/* 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, 4, 0, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([23, 23, -w^2 + 2]) primes_array = [ [4, 2, -w^4 + w^3 + 6*w^2 - w - 3],\ [7, 7, w^4 - w^3 - 6*w^2 + w + 2],\ [8, 2, -2*w^4 + 3*w^3 + 10*w^2 - 5*w - 3],\ [11, 11, w^4 - w^3 - 6*w^2 + 2],\ [13, 13, -w^4 + 2*w^3 + 5*w^2 - 5*w - 3],\ [23, 23, -w^2 + 2],\ [23, 23, -w + 2],\ [31, 31, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\ [43, 43, -w^3 + 2*w^2 + 3*w - 2],\ [47, 47, -w^4 + 2*w^3 + 5*w^2 - 4*w - 4],\ [53, 53, w^2 - w - 4],\ [53, 53, -w^4 + 2*w^3 + 4*w^2 - 4*w - 3],\ [53, 53, -w^3 + w^2 + 4*w - 1],\ [71, 71, w^4 - w^3 - 7*w^2 + 3*w + 6],\ [71, 71, -2*w^4 + 2*w^3 + 11*w^2 - 2],\ [71, 71, -w^4 + 3*w^3 + 3*w^2 - 9*w - 1],\ [73, 73, 2*w^4 - 4*w^3 - 9*w^2 + 10*w + 5],\ [79, 79, 2*w^4 - 3*w^3 - 11*w^2 + 4*w + 8],\ [83, 83, -3*w^4 + 5*w^3 + 15*w^2 - 9*w - 10],\ [89, 89, w^3 - w^2 - 5*w + 1],\ [89, 89, 2*w^4 - 4*w^3 - 8*w^2 + 9*w + 1],\ [121, 11, -w^4 + w^3 + 6*w^2 + w - 4],\ [121, 11, 3*w^4 - 5*w^3 - 15*w^2 + 11*w + 6],\ [131, 131, 4*w^4 - 6*w^3 - 21*w^2 + 9*w + 10],\ [139, 139, -w^4 + 7*w^2 + 4*w - 2],\ [139, 139, w^4 - w^3 - 7*w^2 + w + 7],\ [149, 149, w^3 - 2*w^2 - 5*w + 4],\ [149, 149, -w^3 + 2*w^2 + 4*w - 2],\ [151, 151, 3*w^4 - 5*w^3 - 14*w^2 + 8*w + 2],\ [151, 151, 3*w^4 - 4*w^3 - 16*w^2 + 5*w + 5],\ [151, 151, -3*w^4 + 4*w^3 + 17*w^2 - 5*w - 9],\ [157, 157, w^4 - 8*w^2 - 3*w + 5],\ [157, 157, 3*w^4 - 4*w^3 - 16*w^2 + 4*w + 5],\ [163, 163, 3*w^4 - 5*w^3 - 14*w^2 + 7*w + 6],\ [173, 173, -w - 3],\ [179, 179, w^4 - 2*w^3 - 4*w^2 + 6*w],\ [193, 193, -w^4 + w^3 + 5*w^2 + w - 3],\ [193, 193, w^4 - 2*w^3 - 4*w^2 + 3*w - 1],\ [193, 193, 2*w^4 - 3*w^3 - 9*w^2 + 4*w + 3],\ [197, 197, 2*w^4 - 3*w^3 - 9*w^2 + 4*w],\ [197, 197, -4*w^4 + 6*w^3 + 20*w^2 - 9*w - 8],\ [197, 197, 2*w^4 - 2*w^3 - 10*w^2 - 2*w + 3],\ [211, 211, -4*w^4 + 6*w^3 + 21*w^2 - 10*w - 9],\ [227, 227, 6*w^4 - 8*w^3 - 32*w^2 + 10*w + 15],\ [227, 227, -3*w^4 + 5*w^3 + 15*w^2 - 9*w - 9],\ [227, 227, w^3 - 2*w^2 - 5*w + 2],\ [243, 3, -3],\ [263, 263, 2*w^4 - 2*w^3 - 12*w^2 + w + 4],\ [269, 269, 2*w^4 - 2*w^3 - 11*w^2 - 2*w + 5],\ [271, 271, 4*w^4 - 4*w^3 - 24*w^2 + 2*w + 13],\ [271, 271, 2*w^4 - 2*w^3 - 11*w^2 + 6],\ [277, 277, 3*w^4 - 4*w^3 - 15*w^2 + 4*w + 4],\ [283, 283, 4*w^4 - 6*w^3 - 20*w^2 + 9*w + 9],\ [283, 283, 5*w^4 - 7*w^3 - 26*w^2 + 8*w + 13],\ [307, 307, w^2 - 2*w - 5],\ [313, 313, -4*w^4 + 8*w^3 + 17*w^2 - 18*w - 5],\ [331, 331, -2*w^4 + 4*w^3 + 10*w^2 - 11*w - 7],\ [337, 337, 4*w^4 - 6*w^3 - 19*w^2 + 6*w + 7],\ [337, 337, -w^4 + 2*w^3 + 4*w^2 - 3*w - 5],\ [349, 349, w^4 - 3*w^3 - 4*w^2 + 12*w + 1],\ [353, 353, w^4 - 3*w^3 - 4*w^2 + 12*w + 2],\ [359, 359, -3*w^4 + 5*w^3 + 14*w^2 - 6*w - 7],\ [359, 359, -2*w^4 + 3*w^3 + 11*w^2 - 7*w - 7],\ [367, 367, w^4 - 2*w^3 - 5*w^2 + 7*w + 5],\ [373, 373, 4*w^4 - 5*w^3 - 21*w^2 + 4*w + 7],\ [373, 373, -2*w^4 + 3*w^3 + 9*w^2 - 4*w + 2],\ [373, 373, 3*w^4 - 4*w^3 - 15*w^2 + 3*w + 5],\ [379, 379, w^4 - 8*w^2 - 3*w + 1],\ [379, 379, 3*w^4 - 4*w^3 - 16*w^2 + 6*w + 5],\ [401, 401, -3*w^4 + 4*w^3 + 16*w^2 - 6*w - 6],\ [401, 401, w^2 - 3*w - 4],\ [409, 409, 6*w^4 - 10*w^3 - 28*w^2 + 19*w + 6],\ [419, 419, 2*w^4 - 3*w^3 - 10*w^2 + 2*w + 4],\ [421, 421, -4*w^4 + 7*w^3 + 20*w^2 - 14*w - 8],\ [421, 421, 4*w^4 - 5*w^3 - 21*w^2 + 3*w + 5],\ [433, 433, -3*w^4 + 5*w^3 + 13*w^2 - 7*w],\ [433, 433, 2*w^4 - 3*w^3 - 11*w^2 + 4*w + 9],\ [443, 443, 2*w^4 - 2*w^3 - 12*w^2 + 3*w + 9],\ [443, 443, -2*w^4 + 4*w^3 + 9*w^2 - 8*w - 5],\ [457, 457, -w^4 + w^3 + 7*w^2 - 3*w - 7],\ [461, 461, w^4 - 2*w^3 - 3*w^2 + 4*w - 2],\ [463, 463, 3*w^4 - 3*w^3 - 17*w^2 - w + 4],\ [467, 467, -4*w^4 + 6*w^3 + 21*w^2 - 10*w - 15],\ [467, 467, -4*w^4 + 7*w^3 + 19*w^2 - 12*w - 6],\ [467, 467, -3*w^4 + 4*w^3 + 17*w^2 - 6*w - 8],\ [487, 487, -3*w^4 + 6*w^3 + 14*w^2 - 14*w - 7],\ [487, 487, -2*w^4 + 3*w^3 + 12*w^2 - 9*w - 6],\ [491, 491, -3*w^4 + 4*w^3 + 15*w^2 - 3*w - 9],\ [491, 491, 4*w^4 - 7*w^3 - 20*w^2 + 16*w + 12],\ [499, 499, -4*w^4 + 7*w^3 + 20*w^2 - 14*w - 10],\ [499, 499, 5*w^4 - 8*w^3 - 25*w^2 + 14*w + 8],\ [503, 503, 2*w^4 - 3*w^3 - 10*w^2 + 5*w],\ [503, 503, w^4 - w^3 - 5*w^2 + w - 1],\ [509, 509, -w^4 + 9*w^2 + 2*w - 8],\ [521, 521, -2*w^4 + 4*w^3 + 10*w^2 - 10*w - 5],\ [523, 523, 4*w^4 - 8*w^3 - 18*w^2 + 18*w + 7],\ [523, 523, w^4 - 2*w^3 - 6*w^2 + 6*w + 5],\ [541, 541, -2*w^4 + 3*w^3 + 9*w^2 - 6*w - 5],\ [547, 547, -2*w^4 + 2*w^3 + 12*w^2 + w - 7],\ [547, 547, 4*w^4 - 6*w^3 - 20*w^2 + 11*w + 7],\ [547, 547, -3*w^4 + 6*w^3 + 14*w^2 - 16*w - 8],\ [557, 557, 4*w^4 - 7*w^3 - 19*w^2 + 15*w + 9],\ [563, 563, -3*w^4 + 5*w^3 + 15*w^2 - 7*w - 9],\ [569, 569, -5*w^4 + 9*w^3 + 22*w^2 - 18*w - 4],\ [569, 569, -w^4 + 7*w^2 + 3*w - 3],\ [571, 571, w^2 - 5],\ [571, 571, -w^3 + w^2 + 5*w - 3],\ [577, 577, -2*w^4 + 4*w^3 + 9*w^2 - 8*w - 6],\ [587, 587, -3*w^4 + 3*w^3 + 17*w^2 - w - 3],\ [593, 593, w^4 - 8*w^2 - w + 5],\ [599, 599, -w^4 + 8*w^2 + 4*w - 6],\ [599, 599, w^4 - 2*w^3 - 3*w^2 + 3*w - 3],\ [607, 607, -7*w^4 + 11*w^3 + 35*w^2 - 19*w - 12],\ [607, 607, 4*w^4 - 6*w^3 - 21*w^2 + 8*w + 8],\ [607, 607, 2*w^4 - 4*w^3 - 10*w^2 + 13*w + 5],\ [613, 613, -3*w^4 + 5*w^3 + 16*w^2 - 11*w - 6],\ [613, 613, 4*w^4 - 6*w^3 - 19*w^2 + 9*w + 2],\ [619, 619, -5*w^4 + 9*w^3 + 22*w^2 - 18*w - 3],\ [619, 619, -3*w^4 + 4*w^3 + 16*w^2 - 3*w - 7],\ [619, 619, -3*w^4 + 4*w^3 + 16*w^2 - 4*w - 12],\ [641, 641, -5*w^4 + 8*w^3 + 25*w^2 - 13*w - 9],\ [647, 647, -w^4 + 2*w^3 + 6*w^2 - 7*w - 5],\ [659, 659, -6*w^4 + 11*w^3 + 28*w^2 - 24*w - 12],\ [659, 659, -4*w^4 + 6*w^3 + 22*w^2 - 11*w - 14],\ [661, 661, -4*w^4 + 5*w^3 + 21*w^2 - 2*w - 9],\ [661, 661, -3*w^4 + 5*w^3 + 14*w^2 - 7*w - 2],\ [677, 677, w^3 - w^2 - 2*w + 4],\ [677, 677, 2*w^4 - 2*w^3 - 11*w^2 - 1],\ [683, 683, 3*w^4 - 3*w^3 - 16*w^2 - 2*w + 6],\ [691, 691, -2*w^3 + 3*w^2 + 10*w - 4],\ [701, 701, 2*w^4 - 4*w^3 - 7*w^2 + 7*w],\ [709, 709, 4*w^4 - 6*w^3 - 21*w^2 + 10*w + 8],\ [727, 727, 2*w^3 - 2*w^2 - 11*w + 1],\ [733, 733, 7*w^4 - 12*w^3 - 33*w^2 + 23*w + 7],\ [743, 743, 4*w^4 - 7*w^3 - 20*w^2 + 15*w + 14],\ [751, 751, -5*w^4 + 7*w^3 + 28*w^2 - 12*w - 17],\ [757, 757, -w^2 - w + 4],\ [761, 761, -5*w^4 + 8*w^3 + 24*w^2 - 15*w - 5],\ [769, 769, -w^4 + 7*w^2 + 6*w - 4],\ [769, 769, w^3 - 8*w],\ [773, 773, -5*w^4 + 7*w^3 + 27*w^2 - 9*w - 12],\ [787, 787, 3*w^4 - 5*w^3 - 16*w^2 + 10*w + 9],\ [787, 787, 3*w^4 - 6*w^3 - 12*w^2 + 12*w + 5],\ [797, 797, -2*w^4 + 2*w^3 + 13*w^2 - 8],\ [809, 809, -3*w^4 + 6*w^3 + 12*w^2 - 11*w - 1],\ [823, 823, w^4 - 3*w^3 - 3*w^2 + 9*w + 5],\ [823, 823, 5*w^4 - 8*w^3 - 24*w^2 + 11*w + 11],\ [827, 827, -3*w^4 + 4*w^3 + 15*w^2 - 4*w - 6],\ [827, 827, 2*w^4 - 4*w^3 - 10*w^2 + 12*w + 3],\ [859, 859, w^4 - 3*w^3 - 2*w^2 + 6*w - 3],\ [859, 859, -w^4 + w^3 + 6*w^2 + 2*w - 4],\ [877, 877, 4*w^4 - 7*w^3 - 20*w^2 + 16*w + 8],\ [877, 877, -2*w^4 + 2*w^3 + 11*w^2 - 2*w - 3],\ [883, 883, -3*w^4 + 6*w^3 + 14*w^2 - 14*w - 11],\ [907, 907, 3*w^4 - 5*w^3 - 14*w^2 + 6*w + 6],\ [911, 911, 3*w^4 - 6*w^3 - 12*w^2 + 13*w + 3],\ [929, 929, -3*w^4 + 4*w^3 + 18*w^2 - 7*w - 9],\ [929, 929, -7*w^4 + 11*w^3 + 36*w^2 - 19*w - 18],\ [941, 941, 2*w^4 - 3*w^3 - 12*w^2 + 9*w + 8],\ [947, 947, 3*w^4 - 5*w^3 - 16*w^2 + 10*w + 8],\ [947, 947, -5*w^4 + 7*w^3 + 26*w^2 - 9*w - 8],\ [947, 947, 4*w^4 - 6*w^3 - 21*w^2 + 11*w + 6],\ [953, 953, -w^4 + 2*w^3 + 2*w^2 + 2*w - 3],\ [953, 953, -2*w^4 + 3*w^3 + 11*w^2 - 6*w - 11],\ [971, 971, 4*w^4 - 7*w^3 - 20*w^2 + 15*w + 8],\ [977, 977, -w^4 + 7*w^2 + 2*w - 4],\ [977, 977, -4*w^4 + 6*w^3 + 22*w^2 - 11*w - 11],\ [991, 991, 4*w^4 - 6*w^3 - 19*w^2 + 6*w + 10],\ [997, 997, -5*w^4 + 10*w^3 + 22*w^2 - 22*w - 12],\ [997, 997, 7*w^4 - 12*w^3 - 32*w^2 + 22*w + 7],\ [997, 997, -3*w^4 + 3*w^3 + 19*w^2 - 5*w - 10]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [-1, -4, 3, -4, -2, -1, 8, 0, -8, -4, -2, -2, -6, 12, 12, -8, 2, 0, -4, 6, 6, 10, -10, 12, 12, 0, 10, -6, 8, 16, -20, -14, 14, 4, -18, 20, -14, 14, -6, -10, -18, 18, -20, 0, -20, -12, 16, 16, -18, 28, -8, 2, 4, -4, 20, -10, 16, -2, -14, -10, 2, -24, 8, 0, 30, -22, -26, 4, -12, 14, 10, 14, 0, -10, -10, -18, -14, -24, -24, 18, 6, 4, -20, -8, -28, 16, -12, -32, -36, 24, 12, -16, 0, -14, 18, 28, -4, -30, 12, 36, -12, -10, -4, 6, 6, 40, 8, 30, 24, 18, -24, 16, 24, 16, -32, -34, -42, -4, 20, 8, -6, -32, 36, -40, 10, -42, 6, -6, -28, 20, -26, -14, -12, 6, 0, -40, 10, -2, -14, -10, 22, -12, 12, -30, -26, 40, -32, -12, 20, 36, 12, 42, -30, 36, 16, 52, -34, -50, 42, 20, 20, 52, 6, -2, -20, 26, -42, 48, -38, -22, -46] 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 + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]