/* 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([-51, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9,9,w + 6]) primes_array = [ [3, 3, w],\ [3, 3, w + 2],\ [4, 2, 2],\ [5, 5, -w + 8],\ [7, 7, w + 1],\ [7, 7, w + 5],\ [13, 13, w + 3],\ [13, 13, w + 9],\ [17, 17, w],\ [17, 17, w + 16],\ [31, 31, -w - 4],\ [31, 31, w - 5],\ [41, 41, 3*w - 22],\ [47, 47, w + 19],\ [47, 47, w + 27],\ [53, 53, w + 14],\ [53, 53, w + 38],\ [59, 59, -w - 10],\ [59, 59, w - 11],\ [61, 61, 2*w - 13],\ [61, 61, -2*w - 11],\ [67, 67, w + 32],\ [67, 67, w + 34],\ [97, 97, w + 18],\ [97, 97, w + 78],\ [121, 11, -11],\ [131, 131, -w - 13],\ [131, 131, w - 14],\ [137, 137, w + 21],\ [137, 137, w + 115],\ [139, 139, 9*w - 70],\ [139, 139, 3*w - 26],\ [157, 157, w + 65],\ [157, 157, w + 91],\ [167, 167, w + 23],\ [167, 167, w + 143],\ [193, 193, w + 82],\ [193, 193, w + 110],\ [227, 227, w + 40],\ [227, 227, w + 186],\ [233, 233, w + 55],\ [233, 233, w + 177],\ [241, 241, 3*w - 28],\ [241, 241, -3*w - 25],\ [251, 251, -3*w - 13],\ [251, 251, 3*w - 16],\ [257, 257, w + 62],\ [257, 257, w + 194],\ [263, 263, w + 43],\ [263, 263, w + 219],\ [269, 269, -4*w - 31],\ [269, 269, 4*w - 35],\ [271, 271, 6*w - 49],\ [271, 271, 9*w - 71],\ [293, 293, w + 30],\ [293, 293, w + 262],\ [313, 313, w + 121],\ [313, 313, w + 191],\ [317, 317, w + 141],\ [317, 317, w + 175],\ [347, 347, w + 72],\ [347, 347, w + 274],\ [349, 349, 11*w - 82],\ [349, 349, -7*w + 50],\ [359, 359, 5*w - 43],\ [359, 359, 14*w - 109],\ [361, 19, -19],\ [379, 379, 4*w - 23],\ [379, 379, -4*w - 19],\ [383, 383, w + 155],\ [383, 383, w + 227],\ [389, 389, -3*w - 7],\ [389, 389, 3*w - 10],\ [397, 397, w + 77],\ [397, 397, w + 319],\ [401, 401, -6*w + 41],\ [401, 401, 15*w - 113],\ [409, 409, 3*w - 31],\ [409, 409, -3*w - 28],\ [419, 419, 3*w - 8],\ [419, 419, -3*w - 5],\ [431, 431, -3*w - 4],\ [431, 431, 3*w - 7],\ [449, 449, 3*w - 5],\ [449, 449, -3*w - 2],\ [457, 457, w + 172],\ [457, 457, w + 284],\ [461, 461, 3*w - 2],\ [461, 461, 3*w - 1],\ [463, 463, w + 127],\ [463, 463, w + 335],\ [491, 491, -9*w + 65],\ [491, 491, 12*w - 89],\ [503, 503, w + 39],\ [503, 503, w + 463],\ [529, 23, -23],\ [541, 541, 9*w - 73],\ [541, 541, 12*w - 95],\ [547, 547, w + 160],\ [547, 547, w + 386],\ [557, 557, w + 41],\ [557, 557, w + 515],\ [563, 563, w + 123],\ [563, 563, w + 439],\ [569, 569, 16*w - 125],\ [569, 569, 7*w - 59],\ [577, 577, w + 266],\ [577, 577, w + 310],\ [587, 587, w + 64],\ [587, 587, w + 522],\ [593, 593, w + 263],\ [593, 593, w + 329],\ [599, 599, -w - 25],\ [599, 599, w - 26],\ [619, 619, -7*w - 40],\ [619, 619, 7*w - 47],\ [631, 631, -5*w - 23],\ [631, 631, 5*w - 28],\ [643, 643, w + 44],\ [643, 643, w + 598],\ [653, 653, w + 276],\ [653, 653, w + 376],\ [661, 661, -3*w - 32],\ [661, 661, 3*w - 35],\ [673, 673, w + 45],\ [673, 673, w + 627],\ [683, 683, w + 94],\ [683, 683, w + 588],\ [701, 701, 4*w - 41],\ [701, 701, -4*w - 37],\ [727, 727, w + 217],\ [727, 727, w + 509],\ [739, 739, 4*w - 11],\ [739, 739, -4*w - 7],\ [757, 757, w + 218],\ [757, 757, w + 538],\ [761, 761, -w - 28],\ [761, 761, w - 29],\ [769, 769, 17*w - 127],\ [769, 769, -10*w + 71],\ [773, 773, w + 100],\ [773, 773, w + 672],\ [811, 811, -4*w - 1],\ [811, 811, 4*w - 5],\ [821, 821, 6*w - 35],\ [821, 821, -6*w - 29],\ [823, 823, w + 118],\ [823, 823, w + 704],\ [827, 827, w + 235],\ [827, 827, w + 591],\ [829, 829, 13*w - 95],\ [829, 829, 14*w - 103],\ [841, 29, -29],\ [859, 859, 6*w - 55],\ [859, 859, -6*w - 49],\ [881, 881, -5*w - 44],\ [881, 881, 5*w - 49],\ [883, 883, w + 321],\ [883, 883, w + 561],\ [887, 887, w + 236],\ [887, 887, w + 650],\ [911, 911, 7*w + 55],\ [911, 911, 7*w - 62],\ [937, 937, w + 53],\ [937, 937, w + 883],\ [941, 941, -w - 31],\ [941, 941, w - 32],\ [967, 967, w + 226],\ [967, 967, w + 740],\ [977, 977, w + 460],\ [977, 977, w + 516],\ [997, 997, w + 122],\ [997, 997, w + 874]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^10 + 19*x^8 + 123*x^6 + 301*x^4 + 196*x^2 + 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -1/12*e^8 - 7/6*e^6 - 59/12*e^4 - 13/2*e^2 - 7/3, 1/12*e^8 + 7/6*e^6 + 59/12*e^4 + 11/2*e^2 - 2/3, 1/12*e^9 + 5/3*e^7 + 137/12*e^5 + 59/2*e^3 + 58/3*e, 1/12*e^9 + 5/3*e^7 + 