ [233, 233, -28*w - 123],\ [233, 233, 37*w + 162],\ [277, 277, -6*w + 31],\ [277, 277, 6*w + 31],\ [307, 307, -17*w + 72],\ [307, 307, -17*w - 72],\ [313, 313, 3*w - 22],\ [313, 313, -3*w - 22],\ [331, 331, 5*w - 12],\ [331, 331, -5*w - 12],\ [349, 349, 15*w - 68],\ [349, 349, -15*w - 68],\ [353, 353, 68*w + 297],\ [353, 353, -23*w - 102],\ [379, 379, -10*w + 39],\ [379, 379, -10*w - 39],\ [383, 383, -9*w - 34],\ [383, 383, -9*w + 34],\ [389, 389, -13*w + 60],\ [389, 389, -13*w - 60],\ [397, 397, 9*w + 44],\ [397, 397, 9*w - 44],\ [431, 431, -15*w + 62],\ [431, 431, -15*w - 62],\ [439, 439, -5*w - 6],\ [439, 439, 5*w - 6],\ [457, 457, 21*w + 94],\ [457, 457, 21*w - 94],\ [461, 461, 50*w + 219],\ [461, 461, -41*w - 180],\ [487, 487, -8*w - 27],\ [487, 487, -8*w + 27],\ [523, 523, -26*w + 111],\ [523, 523, -26*w - 111],\ [529, 23, -23],\ [541, 541, -6*w - 35],\ [541, 541, 6*w - 35],\ [547, 547, -22*w - 93],\ [547, 547, -22*w + 93],\ [557, 557, -w - 24],\ [557, 557, w - 24],\ [563, 563, 6*w - 11],\ [563, 563, -6*w - 11],\ [577, 577, 9*w - 46],\ [577, 577, 9*w + 46],\ [593, 593, -17*w - 78],\ [593, 593, -17*w + 78],\ [599, 599, 75*w + 326],\ [599, 599, 36*w + 155],\ [607, 607, -7*w - 18],\ [607, 607, 7*w - 18],\ [613, 613, 3*w - 28],\ [613, 613, -3*w - 28],\ [617, 617, 11*w + 54],\ [617, 617, 11*w - 54],\ [653, 653, 2*w - 27],\ [653, 653, -2*w - 27],\ [659, 659, 6*w - 5],\ [659, 659, -6*w - 5],\ [683, 683, -6*w - 1],\ [683, 683, 6*w - 1],\ [701, 701, 10*w - 51],\ [701, 701, 10*w + 51],\ [709, 709, 75*w + 328],\ [709, 709, -42*w - 185],\ [733, 733, -18*w + 83],\ [733, 733, -18*w - 83],\ [743, 743, 24*w + 101],\ [743, 743, 141*w + 614],\ [751, 751, 65*w + 282],\ [751, 751, 52*w + 225],\ [757, 757, 66*w + 289],\ [757, 757, -51*w - 224],\ [761, 761, -16*w + 75],\ [761, 761, -16*w - 75],\ [769, 769, 60*w + 263],\ [769, 769, -57*w - 250],\ [787, 787, -7*w - 12],\ [787, 787, 7*w - 12],\ [809, 809, -8*w - 45],\ [809, 809, -8*w + 45],\ [811, 811, 10*w - 33],\ [811, 811, -10*w - 33],\ [821, 821, 5*w - 36],\ [821, 821, -5*w - 36],\ [827, 827, -18*w - 73],\ [827, 827, -18*w + 73],\ [839, 839, 84*w + 365],\ [839, 839, 45*w + 194],\ [841, 29, -29],\ [853, 853, -3*w - 32],\ [853, 853, 3*w - 32],\ [863, 863, 9*w - 26],\ [863, 863, -9*w - 26],\ [881, 881, -w - 30],\ [881, 881, w - 30],\ [887, 887, 12*w - 43],\ [887, 887, -12*w - 43],\ [907, 907, -13*w - 48],\ [907, 907, -13*w + 48],\ [911, 911, -15*w - 58],\ [911, 911, -15*w + 58],\ [929, 929, -40*w - 177],\ [929, 929, 103*w + 450],\ [937, 937, -24*w - 109],\ [937, 937, 24*w - 109],\ [971, 971, -30*w - 127],\ [971, 971, -30*w + 127],\ [983, 983, -21*w + 86],\ [983, 983, -21*w - 86],\ [991, 991, 8*w - 15],\ [991, 991, -8*w - 15],\ [997, 997, -6*w - 41],\ [997, 997, 6*w - 41],\ [1013, 1013, 2*w - 33],\ [1013, 1013, -2*w - 33],\ [1019, 1019, 51*w + 220],\ [1019, 1019, 90*w + 391],\ [1033, 1033, 27*w - 122],\ [1033, 1033, -27*w - 122],\ [1039, 1039, -20*w - 81],\ [1039, 1039, -20*w + 81],\ [1049, 1049, -20*w - 93],\ [1049, 1049, -20*w + 93],\ [1061, 1061, 35*w - 156],\ [1061, 1061, -35*w - 156],\ [1063, 1063, 44*w + 189],\ [1063, 1063, 109*w + 474],\ [1069, 1069, -18*w + 85],\ [1069, 