/* 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([-1, 1, 6, -2, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([23, 23, -w^2 + w + 2]) primes_array = [ [5, 5, -w^4 + w^3 + 4*w^2 - 2*w - 1],\ [7, 7, w^5 - w^4 - 5*w^3 + 3*w^2 + 4*w - 3],\ [19, 19, w^5 - w^4 - 5*w^3 + 2*w^2 + 4*w],\ [23, 23, -w^2 + w + 2],\ [25, 5, w^5 + w^4 - 6*w^3 - 7*w^2 + 4*w + 2],\ [41, 41, 2*w^5 - w^4 - 10*w^3 + 6*w - 1],\ [47, 47, w^3 - w^2 - 4*w],\ [53, 53, w^5 - 5*w^3 - 3*w^2 + w + 3],\ [59, 59, 2*w^5 - 11*w^3 - 4*w^2 + 8*w],\ [61, 61, w^2 - 2*w - 2],\ [64, 2, -2],\ [67, 67, -w^4 + 6*w^2 + w - 3],\ [67, 67, -2*w^4 + w^3 + 10*w^2 - 6],\ [73, 73, w^5 - 5*w^3 - 2*w^2 + 2*w - 2],\ [73, 73, -w^5 - w^4 + 7*w^3 + 6*w^2 - 8*w - 2],\ [79, 79, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 1],\ [83, 83, 2*w^5 - 11*w^3 - 4*w^2 + 6*w],\ [89, 89, -2*w^5 + 11*w^3 + 4*w^2 - 7*w - 2],\ [97, 97, w^5 - w^4 - 5*w^3 + 3*w^2 + 2*w - 2],\ [97, 97, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1],\ [97, 97, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 5*w + 4],\ [101, 101, w^5 - 6*w^3 - 3*w^2 + 6*w + 2],\ [103, 103, -w^5 - w^4 + 5*w^3 + 8*w^2 - 5],\ [107, 107, 2*w^4 - w^3 - 9*w^2 + w + 2],\ [109, 109, 2*w^5 - 10*w^3 - 5*w^2 + 4*w + 1],\ [113, 113, -w^4 + 6*w^2 + 2*w - 3],\ [125, 5, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 6],\ [127, 127, 2*w^5 - w^4 - 11*w^3 + 10*w],\ [131, 131, 2*w^5 - w^4 - 11*w^3 + 9*w + 1],\ [137, 137, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 9*w + 5],\ [139, 139, -w^5 - 2*w^4 + 7*w^3 + 11*w^2 - 5*w - 5],\ [157, 157, 2*w^5 - 10*w^3 - 5*w^2 + 5*w + 1],\ [167, 167, -2*w^5 + 10*w^3 + 5*w^2 - 3*w + 1],\ [173, 173, 4*w^5 + 2*w^4 - 22*w^3 - 19*w^2 + 10*w + 5],\ [179, 179, 2*w^5 - w^4 - 11*w^3 + 10*w + 1],\ [179, 179, 2*w^5 - 11*w^3 - 5*w^2 + 8*w + 1],\ [181, 181, -4*w^5 - w^4 + 22*w^3 + 14*w^2 - 12*w - 4],\ [191, 191, w^4 + w^3 - 6*w^2 - 4*w + 2],\ [191, 191, -3*w^5 + 18*w^3 + 5*w^2 - 15*w - 1],\ [193, 193, -w^4 + w^3 + 6*w^2 - 2*w - 7],\ [197, 197, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 12*w + 1],\ [197, 197, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 7*w - 5],\ [197, 197, -w^5 + w^4 + 6*w^3 - 3*w^2 - 9*w + 1],\ [211, 211, -2*w^5 + w^4 + 10*w^3 - w^2 - 4*w + 4],\ [223, 223, 2*w^5 - 10*w^3 - 6*w^2 + 5*w + 4],\ [227, 227, w^5 - 7*w^3 - w^2 + 9*w],\ [227, 227, -2*w^5 + w^4 + 9*w^3 - 2*w + 1],\ [229, 229, w^5 + w^4 - 5*w^3 - 6*w^2 - 2*w - 2],\ [233, 233, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 8*w - 4],\ [269, 269, 3*w^5 - 16*w^3 - 6*w^2 + 8*w],\ [269, 269, -2*w^5 + 12*w^3 + 3*w^2 - 12*w - 1],\ [277, 277, 3*w^5 - w^4 - 17*w^3 - w^2 + 16*w],\ [289, 17, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 10*w - 3],\ [307, 307, w^4 - 5*w^2 - 2*w - 1],\ [307, 307, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 12*w + 3],\ [307, 307, -w^4 - w^3 + 5*w^2 + 6*w - 2],\ [311, 311, -w^4 - w^3 + 7*w^2 + 4*w - 4],\ [311, 311, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w + 2],\ [313, 313, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w],\ [317, 317, 2*w^5 + w^4 - 12*w^3 - 8*w^2 + 7*w],\ [331, 331, 2*w^5 - 2*w^4 - 10*w^3 + 5*w^2 + 8*w - 5],\ [337, 337, w^5 - 6*w^3 - 3*w^2 + 6*w],\ [347, 347, -2*w^5 + 2*w^4 + 10*w^3 - 6*w^2 - 7*w + 2],\ [347, 347, 3*w^5 + w^4 - 17*w^3 - 11*w^2 + 9*w + 2],\ [349, 349, -2*w^5 + 11*w^3 + 5*w^2 - 7*w - 6],\ [367, 367, -w^5 - 2*w^4 + 5*w^3 + 13*w^2 + 2*w - 5],\ [367, 367, 2*w^5 + 2*w^4 - 10*w^3 - 16*w^2 - w + 6],\ [373, 373, 3*w^5 - w^4 - 17*w^3 - w^2 + 15*w + 1],\ [373, 373, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 4],\ [379, 379, 2*w^5 + w^4 - 11*w^3 - 10*w^2 + 5*w + 7],\ [389, 389, w^2 - 5],\ [389, 389, w^5 - 7*w^3 + 8*w],\ [397, 397, -2*w^5 - w^4 + 12*w^3 + 9*w^2 - 7*w - 5],\ [397, 397, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 14*w - 2],\ [397, 397, -2*w^5 - 2*w^4 + 12*w^3 + 13*w^2 - 6*w - 4],\ [419, 419, 2*w^5 + 2*w^4 - 11*w^3 - 14*w^2 + 2*w + 5],\ [419, 419, -w^5 + 4*w^3 + 3*w^2 + w - 2],\ [433, 433, -w^5 + 5*w^3 + 4*w^2 - w - 3],\ [433, 433, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 4],\ [439, 439, 4*w^5 + 2*w^4 - 22*w^3 - 18*w^2 + 9*w + 4],\ [439, 439, w^5 + w^4 - 5*w^3 - 9*w^2 + 6],\ [439, 439, -3*w^5 + 2*w^4 + 16*w^3 - 3*w^2 - 13*w + 2],\ [443, 443, -3*w^5 + w^4 + 15*w^3 + 3*w^2 - 8*w - 2],\ [449, 449, -w^5 + w^4 + 5*w^3 - 4*w^2 - 2*w + 3],\ [449, 449, 3*w^5 - 16*w^3 - 8*w^2 + 9*w + 4],\ [449, 449, w^4 + w^3 - 6*w^2 - 6*w + 5],\ [449, 449, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4],\ [457, 457, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 6*w + 4],\ [457, 457, w^5 - 3*w^4 - 4*w^3 + 12*w^2 + 5*w - 4],\ [461, 461, -2*w^4 + w^3 + 10*w^2 + w - 5],\ [463, 463, -w^5 + w^4 + 6*w^3 - 4*w^2 - 5*w + 4],\ [487, 487, -3*w^4 + 2*w^3 + 14*w^2 - w - 7],\ [487, 487, 3*w^5 + w^4 - 18*w^3 - 10*w^2 + 14*w + 3],\ [491, 491, -3*w^5 + w^4 + 16*w^3 + w^2 - 12*w + 2],\ [491, 491, 3*w^5 - w^4 - 16*w^3 - 2*w^2 + 12*w + 3],\ [499, 499, 4*w^5 - 22*w^3 - 9*w^2 + 15*w + 4],\ [499, 499, w^4 + w^3 - 7*w^2 - 6*w + 6],\ [523, 523, w^4 - 2*w^3 - 4*w^2 + 5*w + 2],\ [529, 23, -w^5 + 7*w^3 + w^2 - 10*w - 1],\ [541, 541, w^5 + 2*w^4 - 8*w^3 - 10*w^2 + 9*w + 3],\ [557, 557, w^5 - w^4 - 5*w^3 + 3*w^2 + 5*w - 5],\ [557, 557, 4*w^5 - 22*w^3 - 8*w^2 + 14*w + 1],\ [563, 563, 2*w^5 - w^4 - 12*w^3 + 2*w^2 + 13*w - 1],\ [563, 563, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 2*w - 6],\ [563, 563, -w^5 + w^4 + 4*w^3 - 2*w^2 + w + 3],\ [569, 569, -3*w^5 + w^4 + 16*w^3 + w^2 - 9*w + 1],\ [599, 599, -w^4 - w^3 + 7*w^2 + 6*w - 5],\ [607, 607, -w^5 + 6*w^3 + 3*w^2 - 7*w - 1],\ [613, 613, -w^5 - 2*w^4 + 7*w^3 + 12*w^2 - 4*w - 7],\ [619, 619, 4*w^5 - w^4 - 22*w^3 - 3*w^2 + 16*w],\ [653, 653, 2*w^5 - 2*w^4 - 10*w^3 + 4*w^2 + 7*w],\ [653, 653, -2*w^5 + w^4 + 12*w^3 - w^2 - 14*w + 1],\ [659, 659, -2*w^5 - w^4 + 11*w^3 + 11*w^2 - 5*w - 7],\ [661, 661, 2*w^5 - 11*w^3 - 3*w^2 + 7*w],\ [661, 661, 2*w^5 - w^4 - 11*w^3 + 11*w + 2],\ [673, 673, -w^4 - w^3 + 5*w^2 + 7*w - 1],\ [677, 677, 3*w^5 - 16*w^3 - 7*w^2 + 10*w + 3],\ [677, 677, w^5 + 3*w^4 - 6*w^3 - 18*w^2 + 12],\ [691, 691, w^5 - w^4 - 4*w^3 + 2*w^2 - 4],\ [709, 709, -w^4 + w^3 + 3*w^2 - 2*w + 2],\ [709, 709, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 6*w - 1],\ [719, 719, 2*w^5 + 2*w^4 - 12*w^3 - 14*w^2 + 8*w + 5],\ [729, 3, -3],\ [739, 739, 4*w^5 - w^4 - 21*w^3 - 5*w^2 + 12*w + 3],\ [743, 743, -2*w^5 - w^4 + 10*w^3 + 11*w^2 - w - 6],\ [751, 751, -3*w^5 + w^4 + 17*w^3 + 2*w^2 - 15*w - 2],\ [751, 751, -2*w^5 + 3*w^4 + 10*w^3 - 11*w^2 - 10*w + 6],\ [757, 757, w^5 - 3*w^4 - 5*w^3 + 13*w^2 + 7*w - 6],\ [761, 761, -w^5 + w^4 + 6*w^3 - 3*w^2 - 6*w - 1],\ [787, 787, -3*w^5 - w^4 + 18*w^3 + 10*w^2 - 15*w - 2],\ [797, 797, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 12*w - 1],\ [797, 797, w^2 - 2*w - 5],\ [809, 809, 2*w^5 - w^4 - 12*w^3 + 3*w^2 + 12*w - 3],\ [827, 827, 4*w^5 - 22*w^3 - 8*w^2 + 15*w + 1],\ [839, 839, 3*w^5 - w^4 - 16*w^3 - 3*w^2 + 10*w + 2],\ [839, 839, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 1],\ [841, 29, -2*w^5 - w^4 + 11*w^3 + 9*w^2 - 7*w - 3],\ [853, 853, w^5 - 7*w^3 - w^2 + 7*w + 1],\ [857, 857, -2*w^5 + w^4 + 10*w^3 - w^2 - 6*w + 4],\ [863, 863, 3*w^5 - w^4 - 17*w^3 - 2*w^2 + 16*w],\ [877, 877, 2*w^5 - 10*w^3 - 5*w^2 + 3*w + 4],\ [887, 887, 2*w^5 + w^4 - 13*w^3 - 8*w^2 + 13*w + 5],\ [887, 887, 3*w^5 - 2*w^4 - 15*w^3 + 3*w^2 + 8*w - 4],\ [907, 907, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 17*w],\ [907, 907, -2*w^5 + 2*w^4 + 10*w^3 - 4*w^2 - 9*w + 1],\ [919, 919, -3*w^5 + 17*w^3 + 5*w^2 - 11*w],\ [929, 929, -3*w^5 + 17*w^3 + 6*w^2 - 11*w],\ [941, 941, 2*w^5 - w^4 - 11*w^3 - w^2 + 9*w + 2],\ [947, 947, -w^4 + 6*w^2 + 4*w - 4],\ [947, 947, -2*w^5 + w^4 + 11*w^3 - 11*w - 3],\ [947, 947, -4*w^5 + w^4 + 22*w^3 + 3*w^2 - 16*w - 1],\ [953, 953, -w^5 - 2*w^4 + 6*w^3 + 13*w^2 - 3*w - 8],\ [953, 953, 2*w^5 + 2*w^4 - 13*w^3 - 14*w^2 + 10*w + 6],\ [961, 31, 2*w^5 + w^4 - 12*w^3 - 10*w^2 + 9*w + 4],\ [967, 967, 2*w^5 - w^4 - 11*w^3 + w^2 + 11*w + 1],\ [967, 967, 2*w^5 - 2*w^4 - 11*w^3 + 6*w^2 + 13*w - 3],\ [983, 983, -4*w^5 + w^4 + 22*w^3 + 4*w^2 - 18*w - 2],\ [991, 991, 3*w^5 - 16*w^3 - 7*w^2 + 7*w + 2],\ [991, 991, -2*w^5 - w^4 + 12*w^3 + 8*w^2 - 11*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + 3*x^2 - 3*x - 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 2, -2*e - 4, 1, e^2 + 2*e - 6, e - 4, -4*e - 6, -e^2 + 4, -2*e^2 - 8*e, -e^2 - 6*e - 6, -2*e^2 - 8*e + 3, -2*e^2 - 4*e + 8, 2*e^2 + 8*e - 4, 4*e^2 + 11*e - 12, 2*e^2 + 9*e - 6, -4*e^2 - 8*e + 12, 4*e^2 + 14*e - 4, -2*e^2 - 5*e + 6, e^2 + 2*e - 10, -5*e^2 - 16*e + 6, e^2 + 6*e - 2, 5*e^2 + 14*e - 6, 2*e^2 + 4*e - 6, -4*e^2 - 10*e, -4*e^2 - 11*e, -5*e^2 - 14*e + 6, 3*e^2 + 8*e - 14, 4*e^2 + 12*e - 8, -2*e^2 - 6*e + 8, -3*e^2 - 8*e + 10, 4*e^2 + 6*e - 12, 2*e^2 - 3*e - 20, 4*e^2 + 14*e - 6, 2*e^2 + 13*e + 6, -6*e^2 - 18*e + 12, -2*e^2 - 2*e + 12, -2*e^2 - 5*e, -22, -4*e^2 - 10*e + 