131/12*e^5 + 25*e^3 + 37/3*e, -1/12*e^9 - 17/12*e^7 - 101/12*e^5 - 79/4*e^3 - 34/3*e, 1/6*e^9 + 37/12*e^7 + 113/6*e^5 + 161/4*e^3 + 50/3*e, 1/6*e^9 + 37/12*e^7 + 119/6*e^5 + 201/4*e^3 + 113/3*e, 1/6*e^9 + 31/12*e^7 + 77/6*e^5 + 83/4*e^3 + 8/3*e, 1/12*e^8 + 5/3*e^6 + 131/12*e^4 + 25*e^2 + 34/3, -1/6*e^8 - 11/6*e^6 - 16/3*e^4 - 5*e^2 - 14/3, 1/4*e^8 + 4*e^6 + 81/4*e^4 + 67/2*e^2 + 8, 1/4*e^9 + 4*e^7 + 89/4*e^5 + 103/2*e^3 + 41*e, 2*e^3 + 12*e, -1/4*e^9 - 19/4*e^7 - 123/4*e^5 - 301/4*e^3 - 49*e, 1/4*e^9 + 21/4*e^7 + 147/4*e^5 + 371/4*e^3 + 50*e, -1/3*e^8 - 14/3*e^6 - 62/3*e^4 - 31*e^2 - 16/3, -1/12*e^8 - 7/6*e^6 - 53/12*e^4 - 4*e^2 - 4/3, -1/6*e^8 - 7/3*e^6 - 31/3*e^4 - 31/2*e^2 - 2/3, 1/3*e^8 + 14/3*e^6 + 115/6*e^4 + 41/2*e^2 - 14/3, 1/2*e^9 + 19/2*e^7 + 123/2*e^5 + 295/2*e^3 + 78*e, 1/4*e^9 + 4*e^7 + 81/4*e^5 + 65/2*e^3 + 3*e, 1/4*e^9 + 19/4*e^7 + 121/4*e^5 + 295/4*e^3 + 59*e, 1/4*e^9 + 21/4*e^7 + 147/4*e^5 + 383/4*e^3 + 69*e, 1/4*e^8 + 4*e^6 + 85/4*e^4 + 81/2*e^2 + 8, 1/6*e^8 + 13/3*e^6 + 191/6*e^4 + 70*e^2 + 62/3, -1/3*e^8 - 25/6*e^6 - 85/6*e^4 - 14*e^2 - 52/3, 1/4*e^9 + 17/4*e^7 + 97/4*e^5 + 197/4*e^3 + 14*e, -1/4*e^9 - 15/4*e^7 - 67/4*e^5 - 73/4*e^3 + 11*e, 7/12*e^8 + 55/6*e^6 + 563/12*e^4 + 82*e^2 + 64/3, 1/12*e^8 + 2/3*e^6 + 11/12*e^4 + e^2 + 4/3, -1/3*e^9 - 83/12*e^7 - 277/6*e^5 - 421/4*e^3 - 133/3*e, -1/12*e^9 - 17/12*e^7 - 89/12*e^5 - 43/4*e^3 + 17/3*e, -5/12*e^9 - 19/3*e^7 - 373/12*e^5 - 109/2*e^3 - 89/3*e, -1/6*e^9 - 29/6*e^7 - 121/3*e^5 - 112*e^3 - 200/3*e, -1/4*e^9 - 19/4*e^7 - 119/4*e^5 - 269/4*e^3 - 42*e, -1/4*e^9 - 15/4*e^7 - 77/4*e^5 - 163/4*e^3 - 32*e, -5/12*e^9 - 28/3*e^7 - 817/12*e^5 - 357/2*e^3 - 344/3*e, 5/6*e^9 + 91/6*e^7 + 559/6*e^5 + 423/2*e^3 + 337/3*e, -1/2*e^9 - 39/4*e^7 - 125/2*e^5 - 573/4*e^3 - 82*e, -1/4*e^9 - 13/4*e^7 - 47/4*e^5 - 35/4*e^3 + 5*e, 3/4*e^8 + 21/2*e^6 + 179/4*e^4 + 57*e^2 + 6, -1/4*e^8 - 7/2*e^6 - 69/4*e^4 - 39*e^2 - 22, 2/3*e^8 + 25/3*e^6 + 82/3*e^4 + 10*e^2 - 28/3, -7/12*e^8 - 23/3*e^6 - 377/12*e^4 - 47*e^2 - 34/3, 3/4*e^7 + 23/2*e^5 + 207/4*e^3 + 61*e, -1/2*e^9 - 35/4*e^7 - 103/2*e^5 - 449/4*e^3 - 68*e, -7/12*e^9 - 67/6*e^7 - 869/12*e^5 - 347/2*e^3 - 262/3*e, 5/12*e^9 + 16/3*e^7 + 223/12*e^5 + 12*e^3 - 1/3*e, 1/4*e^8 + 5/2*e^6 + 17/4*e^4 - 9*e^2 + 2, -e^6 - 9*e^4 - 12*e^2 + 10, 1/3*e^8 + 25/6*e^6 + 97/6*e^4 + 19*e^2 - 44/3, 1/12*e^8 + 11/3*e^6 + 395/12*e^4 + 82*e^2 + 70/3, 5/4*e^7 + 16*e^5 + 227/4*e^3 + 45*e, -3/4*e^9 - 51/4*e^7 - 295/4*e^5 - 655/4*e^3 - 98*e, -3/4*e^9 - 53/4*e^7 - 321/4*e^5 - 751/4*e^3 - 120*e, -1/2*e^9 - 35/4*e^7 - 107/2*e^5 - 509/4*e^3 - 74*e, -1/2*e^9 - 43/4*e^7 - 78*e^5 - 851/4*e^3 - 149*e, 1/2*e^9 + 39/4*e^7 + 129/2*e^5 + 633/4*e^3 + 90*e, 1/12*e^9 + 13/6*e^7 + 239/12*e^5 + 145/2*e^3 + 223/3*e, 1/3*e^9 + 31/6*e^7 + 83/3*e^5 + 113/2*e^3 + 40/3*e, -2/3*e^8 - 47/6*e^6 - 125/6*e^4 + 10*e^2 + 34/3, 5/6*e^8 + 35/3*e^6 + 289/6*e^4 + 52*e^2 - 38/3, 1/2*e^8 + 13/2*e^6 + 28*e^4 + 57*e^2 + 40, -1/2*e^8 - 9/2*e^6 - e^4 + 45*e^2 + 18, 5/12*e^8 + 19/3*e^6 + 349/12*e^4 + 79/2*e^2 + 8/3, 1/2*e^8 + 4*e^6 - 9/2*e^4 - 59*e^2 - 26, -1/4*e^8 - 7/2*e^6 - 69/4*e^4 - 41*e^2 - 26, 1/3*e^9 + 11/3*e^7 + 61/6*e^5 + 19/2*e^3 + 76/3*e, -5/12*e^9 - 19/3*e^7 - 373/12*e^5 - 103/2*e^3 - 17/3*e, -3/4*e^8 - 19/2*e^6 - 125/4*e^4 - 21/2*e^2 + 4, 1/4*e^8 + 9/2*e^6 + 123/4*e^4 + 165/2*e^2 + 32, 2/3*e^9 + 133/12*e^7 + 365/6*e^5 + 495/4*e^3 + 224/3*e, -1/3*e^9 - 89/12*e^7 - 331/6*e^5 - 623/4*e^3 - 394/3*e, 5/12*e^8 + 19/3*e^6 + 319/12*e^4 + 19*e^2 - 82/3, -1/3*e^8 - 17/3*e^6 - 98/3*e^4 - 68*e^2 - 118/3, -1/3*e^8 - 31/6*e^6 - 71/3*e^4 - 65/2*e^2 - 76/3, 5/12*e^8 + 16/3*e^6 + 229/12*e^4 + 37/2*e^2 + 14/3, -1/3*e^8 - 25/6*e^6 - 97/6*e^4 - 25*e^2 - 4/3, 5/12*e^8 + 29/6*e^6 + 193/12*e^4 + 20*e^2 + 50/3, 3/2*e^6 + 35/2*e^4 + 45*e^2 - 10, -1/4*e^8 - 2*e^6 + 5/4*e^4 + 30*e^2 + 40, 7/4*e^8 + 49/2*e^6 + 421/4*e^4 + 281/2*e^2 + 16, 1/4*e^8 + 1/2*e^6 - 77/4*e^4 - 157/2*e^2 - 32, 1/2*e^9 + 41/4*e^7 + 70*e^5 + 693/4*e^3 + 94*e, -1/4*e^9 - 15/4*e^7 - 77/4*e^5 - 147/4*e^3, 1/2*e^8 + 3*e^6 - 35/2*e^4 - 102*e^2 - 40, -1/2*e^8 - 13/2*e^6 - 24*e^4 - 18*e^2 + 22, -e^9 - 37/2*e^7 - 116*e^5 - 555/2*e^3 - 190*e, -1/2*e^9 - 17/2*e^7 - 49*e^5 - 105*e^3 - 47*e, -5/12*e^8 - 29/6*e^6 - 169/12*e^4 - 8*e^2 - 56/3, 7/12*e^8 + 37/6*e^6 + 