1069, -18*w - 85],\ [1091, 1091, -33*w + 140],\ [1091, 1091, -33*w - 140],\ [1109, 1109, -58*w - 255],\ [1109, 1109, 85*w + 372],\ [1123, 1123, -14*w + 51],\ [1123, 1123, -14*w - 51],\ [1171, 1171, -10*w - 27],\ [1171, 1171, 10*w - 27],\ [1201, 1201, 15*w + 74],\ [1201, 1201, 15*w - 74],\ [1213, 1213, -33*w - 148],\ [1213, 1213, 33*w - 148],\ [1217, 1217, -4*w - 39],\ [1217, 1217, 4*w - 39],\ [1231, 1231, 175*w + 762],\ [1231, 1231, 32*w + 135],\ [1277, 1277, -w - 36],\ [1277, 1277, w - 36],\ [1283, 1283, -9*w - 16],\ [1283, 1283, 9*w - 16],\ [1289, 1289, -5*w - 42],\ [1289, 1289, 5*w - 42],\ [1291, 1291, 83*w + 360],\ [1291, 1291, 70*w + 303],\ [1297, 1297, 36*w - 161],\ [1297, 1297, 36*w + 161],\ [1301, 1301, 11*w + 60],\ [1301, 1301, -11*w + 60],\ [1307, 1307, -27*w - 112],\ [1307, 1307, -27*w + 112],\ [1319, 1319, 60*w + 259],\ [1319, 1319, 99*w + 430],\ [1367, 1367, -12*w - 37],\ [1367, 1367, 12*w - 37],\ [1369, 37, -37],\ [1373, 1373, -7*w - 48],\ [1373, 1373, 7*w - 48],\ [1399, 1399, 11*w - 30],\ [1399, 1399, -11*w - 30],\ [1427, 1427, 102*w + 443],\ [1427, 1427, 63*w + 272],\ [1429, 1429, 3*w - 40],\ [1429, 1429, -3*w - 40],\ [1439, 1439, -9*w - 10],\ [1439, 1439, 9*w - 10],\ [1447, 1447, 13*w - 42],\ [1447, 1447, -13*w - 42],\ [1453, 1453, -42*w - 187],\ [1453, 1453, 153*w + 668],\ [1459, 1459, -10*w - 21],\ [1459, 1459, 10*w - 21],\ [1471, 1471, -25*w + 102],\ [1471, 1471, -25*w - 102],\ [1489, 1489, -12*w + 65],\ [1489, 1489, 12*w + 65],\ [1493, 1493, 107*w + 468],\ [1493, 1493, -62*w - 273],\ [1511, 1511, 12*w - 35],\ [1511, 1511, -12*w - 35],\ [1523, 1523, 9*w - 4],\ [1523, 1523, -9*w - 4],\ [1571, 1571, -15*w + 52],\ [1571, 1571, -15*w - 52],\ [1579, 1579, -29*w + 120],\ [1579, 1579, -29*w - 120],\ [1597, 1597, -9*w - 56],\ [1597, 1597, 9*w - 56],\ [1601, 1601, 89*w + 390],\ [1601, 1601, -80*w - 351],\ [1613, 1613, -23*w - 108],\ [1613, 1613, -23*w + 108],\ [1621, 1621, 21*w - 100],\ [1621, 1621, 21*w + 100],\ [1627, 1627, -22*w + 87],\ [1627, 1627, -22*w - 87],\ [1657, 1657, 144*w + 629],\ [1657, 1657, -51*w - 226],\ [1663, 1663, -41*w + 174],\ [1663, 1663, -41*w - 174],\ [1667, 1667, -18*w - 67],\ [1667, 1667, -18*w + 67],\ [1669, 1669, -30*w - 137],\ [1669, 1669, -30*w + 137],\ [1681, 41, -41],\ [1697, 1697, 16*w - 81],\ [1697, 1697, 16*w + 81],\ [1699, 1699, -14*w - 45],\ [1699, 1699, 14*w - 45],\ [1721, 1721, 4*w - 45],\ [1721, 1721, -4*w - 45],\ [1723, 1723, 11*w - 24],\ [1723, 1723, -11*w - 24],\ [1733, 1733, -43*w - 192],\ [1733, 1733, 43*w - 192],\ [1747, 1747, 127*w + 552],\ [1747, 1747, 62*w + 267],\ [1753, 1753, -12*w - 67],\ [1753, 1753, -12*w + 67],\ [1823, 1823, 231*w + 1006],\ [1823, 1823, 36*w + 151],\ [1849, 43, -43],\ [1873, 1873, 132*w + 577],\ [1873, 1873, -63*w - 278],\ [1901, 1901, -14*w + 75],\ [1901, 1901, 14*w + 75],\ [1931, 1931, -18*w - 65],\ [1931, 1931, 18*w - 65],\ [1949, 1949, 2*w - 45],\ [1949, 1949, -2*w - 45],\ [1951, 1951, -23*w - 90],\ [1951, 1951, -23*w + 90],\ [1973, 1973, -13*w + 72],\ [1973, 1973, -13*w - 72],\ [1979, 1979, -21*w - 80],\ [1979, 1979, -21*w + 80],\ [1993, 1993, 123*w + 538],\ [1993, 1993, -72*w - 317],\ [2003, 2003, 39*w + 164],\ [2003, 2003, 234*w + 