/* 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([2, 1, -4, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [2, 2, w],\ [8, 2, -w^3 + w^2 + 4*w - 1],\ [11, 11, w^3 - 2*w^2 - 2*w + 1],\ [23, 23, -w^3 + 4*w + 1],\ [23, 23, -w^2 + 2*w + 3],\ [31, 31, w^3 - 2*w^2 - w + 3],\ [37, 37, -w^3 + 3*w + 3],\ [37, 37, -2*w^3 + 3*w^2 + 6*w - 3],\ [41, 41, w^3 - 2*w^2 - 3*w + 1],\ [41, 41, -2*w^3 + 2*w^2 + 6*w - 1],\ [43, 43, -w^2 + w + 5],\ [47, 47, 2*w^2 - 3*w - 5],\ [53, 53, -w^3 + 3*w^2 + w - 7],\ [53, 53, -2*w^2 + 2*w + 5],\ [59, 59, 2*w^2 - w - 7],\ [61, 61, 2*w^2 - w - 3],\ [61, 61, 2*w^2 - w - 5],\ [67, 67, 2*w^3 - 2*w^2 - 7*w + 3],\ [67, 67, 2*w^3 - 2*w^2 - 7*w - 1],\ [71, 71, 2*w^3 - 4*w^2 - 4*w + 7],\ [81, 3, -3],\ [97, 97, -w^3 + 2*w^2 - 3],\ [101, 101, -w^3 + w^2 + 3*w + 3],\ [101, 101, 3*w^3 - 4*w^2 - 9*w + 3],\ [103, 103, -w^3 + 2*w^2 + w - 5],\ [103, 103, -2*w^3 + 4*w^2 + 4*w - 3],\ [107, 107, -w^3 + w^2 + 5*w - 1],\ [127, 127, 2*w^3 - 2*w^2 - 8*w + 1],\ [127, 127, -w^3 + 5*w - 1],\ [137, 137, w^3 - 6*w - 1],\ [149, 149, w^3 - 2*w - 3],\ [151, 151, 2*w^3 - 8*w - 3],\ [157, 157, 2*w^3 - w^2 - 6*w - 5],\ [163, 163, -w^3 + 6*w - 3],\ [173, 173, -4*w^3 + 5*w^2 + 12*w - 5],\ [181, 181, -2*w^3 + 5*w^2 + 5*w - 9],\ [191, 191, w^3 + w^2 - 3*w - 5],\ [193, 193, 2*w^3 - 3*w^2 - 6*w + 1],\ [197, 197, 3*w^3 - 4*w^2 - 10*w + 3],\ [199, 199, -3*w^2 + 4*w + 3],\ [199, 199, 2*w^3 - 4*w^2 - 6*w + 11],\ [211, 211, -2*w^2 + 5*w + 5],\ [223, 223, 2*w^2 + w - 5],\ [227, 227, w^2 + w - 5],\ [229, 229, w^3 - 4*w^2 - w + 9],\ [229, 229, 2*w^2 - 2*w - 9],\ [233, 233, -w^3 - 2*w^2 + 6*w + 7],\ [251, 251, -w^3 + 3*w + 7],\ [257, 257, -2*w^3 + w^2 + 7*w - 1],\ [269, 269, 3*w^3 - 4*w^2 - 11*w + 7],\ [277, 277, 3*w^3 - 4*w^2 - 7*w + 5],\ [277, 277, 3*w^3 - 2*w^2 - 12*w + 1],\ [277, 277, w^3 - 4*w^2 + 7],\ [277, 277, w^2 - 7],\ [281, 281, 3*w^2 - w - 7],\ [283, 283, -w^3 + 2*w^2 + 2*w - 7],\ [293, 293, -3*w^3 + 2*w^2 + 10*w - 1],\ [307, 307, -3*w^2 + 3*w + 7],\ [307, 307, 2*w^3 - 3*w^2 - 8*w + 5],\ [311, 311, -3*w^3 + 5*w^2 + 7*w - 3],\ [313, 313, w^2 - 2*w + 3],\ [313, 313, 2*w^3 - w^2 - 8*w + 1],\ [317, 317, -w^3 - w^2 + 5*w + 1],\ [317, 317, -w^3 + w^2 + 3*w - 5],\ [331, 331, w^3 - w - 3],\ [331, 331, -w^3 + 2*w^2 - 