/* 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, 0, -6, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([10, 10, -w^3 + 3*w^2 + 3*w]) primes_array = [ [2, 2, w],\ [5, 5, -w^3 + 3*w^2 + 2*w - 1],\ [5, 5, w - 1],\ [11, 11, w^3 - 2*w^2 - 5*w - 1],\ [13, 13, -w^3 + 3*w^2 + 3*w - 1],\ [17, 17, 2*w^3 - 5*w^2 - 9*w + 5],\ [23, 23, -w^3 + 3*w^2 + 4*w - 5],\ [25, 5, -w^2 + 3*w + 1],\ [29, 29, -2*w^3 + 6*w^2 + 5*w - 1],\ [31, 31, -w^2 + 2*w + 1],\ [43, 43, -w^3 + 3*w^2 + 2*w - 3],\ [43, 43, -w^3 + 2*w^2 + 6*w - 3],\ [59, 59, 2*w^3 - 6*w^2 - 7*w + 7],\ [73, 73, 3*w^3 - 6*w^2 - 16*w - 5],\ [79, 79, -5*w^3 + 14*w^2 + 20*w - 17],\ [81, 3, -3],\ [83, 83, -w - 3],\ [83, 83, w^2 - 3*w - 7],\ [89, 89, w^2 - 2*w - 7],\ [101, 101, -2*w^3 + 5*w^2 + 8*w - 3],\ [101, 101, 5*w^3 - 14*w^2 - 19*w + 17],\ [103, 103, -w^3 + w^2 + 8*w + 3],\ [103, 103, 2*w^3 - 5*w^2 - 7*w - 1],\ [109, 109, 3*w^3 - 7*w^2 - 14*w + 1],\ [109, 109, -w^2 + 4*w + 1],\ [127, 127, -3*w^3 + 8*w^2 + 11*w - 7],\ [127, 127, -w^3 + 2*w^2 + 7*w + 1],\ [137, 137, -2*w^3 + 5*w^2 + 9*w - 1],\ [139, 139, -w^3 + 3*w^2 + 2*w - 5],\ [149, 149, w^3 - 3*w^2 - w - 1],\ [149, 149, -2*w^3 + 4*w^2 + 10*w + 1],\ [163, 163, w^3 - 4*w^2 - w + 11],\ [163, 163, 2*w^3 - 4*w^2 - 10*w + 1],\ [173, 173, -4*w^3 + 11*w^2 + 15*w - 13],\ [179, 179, 2*w^2 - 6*w - 3],\ [179, 179, -w^3 + 2*w^2 + 7*w - 1],\ [181, 181, w^3 - 10*w - 9],\ [181, 181, -3*w^3 + 8*w^2 + 14*w - 7],\ [199, 199, w^3 - 8*w - 9],\ [211, 211, 2*w - 3],\ [223, 223, -w^3 + 4*w^2 + w - 5],\ [227, 227, w^3 - 2*w^2 - 5*w - 5],\ [227, 227, -3*w^3 + 7*w^2 + 14*w - 5],\ [227, 227, -4*w^3 + 10*w^2 + 19*w - 7],\ [227, 227, w^2 - 4*w - 5],\ [239, 239, -2*w^3 + 5*w^2 + 7*w - 3],\ [251, 251, -w^3 + 4*w^2 + 2*w - 11],\ [251, 251, 2*w^3 - 6*w^2 - 6*w + 1],\ [257, 257, -3*w^3 + 6*w^2 + 16*w + 3],\ [263, 263, -w^3 + 3*w^2 + 3*w - 7],\ [263, 263, w^3 - 4*w^2 + 7],\ [269, 269, -2*w^3 + 6*w^2 + 6*w - 11],\ [269, 269, -w^3 + 4*w^2 - w - 5],\ [271, 271, -3*w^3 + 9*w^2 + 11*w - 11],\ [271, 271, w^2 - 5*w - 3],\ [277, 277, w^3 - 4*w^2 - 2*w + 7],\ [277, 277, -2*w^3 + 7*w^2 + 8*w - 9],\ [281, 281, 2*w^2 - 4*w - 5],\ [281, 281, 2*w^3 - 4*w^2 - 11*w + 1],\ [283, 283, -3*w^3 + 8*w^2 + 14*w - 11],\ [293, 293, w^3 - 2*w^2 - 6*w + 5],\ [307, 307, -2*w^3 + 5*w^2 + 7*w - 1],\ [313, 313, 2*w^3 - 4*w^2 - 9*w + 3],\ [313, 313, -2*w^3 + 5*w^2 + 6*w - 3],\ [317, 317, -2*w^3 + 5*w^2 + 9*w + 1],\ [347, 347, 4*w^3 - 9*w^2 - 21*w + 3],\ [347, 347, 4*w^3 - 10*w^2 - 18*w + 5],\ [349, 349, -2*w^3 + 6*w^2 + 7*w - 3],\ [353, 353, -w^3 + 12*w + 5],\ [353, 353, w^2 - w - 7],\ [359, 359, -2*w^3 + 5*w^2 + 6*w - 7],\ [359, 359, -w^3 + 4*w^2 - 11],\ [373, 373, -w^3 + 2*w^2 + 6*w + 5],\ [379, 379, -3*w^3 + 8*w^2 + 14*w - 5],\ [397, 397, 6*w^3 - 14*w^2 - 30*w + 7],\ [401, 401, 3*w^3 - 5*w^2 - 20*w - 7],\ [409, 409, 3*w^3 - 7*w^2 - 12*w - 3],\ [409, 409, 2*w^2 - 4*w - 7],\ [421, 421, w^3 - 11*w - 11],\ [421, 421, -w^3 + 3*w^2 + w - 7],\ [431, 431, 2*w^3 - 3*w^2 - 14*w - 3],\ [431, 431, -4*w^3 + 11*w^2 + 15*w - 15],\ [433, 433, 2*w^3 - 5*w^2 - 6*w + 1],\ [439, 439, -4*w^3 + 11*w^2 + 13*w - 9],\ [439, 439, -2*w^3 + 6*w^2 + 8*w - 11],\ [443, 443, 2*w - 5],\ [443, 443, -6*w^3 + 17*w^2 + 24*w - 21],\ [449, 449, w^3 - 2*w^2 - 8*w + 3],\ [449, 449, 4*w^3 - 12*w^2 - 15*w + 17],\ [457, 457, -2*w^3 + 4*w^2 + 9*w + 5],\ [461, 461, -3*w^3 + 6*w^2 + 15*w - 1],\ [461, 461, -3*w^3 + 7*w^2 + 17*w + 1],\ [463, 463, -4*w^3 + 10*w^2 + 17*w - 7],\ [463, 463, -3*w^3 + 10*w^2 + 9*w - 17],\ [467, 467, 4*w^3 - 8*w^2 - 21*w - 7],\ [467, 467, w^3 - 6*w^2 + 3*w + 19],\ [479, 479, -w^3 + w^2 + 8*w + 1],\ [487, 487, -4*w^3 + 11*w^2 + 14*w - 15],\ [487, 487, 3*w^3 - 6*w^2 - 15*w - 5],\ [491, 491, w^2 - 5],\ [503, 503, 2*w^3 - 5*w^2 - 11*w + 3],\ [523, 523, -3*w^3 + 7*w^2 + 13*w + 1],\ [547, 547, 2*w^3 - 3*w^2 - 12*w - 3],\ [547, 547, 2*w^3 - 3*w^2 - 12*w - 9],\ [557, 557, -3*w^3 + 9*w^2 + 9*w - 11],\ [557, 557, -6*w^3 + 15*w^2 + 27*w - 13],\ [563, 563, -3*w^3 + 7*w^2 + 14*w - 7],\ [571, 571, -5*w^3 + 11*w^2 + 26*w - 3],\ [593, 593, -6*w^3 + 15*w^2 + 26*w - 7],\ [599, 599, 7*w^3 - 18*w^2 - 29*w + 17],\ [599, 599, -2*w^3 + 6*w^2 + 8*w - 3],\ [601, 601, -7*w^3 + 20*w^2 + 28*w - 23],\ [601, 601, 3*w^3 - 7*w^2 - 12*w + 5],\ [607, 607, 5*w^3 - 12*w^2 - 24*w + 9],\ [607, 607, w^3 - 5*w^2 + 15],\ [613, 613, -w^3 + 5*w^2 - 2*w - 9],\ [617, 617, w^3 + w^2 - 13*w - 15],\ [617, 617, 2*w^3 - 4*w^2 - 8*w - 1],\ [619, 619, 5*w^3 - 10*w^2 - 27*w - 7],\ [619, 619, 2*w^3 - 4*w^2 - 10*w + 3],\ [643, 643, w^3 - 5*w^2 + 5*w + 5],\ [643, 643, 4*w^3 - 9*w^2 - 20*w - 1],\ [643, 643, 3*w^3 - 7*w^2 - 14*w - 3],\ [643, 643, -3*w^3 + 7*w^2 + 13*w - 3],\ [647, 647, -3*w^3 + 8*w^2 + 12*w - 3],\ [653, 653, -w^3 + 9*w + 13],\ [653, 653, -w^3 + 4*w^2 + 3*w - 9],\ [653, 653, -2*w^3 + 7*w^2 + 4*w - 7],\ [653, 653, w^2 - 4*w - 9],\ [673, 673, -w^3 + 5*w^2 + w - 13],\ [683, 683, w^3 - 3*w^2 - 3*w - 3],\ [701, 701, 5*w^3 - 13*w^2 - 23*w + 9],\ [709, 709, 2*w^3 - w^2 - 19*w - 15],\ [719, 719, -5*w^3 + 15*w^2 + 20*w - 19],\ [727, 727, -w - 5],\ [751, 751, -4*w^3 + 10*w^2 + 15*w - 9],\ [751, 751, 6*w^3 - 17*w^2 - 22*w + 21],\ [757, 757, 3*w^3 - 6*w^2 - 18*w - 1],\ [757, 757, -w^3 + 4*w^2 + 2*w + 1],\ [769, 769, w^2 - 2*w + 3],\ [773, 773, 5*w^3 - 12*w^2 - 23*w + 9],\ [787, 787, 2*w^3 - 4*w^2 - 5*w + 1],\ [797, 797, -3*w^3 + 8*w^2 + 10*w - 3],\ [809, 809, 2*w^2 - 5*w - 1],\ [809, 809, 4*w^3 - 6*w^2 - 28*w - 11],\ [827, 827, 4*w^3 - 8*w^2 - 22*w - 1],\ [827, 827, 2*w^2 - 6*w - 9],\ [829, 829, -w^3 + w^2 + 10*w + 1],\ [839, 839, -2*w^3 + 2*w^2 + 16*w + 7],\ [877, 877, 6*w^3 - 16*w^2 - 25*w + 13],\ [877, 877, w^3 - w^2 - 7*w + 1],\ [881, 881, 2*w^3 - 2*w^2 - 15*w - 7],\ [881, 881, -4*w^3 + 7*w^2 + 26*w + 9],\ [887, 887, 2*w^3 - 4*w^2 - 13*w - 1],\ [907, 907, 5*w^3 - 11*w^2 - 26*w - 1],\ [907, 907, 5*w^3 - 13*w^2 - 22*w + 9],\ [907, 907, w^3 - 5*w^2 + 2*w + 13],\ [907, 907, -2*w^3 + 7*w^2 + 5*w - 7],\ [911, 911, -3*w^3 + 8*w^2 + 9*w - 5],\ [919, 919, -2*w^3 + 7*w^2 + 6*w - 3],\ [919, 919, -3*w^3 + 8*w^2 + 15*w - 7],\ [947, 947, -2*w^3 + 5*w^2 + 12*w - 9],\ [947, 947, -w^2 + w - 3],\ [947, 947, 3*w^3 - 6*w^2 - 14*w - 7],\ [947, 947, -w^3 + 5*w^2 - w - 17],\ [953, 953, 3*w^3 - 9*w^2 - 9*w + 13],\ [953, 953, -4*w^3 + 13*w^2 + 11*w - 23],\ [967, 967, 8*w^3 - 20*w^2 - 33*w + 19]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - 3*x^2 - 6*x + 9 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -1, 1/3*e^2 - 2*e + 1, -2/3*e^2 + e + 6, -1/3*e^2 + e + 2, -5/3*e^2 + 3*e + 7, -1, -1/3*e^2 + 2*e + 2, -1/3*e^2 - e + 7, -1/3*e^2 + 1, 1/3*e^2 - 2*e, e^2 + e - 12, e^2 - e + 2, -1/3*e^2 + 3*e + 4, e^2 - 3*e - 8, e^2 - 5*e - 3, -5/3*e^2 - e + 16, -1/3*e^2 + 3*e + 2, e + 12, 2*e^2 - 3*e - 12, 7/3*e^2 - 6*e - 6, e^2 - 7*e - 1, 4/3*e^2 - e, 8/3*e^2 - 4*e - 14, -1/3*e^2 + 7*e - 2, -1/3*e^2 + 3*e - 14, -5/3*e^2 + 7*e + 7, 4/3*e^2 - 3*e - 21, 1/3*e^2 + 3*e - 5, -7/3*e^2 + 6*e + 14, -1/3*e^2 + 7*e + 1, -e^2 + 4*e + 2, 5/3*e^2 + e - 13, -e - 3, 3*e^2 - 7*e - 15, -1/3*e^2 + 4*e + 4, -10/3*e^2 + 2*e + 25, -e^2 - 3*e + 14, -e^2 + 6*e + 8, 16/3*e^2 - 11*e - 18, -1/3*e^2 + 5*e - 16, -7/3*e^2 + 6*e + 14, 5/3*e^2 - 2*e + 5, e^2 - 5*e + 3, 3*e^2 - 6*e - 3, 14/3*e^2 - 7*e - 16, 