1019],\ [2027, 2027, 78*w + 337],\ [2027, 2027, 117*w + 508],\ [2053, 2053, -78*w - 343],\ [2053, 2053, 117*w + 512],\ [2069, 2069, -10*w - 63],\ [2069, 2069, 10*w - 63],\ [2083, 2083, -34*w - 141],\ [2083, 2083, -34*w + 141],\ [2111, 2111, -12*w - 25],\ [2111, 2111, 12*w - 25],\ [2113, 2113, -87*w - 382],\ [2113, 2113, 108*w + 473],\ [2129, 2129, 47*w - 210],\ [2129, 2129, -47*w - 210],\ [2131, 2131, -50*w + 213],\ [2131, 2131, -50*w - 213],\ [2137, 2137, 99*w + 434],\ [2137, 2137, -96*w - 421],\ [2143, 2143, 136*w + 591],\ [2143, 2143, 71*w + 306],\ [2153, 2153, -32*w - 147],\ [2153, 2153, -32*w + 147],\ [2179, 2179, -46*w - 195],\ [2179, 2179, -46*w + 195],\ [2203, 2203, 14*w - 39],\ [2203, 2203, -14*w - 39],\ [2207, 2207, 12*w - 23],\ [2207, 2207, -12*w - 23],\ [2209, 47, -47],\ [2213, 2213, -37*w + 168],\ [2213, 2213, -37*w - 168],\ [2221, 2221, -30*w - 139],\ [2221, 2221, -30*w + 139],\ [2281, 2281, 24*w + 115],\ [2281, 2281, 24*w - 115],\ [2297, 2297, -4*w - 51],\ [2297, 2297, 4*w - 51],\ [2311, 2311, 13*w - 30],\ [2311, 2311, -13*w - 30],\ [2339, 2339, 15*w - 44],\ [2339, 2339, -15*w - 44],\ [2341, 2341, 6*w - 55],\ [2341, 2341, -6*w - 55],\ [2347, 2347, 158*w + 687],\ [2347, 2347, 67*w + 288],\ [2351, 2351, 177*w + 770],\ [2351, 2351, 60*w + 257],\ [2357, 2357, 19*w - 96],\ [2357, 2357, 19*w + 96],\ [2371, 2371, 110*w + 477],\ [2371, 2371, 97*w + 420],\ [2381, 2381, 151*w + 660],\ [2381, 2381, -70*w - 309],\ [2383, 2383, -31*w + 126],\ [2383, 2383, -31*w - 126],\ [2417, 2417, 191*w + 834],\ [2417, 2417, -56*w - 249],\ [2423, 2423, -36*w + 149],\ [2423, 2423, -36*w - 149],\ [2437, 2437, 21*w + 104],\ [2437, 2437, 21*w - 104],\ [2441, 2441, -5*w - 54],\ [2441, 2441, 5*w - 54],\ [2447, 2447, -12*w - 17],\ [2447, 2447, 12*w - 17],\ [2459, 2459, -30*w - 121],\ [2459, 2459, -30*w + 121],\ [2477, 2477, -31*w + 144],\ [2477, 2477, -31*w - 144],\ [2503, 2503, -19*w + 66],\ [2503, 2503, -19*w - 66],\ [2539, 2539, -35*w - 144],\ [2539, 2539, -35*w + 144],\ [2557, 2557, 9*w - 64],\ [2557, 2557, -9*w - 64],\ [2579, 2579, 129*w + 560],\ [2579, 2579, 90*w + 389],\ [2593, 2593, 12*w - 73],\ [2593, 2593, -12*w - 73],\ [2609, 2609, -17*w + 90],\ [2609, 2609, 17*w + 90],\ [2633, 2633, -88*w - 387],\ [2633, 2633, 133*w + 582],\ [2657, 2657, -41*w - 186],\ [2657, 2657, 41*w - 186],\ [2659, 2659, -25*w - 96],\ [2659, 2659, -25*w + 96],\ [2663, 2663, -24*w + 91],\ [2663, 2663, -24*w - 91],\ [2677, 2677, 54*w - 241],\ [2677, 2677, -54*w - 241],\ [2687, 2687, 12*w - 7],\ [2687, 2687, -12*w - 7],\ [2711, 2711, -12*w - 5],\ [2711, 2711, 12*w - 5],\ [2719, 2719, -55*w - 234],\ [2719, 2719, -55*w + 234],\ [2731, 2731, -43*w + 180],\ [2731, 2731, -43*w - 180],\ [2741, 2741, -106*w - 465],\ [2741, 2741, 115*w + 504],\ [2753, 2753, -8*w - 63],\ [2753, 2753, 8*w - 63],\ [2767, 2767, 268*w + 1167],\ [2767, 2767, 47*w + 198],\ [2797, 2797, -6*w - 59],\ [2797, 2797, 6*w - 59],\ [2803, 2803, 119*w + 516],\ [2803, 2803, 106*w + 459],\ [2809, 53, -53],\ [2837, 2837, -14*w - 81],\ [2837, 2837, 14*w - 81],\ [2843, 2843, 186*w + 809],\ [2843, 2843, 69*w + 296],\ [2857, 2857, 21*w - 106],\ [2857, 2857, 21*w + 106],\ [2861, 2861, -10*w - 69],\ [2861, 