/* 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, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([7, 7, -w + 2]) primes_array = [ [5, 5, w + 1],\ [5, 5, w - 1],\ [7, 7, -w + 2],\ [8, 2, 2],\ [11, 11, -w^2 + 2*w + 2],\ [17, 17, -w - 3],\ [17, 17, -w^2 + 2*w + 1],\ [17, 17, -w^2 + w + 4],\ [23, 23, -w^2 + 3],\ [27, 3, -3],\ [29, 29, w^2 - w - 8],\ [29, 29, w^2 - w - 3],\ [29, 29, w^2 - w - 1],\ [31, 31, w^2 - 2],\ [37, 37, 2*w - 5],\ [43, 43, w^2 - 2*w - 5],\ [47, 47, w^2 - 3*w - 2],\ [49, 7, w^2 + w - 4],\ [53, 53, w^2 - 2*w - 6],\ [71, 71, w - 5],\ [79, 79, w^2 - 10],\ [83, 83, -2*w^2 + 3*w + 7],\ [89, 89, 2*w^2 - w - 9],\ [97, 97, 3*w^2 - 4*w - 15],\ [107, 107, w^2 - 2*w - 10],\ [121, 11, w^2 - 5*w + 3],\ [127, 127, 2*w^2 - 3*w - 13],\ [131, 131, 3*w + 4],\ [137, 137, 2*w^2 - 3*w - 12],\ [139, 139, 2*w^2 - 3*w - 6],\ [149, 149, w^2 + w - 9],\ [151, 151, 2*w^2 - 11],\ [157, 157, w^2 + w - 8],\ [163, 163, 2*w^2 + w - 4],\ [173, 173, 3*w - 4],\ [179, 179, -w^2 + 2*w - 3],\ [193, 193, 3*w - 5],\ [197, 197, -w^2 + 3*w - 4],\ [197, 197, w^2 - 3*w - 7],\ [199, 199, 2*w^2 - w - 7],\ [211, 211, w^2 + 2*w - 5],\ [223, 223, 2*w^2 - 3*w - 3],\ [227, 227, w^2 - 2*w - 11],\ [227, 227, 2*w^2 + w - 9],\ [227, 227, 2*w^2 - 5],\ [229, 229, 2*w^2 - 13],\ [233, 233, 4*w - 3],\ [241, 241, 3*w^2 - w - 18],\ [257, 257, 2*w^2 + 2*w - 7],\ [269, 269, w^2 - w - 11],\ [271, 271, 2*w^2 - 2*w - 3],\ [277, 277, 2*w^2 - 4*w - 9],\ [277, 277, 2*w^2 - w - 5],\ [277, 277, w^2 + 2*w - 6],\ [281, 281, w^2 + 4*w - 3],\ [293, 293, 2*w^2 - 3*w - 17],\ [311, 311, w^2 - 12],\ [311, 311, 2*w^2 - 5*w - 6],\ [311, 311, 3*w^2 - 6*w - 10],\ [313, 313, -w^2 + 2*w - 4],\ [337, 337, -w^2 - 2*w - 4],\ [337, 337, w^2 + 2*w - 7],\ [337, 337, w^2 + 2*w - 12],\ [347, 347, -w^2 + 6*w - 6],\ [349, 349, -w - 7],\ [353, 353, 2*w^2 - 19],\ [359, 359, 2*w - 9],\ [383, 383, w^2 + 2*w - 16],\ [383, 383, w^2 + 3*w - 16],\ [383, 383, w^2 + 3*w - 5],\ [389, 389, -3*w^2 + 6*w + 5],\ [397, 397, 3*w^2 - 2*w - 13],\ [397, 397, 2*w^2 - 2*w - 17],\ [397, 397, w^2 + 2*w - 11],\ [401, 401, 3*w - 11],\ [401, 401, w^2 + w - 14],\ [401, 401, w - 8],\ [409, 409, -w^2 - w - 4],\ [409, 409, w^2 - 4*w - 6],\ [409, 409, 3*w^2 - 5*w - 14],\ [421, 421, w^2 + 2*w - 10],\ [433, 433, 4*w^2 - 3*w - 20],\ [439, 439, w^2 - 4*w - 9],\ [443, 443, 3*w^2 - 16],\ [457, 457, w^2 - 4*w - 8],\ [461, 461, 4*w - 7],\ [487, 487, w^2 - 5*w - 13],\ [491, 491, 4*w^2 - 6*w - 19],\ [491, 491, 4*w^2 - 4*w - 19],\ [491, 491, 3*w^2 + w - 8],\ [503, 503, -5*w - 3],\ [509, 509, 2*w^2 + w - 12],\ [523, 523, w^2 - 5*w - 4],\ [523, 523, 2*w^2 - 6*w - 5],\ [523, 523, 2*w^2 - 3*w - 18],\ [529, 23, 3*w^2 - 5*w - 7],\ [541, 541, -w^2 - 2*w - 5],\ [557, 557, 3*w^2 - w - 11],\ [563, 563, 3*w^2 - 5*w - 21],\ [571, 571, 3*w^2 - 17],\ [587, 587, -5*w - 7],\ [599, 599, w^2 + 3*w - 7],\ [607, 607, 3*w^2 - 5*w - 6],\ [613, 613, 4*w^2 - 3*w - 19],\ [617, 617, 2*w^2 + w - 14],\ [631, 631, w^2 - 5*w - 5],\ [631, 631, 4*w^2 - 8*w - 13],\ [631, 631, 2*w^2 + w - 22],\ [641, 641, -w^2 - 5],\ [647, 647, 4*w^2 - w - 17],\ [647, 647, 2*w^2 - 5*w - 9],\ [647, 647, 5*w^2 - 6*w - 23],\ [653, 653, 3*w^2 - 5*w - 5],\ [659, 659, w^2 - 5*w - 12],\ [661, 661, w^2 - 6*w - 3],\ [661, 661, 3*w^2 - 5*w - 4],\ [661, 661, 3*w^2 + w - 6],\ [683, 683, 3*w^2 + w - 5],\ [691, 691, 2*w^2 + 2*w - 21],\ [691, 691, 3*w^2 - 5*w - 17],\ [691, 691, 3*w^2 + 2*w - 12],\ [701, 701, 3*w^2 - 4*w - 8],\ [709, 709, 3*w^2 - 7*w - 9],\ [719, 719, -2*w^2 - w - 3],\ [727, 727, w^2 - 14],\ [733, 733, -3*w - 10],\ [739, 739, 2*w^2 + 3*w - 8],\ [739, 739, 5*w - 6],\ [739, 739, 2*w^2 - w - 19],\ [743, 743, 4*w^2 - 5*w - 16],\ [743, 743, 5*w^2 - 3*w - 25],\ [743, 743, 2*w^2 - 5*w - 10],\ [761, 761, w^2 + 4*w - 6],\ [773, 773, -5*w^2 + 7*w + 25],\ [787, 787, 3*w^2 - 20],\ [797, 797, 3*w^2 + w - 15],\ [809, 809, 4*w^2 - 3*w - 18],\ [821, 821, 2*w^2 - 5*w - 11],\ [821, 821, 2*w^2 + 2*w - 11],\ [821, 821, 3*w^2 - 4*w - 6],\ [827, 827, 5*w - 7],\ [853, 853, -w^2 + w - 6],\ [859, 859, w^2 + 3*w - 12],\ [863, 863, 3*w^2 - 3*w - 8],\ [863, 863, w^2 + 3*w - 11],\ [863, 863, w^2 - 5*w - 9],\ [881, 881, w^2 + 4*w - 18],\ [881, 881, w^2 - 5*w - 17],\ [881, 881, 4*w^2 - 4*w - 17],\ [883, 883, 5*w - 8],\ [883, 883, 3*w^2 - 3*w - 4],\ [883, 883, 4*w^2 - 6*w - 27],\ [887, 887, w^2 - w - 14],\ [887, 887, 3*w^2 - w - 26],\ [887, 887, 2*w^2 - 5*w - 14],\ [911, 911, 2*w^2 + 4*w - 7],\ [919, 919, 3*w^2 - 3*w - 7],\ [929, 929, 3*w^2 - w - 5],\ [941, 941, 5*w^2 - 24],\ [947, 947, 3*w^2 - w - 6],\ [947, 947, 3*w^2 + 