16, e^2 - 2*e - 6, 3*e - 8, -7*e^2 - 26*e + 16, 7*e, 2*e^2 + 2*e - 20, 2*e - 16, 2*e^2 + 6*e - 8, -6*e - 20, e^2 - 2, 2*e^2 + 7*e, 5*e^2 + 4*e - 30, 5*e^2 + 24*e - 8, 2*e^2 + 7*e + 14, -e^2 - 6*e + 10, 10*e + 4, -2*e^2 + 4*e + 12, 10*e^2 + 28*e - 16, -6*e^2 - 18*e + 24, -2*e^2 - 2*e + 12, -8*e^2 - 20*e + 10, 2*e^2 + 9*e - 12, -6*e^2 - 20*e + 16, e^2 - 10, 4*e^2 + 18*e + 2, 4*e^2 + 14*e - 12, -4*e^2 - 7*e + 24, 2*e + 10, 2*e^2 + 4*e + 16, 4*e^2 + 8*e - 34, -7*e^2 - 20*e + 8, 2*e^2 + 10*e + 12, 2*e^2 + 5*e - 16, -11*e^2 - 26*e + 24, -6*e^2 - 19*e + 4, -5*e^2 - 8*e + 8, -2*e^2 - 7*e + 16, -2*e^2 - 8*e + 2, 4*e^2 + 16*e - 12, 4*e^2 + 8*e - 26, 4*e^2 + 17*e - 4, 8*e^2 + 22*e - 28, 2*e^2 + 4*e - 4, -4*e^2 - 8*e + 24, -2*e^2 + 2*e + 22, -e^2 + 6*e - 2, -7*e - 4, -e^2 - 4*e + 6, -2*e^2 - 5*e + 12, 4*e^2 + 5*e - 20, 9*e^2 + 20*e - 26, -4*e^2 - 8*e - 6, 2*e^2 + 6*e - 14, -8*e^2 - 22*e + 8, -2*e^2 - 18*e - 8, -4*e^2 - 16*e + 16, 6*e^2 + 26*e + 4, 6*e + 8, -2*e^2 - 2*e + 16, 4*e^2 - 4*e - 30, 2*e^2 + e - 4, 8*e^2 + 13*e - 26, 2*e^2 + 11*e - 16, 3*e^2 - 2*e - 26, 4*e^2 + 12*e - 36, 6*e^2 + 14*e - 16, -4*e^2 + 34, 9*e^2 + 22*e - 2, 4*e^2 + 12*e - 24, 4*e^2 + 6*e - 24, -6*e^2 - 21*e + 12, -2*e^2 - 4*e - 12, -3*e + 6, -3*e^2 - 18*e + 12, 2*e^2 + 6*e - 14, 7*e + 8, 8*e^2 + 29*e - 16, -8*e^2 - 28*e + 26, -6*e^2 - 22*e + 28, 6*e^2 + 23*e + 2, 10*e^2 + 28*e - 38, 11*e^2 + 30*e - 22, 8*e^2 + 19*e - 36, 4*e^2 + 10*e - 28, 8*e^2 + 17*e - 16, -2*e^2 - 12*e + 24, -2*e^2 - 4*e + 4, -6*e^2 - 26*e + 18, 6*e^2 + 22*e - 16, -7*e^2 - 20*e, -12*e^2 - 37*e + 22, -2*e^2 - 6*e + 42, -10*e^2 - 26*e, -4*e^2 - 5*e, 3*e^2 - 4*e - 22, 12*e^2 + 26*e - 24, 2*e^2 + 8*e - 4, 6*e^2 + 28*e - 20, 4*e^2 + 20*e + 2, -8*e^2 - 23*e - 12, 4*e^2 + 15*e - 44, 2*e^2 - 10*e - 24, 4*e^2 + e - 12, 6*e^2 + 28*e - 24, 6*e^2 + 22*e - 10, -10*e^2 - 8*e + 52, 4*e + 44, 6*e^2 + 6*e - 20, -2*e^2 - 17*e + 6, 12*e^2 + 27*e - 38, 10*e^2 + 34*e - 24, -10*e^2 - 26*e + 26, -10*e^2 - 24*e + 24, -8*e^2 - 3*e + 56, -3*e^2 - 8*e - 10, -6*e^2 - 27*e + 10, 10*e^2 + 38*e - 28, 2*e^2 + 8*e - 16, -10*e^2 - 18*e + 36, -12*e + 4, 8*e^2 + 12*e - 4] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([23, 23, -w^2 + w + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]