3*e + 6, -3*e^2 + 8*e + 18, -13/3*e^2 + 8*e + 23, -e^2 + 4*e - 9, -4*e^2 + 2*e + 33, -4/3*e^2 + 5*e - 4, -10/3*e^2 - e + 28, -5/3*e^2 - 2*e + 15, -10/3*e^2 - 2*e + 37, e^2 + 6*e - 10, 5/3*e^2 - 2*e - 4, -2/3*e^2 - 3*e + 22, -e^2 + 8, 8/3*e^2 - 9*e - 19, 1/3*e^2 + 3, 2/3*e^2 + 22, -5*e^2 + 3*e + 35, -5*e^2 + 7*e + 27, 4/3*e^2 - 7*e + 4, 13/3*e^2 - 9*e - 14, -1/3*e^2 - 3*e - 5, -2*e + 27, -3*e^2 + 4*e + 9, 1/3*e^2 - e - 5, -4/3*e^2 + 12*e + 5, 8/3*e^2 - 10*e + 1, 2*e^2 + 6*e - 28, -e^2 - e + 5, -1/3*e^2 + e - 22, 2*e^2 - 2*e + 2, -e^2 + 5*e - 16, 1/3*e^2 + e - 27, 8/3*e^2 - 10*e + 1, -14/3*e^2 + 5*e + 25, 22/3*e^2 - 17*e - 29, 5/3*e^2 + 2*e - 14, -10/3*e^2 + 9*e - 2, 10/3*e^2 - 12*e - 3, -7/3*e^2 + 6*e + 8, -5/3*e^2 + 7*e + 13, -10*e + 18, -5*e^2 + 4*e + 21, -1/3*e^2 - 4*e - 8, -2/3*e^2 - 3*e + 4, -e^2 + 12*e - 6, 4*e^2 - 5*e - 13, 10/3*e^2 - 6*e, -5/3*e^2 + 6*e - 11, -4/3*e^2 + 8*e - 13, -4*e^2 + 5*e + 24, 13/3*e^2 - 6*e - 18, 5*e^2 - 14*e - 7, 4*e^2 - 14*e - 15, -11/3*e^2 + 16*e + 13, -14/3*e^2 + 2*e + 39, -e^2 - 3*e + 29, 3*e^2 - 18*e - 10, e^2 - 11*e + 9, 4/3*e^2 + 4*e - 5, 8/3*e^2 - 10*e - 7, 5*e^2 - 9*e - 25, -6*e^2 + 14*e + 9, 2/3*e^2 + 11, -e^2 - 4*e + 30, -5/3*e^2 + 4*e - 6, 2*e^2 - 7*e - 34, 17/3*e^2 - 8*e - 20, -7/3*e^2 + 2*e - 17, 11/3*e^2 - 13*e + 7, -e^2 + e + 3, 2*e^2 + e - 18, -7/3*e^2 + 11*e - 5, 5/3*e^2 - 10*e + 16, 5/3*e^2 + 5*e - 23, 4/3*e^2 + 4*e - 15, 23/3*e^2 - 18*e - 32, -3*e^2 + 5*e + 11, -3*e^2 - 2*e + 24, -2*e^2 + 3*e - 6, -4*e^2 + 11*e + 24, -4*e^2 + 13*e + 15, -e^2 + 4*e, 5*e^2 - 14*e - 22, -5*e^2 + 4*e + 18, 11/3*e^2 - 5*e - 28, 26/3*e^2 - 21*e - 32, 16/3*e^2 - 12*e - 8, -1/3*e^2 + 11*e - 8, 1/3*e^2 + 2*e + 18, 5/3*e^2 - 6*e - 44, -7/3*e^2 - e + 7, 14/3*e^2 - 5*e - 26, -10/3*e^2 + 4*e + 22, -6*e^2 + 11*e + 24, 20/3*e^2 - 20*e - 38, -5/3*e^2 - 3*e + 16, -4*e^2 + 17*e, -2/3*e^2 - 6*e + 19, 19/3*e^2 - 8*e - 29, -2*e^2 + 2*e - 15, 5*e^2 - 12*e - 25, 14/3*e^2 - 13*e - 10, 7*e - 28, 1/3*e^2 + 14*e - 6, -1/3*e^2 + 9*e + 5, -8*e^2 + 16*e + 21, 23/3*e^2 - 15*e - 43, -3*e^2 + 15*e - 13, 2/3*e^2 - e + 22, -2*e^2 + 15*e + 14, 5/3*e^2 + 1, -e^2 + 18*e - 3, 6*e^2 - 12*e + 8, e^2 + 4*e - 16, -17/3*e^2 + 2*e + 46, 3*e^2 - 15, -6*e^2 + 9*e + 21, -13/3*e^2 + 16*e + 41, 7*e - 6, 3*e^2 - 4*e - 3, -5*e^2 + 12*e + 32] 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([5, 5, w - 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]