2861, 10*w - 69],\ [2879, 2879, 96*w + 415],\ [2879, 2879, 135*w + 586],\ [2887, 2887, 13*w - 18],\ [2887, 2887, -13*w - 18],\ [2897, 2897, -w - 54],\ [2897, 2897, w - 54],\ [2903, 2903, -21*w - 74],\ [2903, 2903, 21*w - 74],\ [2939, 2939, -30*w + 119],\ [2939, 2939, -30*w - 119],\ [2963, 2963, -54*w - 229],\ [2963, 2963, -54*w + 229],\ [2969, 2969, -35*w + 162],\ [2969, 2969, -35*w - 162],\ [3023, 3023, -24*w + 89],\ [3023, 3023, -24*w - 89],\ [3037, 3037, 6*w - 61],\ [3037, 3037, -6*w - 61],\ [3041, 3041, 55*w + 246],\ [3041, 3041, 55*w - 246],\ [3049, 3049, 27*w + 130],\ [3049, 3049, 27*w - 130],\ [3067, 3067, 13*w - 12],\ [3067, 3067, -13*w - 12],\ [3089, 3089, -40*w + 183],\ [3089, 3089, -40*w - 183],\ [3119, 3119, -15*w - 34],\ [3119, 3119, 15*w - 34],\ [3121, 3121, -15*w - 86],\ [3121, 3121, 15*w - 86],\ [3167, 3167, -57*w + 242],\ [3167, 3167, -57*w - 242],\ [3187, 3187, 17*w - 48],\ [3187, 3187, -17*w - 48],\ [3191, 3191, 192*w + 835],\ [3191, 3191, 75*w + 322],\ [3209, 3209, 155*w + 678],\ [3209, 3209, -92*w - 405],\ [3217, 3217, 24*w + 119],\ [3217, 3217, 24*w - 119],\ [3251, 3251, 15*w - 32],\ [3251, 3251, -15*w - 32],\ [3253, 3253, 18*w + 97],\ [3253, 3253, -18*w + 97],\ [3259, 3259, 19*w - 60],\ [3259, 3259, -19*w - 60],\ [3271, 3271, 128*w + 555],\ [3271, 3271, 115*w + 498],\ [3299, 3299, -39*w - 160],\ [3299, 3299, -39*w + 160],\ [3313, 3313, -48*w - 217],\ [3313, 3313, 48*w - 217],\ [3319, 3319, 85*w + 366],\ [3319, 3319, 176*w + 765],\ [3329, 3329, 28*w - 135],\ [3329, 3329, 28*w + 135],\ [3343, 3343, -16*w - 39],\ [3343, 3343, 16*w - 39],\ [3347, 3347, 18*w - 53],\ [3347, 3347, -18*w - 53],\ [3359, 3359, 105*w + 454],\ [3359, 3359, 144*w + 625],\ [3361, 3361, 9*w - 70],\ [3361, 3361, -9*w - 70],\ [3371, 3371, 78*w + 335],\ [3371, 3371, 195*w + 848],\ [3389, 3389, -110*w - 483],\ [3389, 3389, 137*w + 600],\ [3469, 3469, 15*w - 88],\ [3469, 3469, -15*w - 88],\ [3491, 3491, -15*w - 28],\ [3491, 3491, 15*w - 28],\ [3499, 3499, -14*w - 15],\ [3499, 3499, 14*w - 15],\ [3511, 3511, -37*w + 150],\ [3511, 3511, -37*w - 150],\ [3527, 3527, 147*w + 638],\ [3527, 3527, 108*w + 467],\ [3541, 3541, -6*w - 65],\ [3541, 3541, 6*w - 65],\ [3547, 3547, 98*w + 423],\ [3547, 3547, 163*w + 708],\ [3557, 3557, 59*w - 264],\ [3557, 3557, 59*w + 264],\ [3571, 3571, -22*w - 75],\ [3571, 3571, -22*w + 75],\ [3581, 3581, -w - 60],\ [3581, 3581, w - 60],\ [3617, 3617, -44*w - 201],\ [3617, 3617, -44*w + 201],\ [3623, 3623, 324*w + 1411],\ [3623, 3623, 51*w + 214],\ [3631, 3631, -20*w - 63],\ [3631, 3631, 20*w - 63],\ [3643, 3643, 14*w - 9],\ [3643, 3643, -14*w - 9],\ [3673, 3673, -3*w - 62],\ [3673, 3673, 3*w - 62],\ [3697, 3697, 24*w - 121],\ [3697, 3697, 24*w + 121],\ [3709, 3709, -90*w - 397],\ [3709, 3709, 183*w + 800],\ [3719, 3719, 24*w - 85],\ [3719, 3719, -24*w - 85],\ [3727, 3727, 17*w - 42],\ [3727, 3727, -17*w - 42],\ [3733, 3733, -42*w - 193],\ [3733, 3733, -42*w + 193],\ [3739, 3739, -34*w + 135],\ [3739, 3739, -34*w - 135],\ [3769, 3769, -93*w - 410],\ [3769, 3769, 180*w + 787],\ [3797, 3797, 26*w + 129],\ [3797, 3797, 26*w - 129],\ [3803, 3803, -66*w + 281],\ [3803, 3803, -66*w - 281],\ [3851, 3851, -27*w - 100],\ [3851, 3851, -27*w + 100],\ [3877, 3877, -99*w - 436],\ [3877, 3877, 174*w + 761],\ [3881, 3881, 5*w - 66],\ [3881, 3881, -5*w - 66],\ [3907, 3907, 73*w + 312],\ [3907, 3907, 242*w + 1053],\ [3943, 3943, -19*w - 54],\ [3943, 3943, 19*w - 54],\ [3947, 3947, -18*w - 47],\ [3947, 3947, 18*w - 47],\ [3967, 3967, -23*w + 78],\ [3967, 3967, -23*w - 78],\ [4001, 4001, 25*w - 126],\ [4001, 4001, 25*w + 126],\ [4003, 4003, -49*w + 204],\ [4003, 4003, -49*w - 204],\ [4013, 4013, -82*w - 363],\ [4013, 4013, 217*w + 948],\ [4019, 4019, -15*w - 16],\ [4019, 4019, 15*w - 16],\ [4027, 4027, -38*w - 153],\ [4027, 4027, -38*w + 153],\ [4073, 4073, -43*w + 198],\ [4073, 4073, -43*w - 198],\ [4079, 4079, 15*w - 14],\ [4079, 4079, -15*w - 14],\ [4099, 4099, -65*w + 276],\ [4099, 4099, -65*w - 276],\ [4129, 4129, 153*w + 670],\ [4129, 4129, -120*w - 527],\ [4153, 4153, -12*w - 83],\ [4153, 4153, 12*w - 83],\ [4177, 4177, -129*w - 566],\ [4177, 4177, 144*w + 631],\ [4211, 4211, -15*w - 8],\ [4211, 4211, 15*w - 8],\ [4229, 4229, -53*w - 240],\ [4229, 4229, 53*w - 240],\ [4231, 4231, -35*w - 138],\ [4231, 4231, -35*w + 138],\ [4241, 4241, 31*w - 150],\ [4241, 4241, 31*w + 150],\ [4253, 4253, -7*w - 72],\ [4253, 4253, 7*w - 72],\ [4259, 4259, -15*w - 4],\ [4259, 4259, 15*w - 4],\ [4261, 4261, -66*w - 295],\ [4261, 4261, 66*w - 295],\ [4271, 4271, 15*w - 2],\ [4271, 4271, -15*w - 2],\ [4273, 4273, 33*w - 158],\ [4273, 4273, 33*w + 158],\ [4283, 4283, -21*w - 64],\ [4283, 4283, 21*w - 64],\ [4327, 4327, -32*w + 123],\ [4327, 4327, -32*w - 123],\ [4337, 4337, -w - 66],\ [4337, 4337, w - 66],\ [4349, 4349, 23*w + 120],\ [4349, 4349, 23*w - 120],\ [4357, 4357, -6*w - 71],\ [4357, 4357, 6*w - 71],\ [4363, 4363, 86*w + 369],\ [4363, 4363, 229*w + 996],\ [4391, 4391, 333*w + 1450],\ [4391, 4391, 60*w + 253],\ [4409, 4409, 8*w - 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8363, 21*w - 4],\ [8363, 8363, -21*w - 4],\ [8369, 8369, 260*w + 1137],\ [8369, 8369, -143*w - 630],\ [8377, 8377, -24*w - 139],\ [8377, 8377, 24*w - 139],\ [8387, 8387, -87*w + 368],\ [8387, 8387, -87*w - 368],\ [8419, 8419, -70*w + 291],\ [8419, 8419, -70*w - 291],\ [8431, 8431, 32*w - 105],\ [8431, 8431, -32*w - 105],\ [8461, 8461, 9*w - 100],\ [8461, 8461, -9*w - 100],\ [8467, 8467, -22*w - 27],\ [8467, 8467, 22*w - 27],\ [8513, 8513, -28*w + 153],\ [8513, 8513, 28*w + 153],\ [8521, 8521, -21*w - 130],\ [8521, 8521, 21*w - 130],\ [8527, 8527, -52*w + 207],\ [8527, 8527, -52*w - 207],\ [8537, 8537, -56*w - 261],\ [8537, 8537, -56*w + 261],\ [8539, 8539, -59*w - 240],\ [8539, 8539, -59*w + 240],\ [8543, 8543, -24*w - 49],\ [8543, 8543, 24*w - 49],\ [8563, 8563, -74*w + 309],\ [8563, 8563, -74*w - 309],\ [8573, 8573, 2*w - 93],\ [8573, 8573, -2*w - 93],\ [8597, 8597, -14*w - 111],\ [8597, 8597, 14*w - 111],\ [8647, 8647, 157*w + 678],\ [8647, 8647, 248*w + 1077],\ [8663, 8663, -69*w - 286],\ [8663, 8663, -69*w + 286],\ [8669, 8669, -50*w + 237],\ [8669, 8669, -50*w - 237],\ [8681, 8681, 101*w - 450],\ [8681, 8681, -101*w - 450],\ [8689, 8689, -63*w + 290],\ [8689, 8689, -63*w - 290],\ [8713, 8713, -12*w - 107],\ [8713, 8713, 12*w - 107],\ [8731, 8731, 82*w + 345],\ [8731, 8731, 485*w + 2112],\ [8737, 8737, 39*w + 194],\ [8737, 8737, 39*w - 194],\ [8741, 8741, 5*w - 96],\ [8741, 8741, -5*w - 96],\ [8807, 8807, -48*w - 187],\ [8807, 8807, -48*w + 187],\ [8819, 8819, -30*w - 91],\ [8819, 8819, 30*w - 91],\ [8821, 8821, -30*w + 161],\ [8821, 8821, 30*w + 161],\ [8831, 8831, -57*w + 230],\ [8831, 8831, -57*w - 230],\ [8861, 8861, 25*w - 144],\ [8861, 8861, -25*w - 144],\ [8867, 8867, -42*w - 157],\ [8867, 8867, -42*w + 157],\ [8887, 8887, -43*w + 162],\ [8887, 8887, -43*w - 162],\ [8893, 8893, 33*w + 172],\ [8893, 8893, 33*w - 172],\ [8923, 8923, 29*w - 84],\ [8923, 8923, -29*w - 84],\ [8941, 8941, 294*w + 1285],\ [8941, 8941, -135*w - 596],\ [8951, 8951, 27*w - 70],\ [8951, 8951, -27*w - 70],\ [8963, 8963, 321*w + 1396],\ [8963, 8963, 126*w + 541],\ [8969, 8969, 43*w - 210],\ [8969, 8969, 43*w + 210],\ [8971, 8971, -22*w - 15],\ [8971, 8971, 22*w - 15],\ [8999, 8999, -36*w - 125],\ [8999, 8999, -36*w + 125],\ [9013, 9013, -78*w - 353],\ [9013, 9013, 78*w - 353],\ [9029, 9029, -22*w - 135],\ [9029, 9029, 22*w - 135],\ [9041, 9041, 20*w - 129],\ [9041, 9041, -20*w - 129],\ [9043, 9043, 31*w - 96],\ [9043, 9043, -31*w - 96],\ [9049, 9049, 45*w + 218],\ [9049, 9049, 45*w - 218],\ [9059, 9059, 510*w + 2221],\ [9059, 9059, 81*w + 340],\ [9103, 9103, 209*w + 906],\ [9103, 9103, 196*w + 849],\ [9137, 9137, 49*w + 234],\ [9137, 9137, 49*w - 234],\ [9151, 9151, -23*w - 30],\ [9151, 9151, 23*w - 30],\ [9181, 9181, -15*w - 116],\ [9181, 9181, 15*w - 116],\ [9187, 9187, 22*w - 3],\ [9187, 9187, -22*w - 3],\ [9199, 9199, 305*w + 1326],\ [9199, 9199, 136*w + 585],\ [9221, 9221, 371*w + 1620],\ [9221, 9221, -110*w - 489],\ [9227, 9227, -27*w - 68],\ [9227, 9227, 27*w - 68],\ [9241, 9241, -144*w - 635],\ [9241, 9241, 285*w + 1246],\ [9257, 9257, -124*w - 549],\ [9257, 9257, 331*w + 1446],\ [9277, 9277, 9*w - 104],\ [9277, 9277, -9*w - 104],\ [9281, 9281, -76*w + 345],\ [9281, 9281, -76*w - 345],\ [9323, 9323, -66*w - 271],\ [9323, 9323, -66*w + 271],\ [9343, 9343, -71*w + 294],\ [9343, 9343, -71*w - 294],\ [9349, 9349, -102*w - 455],\ [9349, 9349, 102*w - 455],\ [9397, 9397, -69*w + 316],\ [9397, 9397, -69*w - 316],\ [9409, 97, -97],\ [9419, 9419, -51*w + 200],\ [9419, 9419, -51*w - 200],\ [9421, 9421, -150*w - 661],\ [9421, 9421, 279*w + 1220],\ [9433, 9433, -3*w - 98],\ [9433, 9433, 3*w - 98],\ [9439, 9439, -41*w + 150],\ [9439, 9439, -41*w - 150],\ [9473, 9473, 7*w - 102],\ [9473, 9473, -7*w - 102],\ [9491, 9491, -42*w + 155],\ [9491, 9491, -42*w - 155],\ [9497, 9497, -4*w - 99],\ [9497, 9497, 4*w - 99],\ [9551, 9551, -48*w + 185],\ [9551, 9551, -48*w - 185],\ [9601, 9601, 36*w + 185],\ [9601, 9601, 36*w - 185],\ [9643, 9643, 34*w - 111],\ [9643, 9643, -34*w - 111],\ [9649, 9649, 15*w - 118],\ [9649, 9649, -15*w - 118],\ [9661, 9661, -159*w - 700],\ [9661, 9661, 270*w + 1181],\ [9677, 9677, 41*w + 204],\ [9677, 9677, 41*w - 204],\ [9679, 9679, -79*w - 330],\ [9679, 9679, -79*w + 330],\ [9697, 9697, 9*w - 106],\ [9697, 9697, -9*w - 106],\ [9719, 9719, -24*w - 35],\ [9719, 9719, 24*w - 35],\ [9733, 9733, 267*w + 1168],\ [9733, 9733, -162*w - 713],\ [9743, 9743, 159*w + 686],\ [9743, 9743, 