179/12*e^4 + e^2 - 80/3, -3/4*e^9 - 15*e^7 - 401/4*e^5 - 245*e^3 - 156*e, -3/4*e^9 - 29/2*e^7 - 373/4*e^5 - 423/2*e^3 - 83*e, -1/4*e^8 - 1/2*e^6 + 63/4*e^4 + 55*e^2 + 28, 1/3*e^8 + 23/3*e^6 + 325/6*e^4 + 237/2*e^2 + 82/3, -1/6*e^8 - 7/3*e^6 - 25/3*e^4 - 9/2*e^2 - 14/3, -5/12*e^9 - 41/6*e^7 - 433/12*e^5 - 67*e^3 - 104/3*e, 5/6*e^9 + 73/6*e^7 + 170/3*e^5 + 93*e^3 + 139/3*e, 7/12*e^9 + 161/12*e^7 + 1199/12*e^5 + 1055/4*e^3 + 454/3*e, -7/6*e^9 - 241/12*e^7 - 713/6*e^5 - 1097/4*e^3 - 545/3*e, 1/2*e^7 + 7*e^5 + 39/2*e^3 - 22*e, -3/4*e^9 - 15*e^7 - 415/4*e^5 - 547/2*e^3 - 187*e, 3/2*e^8 + 39/2*e^6 + 141/2*e^4 + 105/2*e^2 - 12, -1/4*e^8 - 1/2*e^6 + 85/4*e^4 + 189/2*e^2 + 38, 5/12*e^9 + 97/12*e^7 + 649/12*e^5 + 547/4*e^3 + 215/3*e, -1/3*e^9 - 113/12*e^7 - 469/6*e^5 - 879/4*e^3 - 415/3*e, -1/3*e^9 - 26/3*e^7 - 415/6*e^5 - 387/2*e^3 - 427/3*e, 2/3*e^9 + 77/6*e^7 + 503/6*e^5 + 209*e^3 + 458/3*e, -5/4*e^9 - 95/4*e^7 - 601/4*e^5 - 1363/4*e^3 - 169*e, e^9 + 65/4*e^7 + 175/2*e^5 + 677/4*e^3 + 70*e, -3/4*e^8 - 15/2*e^6 - 47/4*e^4 + 31*e^2 + 40, -3/2*e^6 - 25/2*e^4 - 17*e^2 + 4, -1/12*e^8 - 8/3*e^6 - 227/12*e^4 - 31*e^2 + 14/3, -13/12*e^8 - 41/3*e^6 - 611/12*e^4 - 58*e^2 + 20/3, -e^8 - 15*e^6 - 73*e^4 - 127*e^2 - 46, 3/4*e^8 + 19/2*e^6 + 127/4*e^4 + 11*e^2 - 18, 7/12*e^9 + 41/3*e^7 + 1235/12*e^5 + 565/2*e^3 + 652/3*e, -11/12*e^9 - 52/3*e^7 - 1339/12*e^5 - 531/2*e^3 - 437/3*e, -e^9 - 63/4*e^7 - 171/2*e^5 - 771/4*e^3 - 156*e, 3/4*e^9 + 51/4*e^7 + 283/4*e^5 + 535/4*e^3 + 33*e, 1/12*e^8 - 1/3*e^6 - 199/12*e^4 - 145/2*e^2 - 62/3, 7/12*e^8 + 31/6*e^6 + 65/12*e^4 - 19/2*e^2 + 106/3, 11/12*e^9 + 193/12*e^7 + 1159/12*e^5 + 879/4*e^3 + 389/3*e, 7/6*e^9 + 253/12*e^7 + 767/6*e^5 + 1141/4*e^3 + 497/3*e, -1/3*e^9 - 49/6*e^7 - 188/3*e^5 - 325/2*e^3 - 190/3*e, 11/12*e^9 + 95/6*e^7 + 1087/12*e^5 + 181*e^3 + 143/3*e, 2/3*e^8 + 53/6*e^6 + 109/3*e^4 + 97/2*e^2 - 64/3, 5/12*e^8 + 23/6*e^6 + 79/12*e^4 + 13/2*e^2 + 122/3, 1/4*e^9 + 5*e^7 + 119/4*e^5 + 49*e^3 - 6*e, 1/4*e^9 + 13/2*e^7 + 215/4*e^5 + 331/2*e^3 + 161*e, -1/3*e^8 - 13/6*e^6 + 23/6*e^4 + 22*e^2 + 38/3, -1/3*e^8 - 13/6*e^6 + 59/6*e^4 + 58*e^2 + 20/3, -13/12*e^9 - 209/12*e^7 - 1103/12*e^5 - 673/4*e^3 - 133/3*e, 5/12*e^9 + 49/12*e^7 + 49/12*e^5 - 121/4*e^3 - 31/3*e, 1/2*e^8 + 15/2*e^6 + 83/2*e^4 + 195/2*e^2 + 54, -e^8 - 27/2*e^6 - 50*e^4 - 57/2*e^2 + 30, -1/4*e^8 - 9/2*e^6 - 135/4*e^4 - 207/2*e^2 - 42, -5/4*e^8 - 17*e^6 - 269/4*e^4 - 147/2*e^2 - 34, -7/6*e^9 - 241/12*e^7 - 349/3*e^5 - 1011/4*e^3 - 473/3*e, -11/12*e^9 - 199/12*e^7 - 1189/12*e^5 - 831/4*e^3 - 248/3*e, -7/2*e^6 - 77/2*e^4 - 103*e^2 - 36, 5/4*e^8 + 20*e^6 + 411/4*e^4 + 176*e^2 + 44, 7/12*e^8 + 29/3*e^6 + 593/12*e^4 + 76*e^2 + 58/3, -5/12*e^8 - 35/6*e^6 - 313/12*e^4 - 35*e^2 + 64/3, 3/4*e^9 + 15*e^7 + 411/4*e^5 + 523/2*e^3 + 151*e, -1/4*e^9 - 7/2*e^7 - 59/4*e^5 - 43/2*e^3 - 14*e, -13/12*e^9 - 62/3*e^7 - 1601/12*e^5 - 653/2*e^3 - 697/3*e, 2/3*e^9 + 34/3*e^7 + 190/3*e^5 + 123*e^3 + 134/3*e, 5/3*e^8 + 131/6*e^6 + 515/6*e^4 + 106*e^2 + 38/3, -13/12*e^8 - 73/6*e^6 - 341/12*e^4 + 28*e^2 + 68/3, 1/4*e^8 + 9/2*e^6 + 121/4*e^4 + 83*e^2 + 34, 3/4*e^8 + 25/2*e^6 + 255/4*e^4 + 95*e^2 + 4, -7/4*e^8 - 51/2*e^6 - 467/4*e^4 - 169*e^2 - 24, -1/3*e^8 - 31/6*e^6 - 169/6*e^4 - 54*e^2 - 22/3, 11/12*e^8 + 28/3*e^6 + 205/12*e^4 - 21*e^2 + 38/3, 3/2*e^9 + 55/2*e^7 + 172*e^5 + 403*e^3 + 230*e, 3/4*e^9 + 13*e^7 + 299/4*e^5 + 313/2*e^3 + 91*e, e^9 + 20*e^7 + 263/2*e^5 + 605/2*e^3 + 132*e, -e^7 - 19/2*e^5 - 39/2*e^3 - 7*e, 5/12*e^8 + 10/3*e^6 - 65/12*e^4 - 49*e^2 + 56/3, -7/12*e^8 - 14/3*e^6 + 55/12*e^4 + 65*e^2 + 134/3, -9/4*e^9 - 165/4*e^7 - 1019/4*e^5 - 2311/4*e^3 - 288*e, -1/4*e^9 - 29/4*e^7 - 249/4*e^5 - 761/4*e^3 - 166*e, 2/3*e^8 + 25/3*e^6 + 76/3*e^4 - 7*e^2 - 52/3, -7/12*e^8 - 23/3*e^6 - 401/12*e^4 - 70*e^2 - 184/3, -13/12*e^9 - 62/3*e^7 - 1649/12*e^5 - 715/2*e^3 - 787/3*e, -7/12*e^9 - 38/3*e^7 - 1067/12*e^5 - 441/2*e^3 - 406/3*e, 19/12*e^9 + 347/12*e^7 + 2183/12*e^5 + 1757/4*e^3 + 793/3*e, 19/12*e^9 + 329/12*e^7 + 1883/12*e^5 + 1255/4*e^3 + 358/3*e, 11/12*e^9 + 157/12*e^7 + 691/12*e^5 + 319/4*e^3 + 44/3*e, 17/12*e^9 + 349/12*e^7 + 2395/12*e^5 + 1977/4*e^3 + 782/3*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([3,3,-w + 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]