5],\ [337, 337, -2*w^3 + 4*w^2 + 3*w - 7],\ [347, 347, 3*w^2 - w - 5],\ [349, 349, 2*w^2 - 7],\ [353, 353, 3*w^2 - 2*w - 5],\ [367, 367, 3*w^3 - 2*w^2 - 9*w - 1],\ [389, 389, 2*w^3 - 5*w^2 - 6*w + 11],\ [401, 401, -w^3 + 4*w^2 + w - 7],\ [401, 401, 3*w^2 - 2*w - 7],\ [409, 409, 3*w^3 - 4*w^2 - 7*w + 3],\ [409, 409, -2*w^3 + 6*w^2 + w - 7],\ [421, 421, 2*w^3 - 7*w - 1],\ [421, 421, 4*w^3 - 7*w^2 - 11*w + 9],\ [433, 433, w^3 - 3*w^2 - 5*w + 7],\ [439, 439, -w^3 + 7*w - 1],\ [457, 457, -2*w^3 + 4*w^2 + 5*w - 1],\ [457, 457, 2*w^3 - 4*w^2 - 6*w + 3],\ [467, 467, -2*w^3 - w^2 + 12*w + 5],\ [467, 467, -3*w^3 + 5*w^2 + 9*w - 5],\ [479, 479, w^2 + 2*w - 5],\ [479, 479, 3*w^3 - 2*w^2 - 11*w + 1],\ [499, 499, 2*w^3 - w^2 - 7*w + 3],\ [499, 499, w^3 - 4*w^2 + 2*w + 7],\ [499, 499, 4*w^3 - 3*w^2 - 13*w - 1],\ [499, 499, -w^2 + w - 3],\ [503, 503, 4*w^3 - 6*w^2 - 11*w + 5],\ [509, 509, 3*w^3 - 12*w - 5],\ [509, 509, w^2 - w - 7],\ [521, 521, -4*w^3 + 8*w^2 + 11*w - 11],\ [521, 521, w^3 - 2*w - 7],\ [523, 523, 4*w^3 - 7*w^2 - 7*w + 5],\ [523, 523, 2*w^3 - 3*w^2 - 3*w + 7],\ [529, 23, -2*w^3 + 3*w^2 + 2*w + 3],\ [557, 557, -4*w^3 + 4*w^2 + 13*w - 5],\ [557, 557, -2*w^3 + 2*w^2 + 9*w - 3],\ [563, 563, 4*w^3 - 4*w^2 - 15*w + 3],\ [569, 569, 2*w^2 - 4*w - 7],\ [571, 571, 3*w^3 - 4*w^2 - 6*w + 3],\ [571, 571, w^3 - 7*w - 1],\ [577, 577, -w^2 - 3],\ [593, 593, 2*w^3 - 2*w^2 - 9*w + 1],\ [593, 593, 2*w^2 - 9],\ [599, 599, 5*w^3 - 6*w^2 - 15*w + 5],\ [601, 601, -2*w^3 + 6*w^2 + 4*w - 5],\ [601, 601, 3*w^3 - 5*w^2 - 11*w + 7],\ [613, 613, -5*w^3 + 7*w^2 + 13*w - 3],\ [613, 613, 4*w^3 - 5*w^2 - 10*w + 7],\ [617, 617, 4*w^3 - 2*w^2 - 13*w - 3],\ [625, 5, -5],\ [631, 631, 4*w^2 - 2*w - 5],\ [631, 631, 4*w^3 - 5*w^2 - 11*w + 5],\ [641, 641, w^3 + 2*w^2 - 4*w - 9],\ [641, 641, 2*w^3 - 5*w^2 - 2*w + 9],\ [643, 643, w^3 - 4*w - 7],\ [647, 647, -w - 5],\ [653, 653, -w^3 + w^2 + 5*w - 7],\ [673, 673, -2*w^3 + w^2 + 8*w - 3],\ [673, 673, -w^3 + 5*w - 5],\ [673, 673, -3*w^3 + 7*w^2 + 5*w - 13],\ [673, 673, -w^2 + 5*w - 1],\ [677, 677, -2*w^3 + 6*w^2 + 5*w - 9],\ [701, 701, 3*w^3 - 2*w^2 - 7*w - 1],\ [701, 701, 4*w^3 - 6*w^2 - 9*w + 7],\ [709, 709, 4*w^3 - 5*w^2 - 10*w + 1],\ [719, 719, w^3 - 2*w^2 - 3*w - 3],\ [727, 727, -3*w^3 + 4*w^2 + 11*w - 1],\ [727, 727, -2*w^3 + 6*w^2 + w - 11],\ [733, 733, -2*w^3 + 2*w^2 + 7*w + 5],\ [733, 733, -w^3 + 6*w^2 - 4*w - 9],\ [743, 743, -4*w^3 + 6*w^2 + 8*w - 7],\ [743, 743, -2*w^3 + 11*w + 3],\ [757, 757, -3*w^3 + 4*w^2 + 9*w + 1],\ [769, 769, -2*w^3 + 9*w + 1],\ [787, 787, -4*w^3 + 7*w^2 + 10*w - 7],\ [787, 787, -3*w^2 + w + 11],\ [787, 787, w^3 - w^2 - 3*w - 5],\ [787, 787, -2*w^3 + w^2 + 10*w + 1],\ [797, 797, 4*w^3 - 6*w^2 - 9*w + 5],\ [797, 797, -2*w^3 + 3*w^2 + 5*w - 9],\ [809, 809, -w^3 + 4*w^2 + 4*w - 9],\ [811, 811, -w^3 + 3*w^2 - w - 7],\ [811, 811, w^2 - 5*w - 1],\ [829, 829, -2*w^3 + 4*w^2 + 2*w - 7],\ [839, 839, 2*w^2 + w - 7],\ [839, 839, 4*w^3 - 4*w^2 - 12*w + 7],\ [841, 29, 2*w^2 - 6*w - 3],\ [841, 29, -4*w^3 + 6*w^2 + 14*w - 9],\ [857, 857, -2*w^3 + 6*w^2 + 3*w - 11],\ [881, 881, 4*w^2 - 4*w - 7],\ [883, 883, 4*w^3 - 4*w^2 - 11*w - 1],\ [883, 883, -3*w^3 + 6*w^2 + 7*w - 5],\ [919, 919, -w^3 + 5*w^2 - w - 7],\ [929, 929, -w^3 + 5*w^2 + w - 11],\ [941, 941, -3*w^3 + w^2 + 13*w + 1],\ [947, 947, 2*w^3 - 3*w^2 - 9*w + 3],\ [947, 947, 2*w^3 - 6*w^2 - 4*w + 11],\ [953, 953, -2*w^3 + 7*w - 1],\ [971, 971, 3*w^3 - w^2 - 11*w - 1],\ [977, 977, -w^3 + 5*w^2 - w - 9],\ [983, 983, w^3 - w - 5],\ [983, 983, 3*w^3 - 2*w^2 - 10*w + 3],\ [991, 991, -2*w^3 - 2*w^2 + 9*w + 5],\ [991, 991, -6*w^3 + 8*w^2 + 17*w - 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, 1, -2, -4, -4, 4, 10, 10, 0, 0, 2, -8, -2, -2, -10, 6, 6, -2, -2, -12, -2, 0, 18, -10, -8, 20, -4, -12, -12, -16, -18, 12, 18, -4, -22, -14, -4, -2, 2, -24, 4, 2, -28, 4, 6, -22, 24, 28, -8, 18, 26, -2, -2, -2, 16, 4, -14, 28, -14, -24, 6, 6, -18, -18, -4, -4, 30, 12, 14, 18, -24, -30, -4, -18, -10, 4, 30, -26, 0, -8, 10, 24, 20, -36, -24, 4, -4, 10, 10, -4, 0, 6, 34, -10, 18, 6, -22, -2, -2, -2, 18, 38, 26, -44, 32, -36, 34, 12, 28, 0, -2, -2, -40, 24, 44, -40, 26, -44, -14, 4, -46, -26, 44, -40, 30, 6, 2, 2, 10, -8, -28, -28, -22, 34, 16, 44, -26, 28, -10, 46, 4, 4, 42, 42, -30, 42, 28, -10, -28, 0, -40, 58, -24, 42, 2, 44, -4, -36, 18, -18, 10, 2, 6, 12, 4, -24, -16, -16] 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, w])] = -1 AL_eigenvalues[ZF.ideal([8, 2, -w^3 + w^2 + 4*w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]