276*w + 1199],\ [9787, 9787, 83*w + 348],\ [9787, 9787, 538*w + 2343],\ [9803, 9803, -69*w - 284],\ [9803, 9803, -69*w + 284],\ [9829, 9829, 3*w - 100],\ [9829, 9829, -3*w - 100],\ [9871, 9871, -95*w - 402],\ [9871, 9871, -95*w + 402],\ [9883, 9883, 91*w + 384],\ [9883, 9883, 494*w + 2151],\ [9907, 9907, -23*w - 12],\ [9907, 9907, 23*w - 12],\ [9929, 9929, -5*w - 102],\ [9929, 9929, 5*w - 102],\ [9931, 9931, 218*w + 945],\ [9931, 9931, 205*w + 888],\ [9941, 9941, -70*w - 321],\ [9941, 9941, -70*w + 321],\ [9973, 9973, 18*w - 127],\ [9973, 9973, -18*w - 127]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -1, -1, -4, -4, 3, 3, -1, -8, -8, -5, 15, 15, 2, 2, 3, 3, 2, 2, 9, 9, -10, -10, 2, 2, -6, -6, -7, -7, -18, 18, 18, -17, -17, 0, 0, 2, 2, -2, -2, -12, -12, -25, 0, 0, 8, 8, 27, 27, 14, 14, -17, -17, -10, -10, -6, -6, 28, 28, -12, -12, 29, 29, 17, 17, 10, 10, 9, 9, 15, 15, -26, -26, -30, -30, 8, 8, -18, -18, 20, 20, -7, -7, -28, -28, -2, -2, 29, 29, -45, 2, 2, 28, 28, 28, 28, -36, -36, -37, -37, 34, 34, 0, 0, -22, -22, 34, 34, 18, 18, -36, -36, 5, 5, 4, 4, -28, -28, -30, -30, -36, -36, -16, -16, 32, 32, -2, -2, 27, 27, -35, -35, -17, -17, -15, -15, -3, -3, 12, 12, 23, 23, 20, 20, -33, -6, -6, 54, 54, -18, -18, -2, -2, 53, 53, 12, 12, -55, -55, -7, -7, -28, -28, -6, -6, -8, -8, 28, 28, -36, -36, -15, -15, -1, -1, 0, 0, 30, 30, -18, -18, 4, 4, -40, -40, -23, -23, -30, -30, -56, -56, -8, -8, 17, 17, -36, -36, -37, -37, -68, -68, 8, 8, 39, 39, 45, 45, -13, -13, 53, 53, -28, -28, -27, -27, 0, 0, 58, 58, -70, -6, -6, 50, 50, 53, 53, 30, 30, -30, -30, 38, 38, -6, -6, 35, 35, -48, -48, 30, 30, -16, -16, 12, 12, 4, 4, -28, -28, 20, 20, -2, -2, -13, -13, 44, 44, -18, -18, -52, -52, -2, -2, 64, 64, -12, -12, 0, 0, -18, 18, 18, -65, -65, 27, 27, 29, 29, -6, -6, -47, -47, 9, 9, 54, 54, -70, 9, 9, -18, -18, -28, -28, 80, 80, 2, 2, -46, -46, -35, -35, -26, -26, 59, 59, -57, -57, -16, -16, -30, -30, -36, -36, -38, -38, -61, -61, -50, -50, 37, 37, -57, -57, 14, 14, 39, 39, -65, -65, 49, 49, -32, -32, -30, 44, 44, 52, 52, -73, -73, -57, -57, -58, -58, -20, -20, -18, -18, -27, -27, 2, 2, 48, 48, -8, -8, 32, 32, -36, -36, -17, -17, -66, -66, 88, 88, -58, -58, 78, 78, 95, 95, 18, 18, -6, -6, 45, 45, 8, 8, 5, 5, 14, 14, -75, -75, 74, 74, 78, 78, -45, -45, 74, 74, -32, -32, -32, -32, 12, 12, -50, -50, -68, -68, 72, 72, -86, -86, 18, 18, -2, -2, -71, -71, -105, 18, 18, 19, 19, 13, 13, -48, -48, 90, 90, 8, 8, 3, 3, -56, -56, -35, -35, -61, -61, -5, -5, -16, -16, 58, 58, 97, 97, -70, -70, -52, -52, -75, -75, -80, -80, -53, -53, -42, -42, 83, 83, 12, 12, 10, 10, 78, 78, 27, 27, -6, -6, -65, -65, -8, -8, 45, 45, 29, 29, -20, -20, 35, 35, 4, 4, 73, 73, -40, -40, -113, -113, -68, -68, -80, -80, -40, -40, 87, 87, -100, -100, 42, 42, 68, 68, 92, 92, -52, -52, -112, -112, -33, -33, -78, -78, -87, -87, -26, -26, 102, 102, -61, -61, 9, 9, -7, -7, 80, 80, -90, -90, 28, 28, -66, -66, -15, -15, 70, 70, -42, -42, -21, -21, 52, 52, 58, 58, -18, -18, 3, 3, 34, 34, -17, -17, 68, 68, -13, -13, 44, 44, 4, 4, 15, 15, -17, -17, -51, -51, -50, -50, 60, 60, 35, 35, 79, 79, -42, -42, 87, 87, -120, -120, 72, 72, -78, -78, -66, -66, -95, -95, 62, 62, -8, -8, 69, 69, 44, 44, 58, 58, 103, 103, 70, 70, 28, 28, -16, -16, 42, 42, -75, -75, -96, -96, 58, 58, -53, -53, 59, 59, -106, -106, 17, 17, 32, 32, 32, 32, 28, 28, -50, -50, -53, -53, -76, -76, -16, -16, 27, 27, -35, -35, -36, -36, 68, 68, 110, 110, 49, 49, 4, 4, -38, -38, 10, 10, -50, -50, -43, -43, 123, 123, 114, 114, 118, 118, -72, -72, -30, -30, 112, 112, 22, 22, -62, -62, -122, -122, 12, 12, 53, 53, 10, 10, 9, 9, 128, 128, 92, 92, -87, -87, 42, 42, 24, 24, -31, -31, 13, 13, 72, 72, -18, -18, -87, -87, 9, 9, -40, -40, -26, -26, -33, -33, -78, -78, -42, -42, 104, 104, 122, 122, 27, 27, -22, -22, -116, -116, 14, 14, -56, -56, -20, -20, -58, -58, 27, 27, -30, -30, 89, 89, -8, -8, -92, -92, 120, 120, 140, 140, 52, 52, -3, -3, -42, -42, 72, 72, 68, 68, -36, -36, 75, 75, 92, 92, 98, 98, 12, 12, -47, -47, 130, 130, -140, -140, -27, -27, 74, 74, 29, 29, -52, -52, -1, -1, 43, 43, -12, -12, 20, 20, 4, 4, 90, 90, 45, 45, 97, 97, 52, 52, 84, 84, 13, 13, -90, -90, 38, 38, 33, 33, 74, 74, -82, -82, -25, -25, -138, -138, 58, 58, -76, -76, 39, 39, -20, -20, -42, -42, 15, 15, -30, -30, 17, 17, -10, -10, 97, 97, -23, -23, 102, 102, 149, 149, 99, 99, -68, -68, 158, 158, -112, -112, -128, -128, -41, -41, 40, 40, 95, 95, 107, 107, 14, 14, 30, 30, -86, -86, -27, -27, -63, -63, -116, -116, -135, -135, -68, -68, -58, -58, -56, -56, 62, 62, -37, -37, -128, -128, -130, 60, 60, -132, -132, -118, -118, -122, -122, -92, -92, -73, -73, 88, 88, -147, -147, -36, -36, -28, -28, 148, 148, 97, 97, 125, 125, -161, -161, 160, 160, -132, -132, -95, -95, 90, 90, -106, -106, -153, -153, -20, -20, -70, -70, -132, -132, 78, 78, 18, 18, -18, -18, -15, -15, 103, 103, 9, 9, -153, -153, -22, -22, 15, 15, -72, -72, -31, -31, 30, 30, 102, 102, 130, 130, 114, 114, -66, -66, 22, 22, -41, -41, -45, -45, 54, 54, -38, -38, 87, 87, -86, -86, 158, 158, -111, -111, -52, -52, 44, 44, -50, -50, 174, 174, -106, -106, -172, -172, -20, -20, 92, 92, 33, 33, 140, 140, -178, 70, 70, -1, -1, 52, 52, 10, 10, -116, -116, 155, 155, 27, 27, -82, -82, 72, 72, 108, 108, 49, 49, 88, 88, 92, 92, -5, -5, -71, -71, 100, 100, -112, -112, -86, -86, 172, 172, -55, -55, 84, 84, -165, -165, 13, 13, 108, 108, 95, 95, -68, -68, -38, -38, 83, 83, -51, -51, 107, 107, 8, 8, -22, -22, -145, -145, 134, 134, -111, -111, 104, 104, -72, -72, 38, 38, -136, -136, -70, -70, 57, 57, 15, 15, -151, -151, -48, -48, 23, 23, 2, 2, 88, 88, -105, -105, -58, -58, -68, -68, -118, -118, 33, 33, -112, -112, -106, -106, -131, -131, -98, -98, -118, -118, 129, 129, 150, 150, 37, 37, -160, -160, -66, -66, -120, -120, -33, -33, -156, -156, 55, 55, -75, -75, -106, -106, 133, 133, -118, -118, 42, 42, -32, -32, 90, 90, 42, 42, 68, 68, 77, 77, -102, -102, 148, 148, 57, 57, 179, 179, 44, 44, -150, -150, 48, 48, -190, -60, -60, 142, 142, 89, 89, -80, -80, -6, -6, 87, 87, 153, 153, -78, -78, -163, -163, 44, 44, -110, -110, -88, -88, 68, 68, -20, -20, 178, 178, -30, -30, 14, 14, -156, -156, 163, 163, 79, 79, 70, 70, -38, -38, -181, -181, -7, -7, -25, -25, -43, -43, 72, 72, 74, 74] 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, -3*w - 13])] = -1 AL_eigenvalues[ZF.ideal([19, 19, w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]