2*w - 13],\ [947, 947, 3*w^2 - 5*w - 26],\ [961, 31, 4*w^2 - 5*w - 15],\ [967, 967, -5*w - 13],\ [967, 967, 4*w^2 - 21],\ [967, 967, 3*w^2 - 2*w - 7],\ [971, 971, w^2 + w - 16]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 + 4*x^4 - 11*x^3 - 38*x^2 + 28*x + 24 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/8*e^4 - 1/4*e^3 + 11/8*e^2 + 3/2*e - 5/2, -1, 1/8*e^4 + 1/4*e^3 - 15/8*e^2 - 3*e + 9/2, 1/2*e^3 + e^2 - 9/2*e - 3, e - 2, 1/2*e^2 - 1/2*e - 5, -4, -1/2*e^3 + 11/2*e - 5, 1/8*e^4 - 1/4*e^3 - 15/8*e^2 + 9/2*e + 5/2, -1/4*e^4 - 3/2*e^3 + 7/4*e^2 + 12*e - 3, 1/4*e^4 + 1/2*e^3 - 13/4*e^2 - 5/2*e + 6, -1/8*e^4 - 3/4*e^3 - 1/8*e^2 + 9/2*e + 11/2, 1/2*e^3 - 11/2*e + 5, 1/4*e^4 + e^3 - 13/4*e^2 - 10*e + 5, -1/8*e^4 + 1/4*e^3 + 15/8*e^2 - 5/2*e + 3/2, -1/2*e^4 - e^3 + 11/2*e^2 + 4*e - 10, -1/8*e^4 - 3/4*e^3 + 3/8*e^2 + 4*e - 3/2, 1/4*e^4 + 1/2*e^3 - 7/4*e^2 - 2*e - 9, 1/8*e^4 + 3/4*e^3 + 1/8*e^2 - 9/2*e - 11/2, 1/8*e^4 + 7/4*e^3 + 1/8*e^2 - 35/2*e + 9/2, -1/4*e^4 - 3/2*e^3 + 7/4*e^2 + 15*e - 9, 3/8*e^4 + 3/4*e^3 - 29/8*e^2 - 5*e - 7/2, 1/2*e^4 + e^3 - 9/2*e^2 - 6*e - 8, 3/8*e^4 + 1/4*e^3 - 29/8*e^2 + 5/2*e - 17/2, -1/4*e^4 - 3/2*e^3 + 7/4*e^2 + 15*e - 5, -1/2*e^4 - e^3 + 11/2*e^2 + 4*e - 14, -5/8*e^4 - 3/4*e^3 + 75/8*e^2 + 15/2*e - 41/2, -1/2*e^4 - e^3 + 11/2*e^2 + 3*e - 12, 1/2*e^3 + 3*e^2 - 5/2*e - 19, -1/8*e^4 - 5/4*e^3 - 5/8*e^2 + 29/2*e + 15/2, 1/2*e^4 + 3/2*e^3 - 13/2*e^2 - 29/2*e + 9, 1/4*e^4 - 15/4*e^2 + 11/2*e + 12, 1/2*e^4 + 3/2*e^3 - 7/2*e^2 - 15/2*e + 1, 5/8*e^4 + 3/4*e^3 - 55/8*e^2 - 2*e - 9/2, 1/8*e^4 - 1/4*e^3 - 15/8*e^2 + 13/2*e + 5/2, 1/8*e^4 + 1/4*e^3 - 31/8*e^2 - 3*e + 27/2, -1/4*e^4 - 1/2*e^3 + 3/4*e^2 + 15, 1/4*e^4 + 5/2*e^3 + 5/4*e^2 - 20*e - 3, 1/2*e^3 + e^2 - 17/2*e + 1, -3/4*e^4 - 5/2*e^3 + 33/4*e^2 + 20*e - 13, -5/4*e^4 - 5/2*e^3 + 59/4*e^2 + 18*e - 19, 4, -3/4*e^4 - e^3 + 33/4*e^2 + 7/2*e - 2, 1/2*e^4 + 3/2*e^3 - 9/2*e^2 - 13/2*e - 5, -3/8*e^4 - 3/4*e^3 + 33/8*e^2 + 17/2*e + 17/2, 1/2*e^2 + 7/2*e - 21, -3/4*e^4 - 2*e^3 + 43/4*e^2 + 18*e - 19, -3/4*e^4 - 1/2*e^3 + 35/4*e^2 - 5/2*e - 10, e^3 - e^2 - 19*e + 8, 1/2*e^4 - e^3 - 15/2*e^2 + 18*e + 18, 1/2*e^4 + 2*e^3 - 13/2*e^2 - 15*e + 26, 5/8*e^4 + 7/4*e^3 - 71/8*e^2 - 19*e + 35/2, 3/8*e^4 + 3/4*e^3 - 57/8*e^2 - 19/2*e + 35/2, 1/8*e^4 + 9/4*e^3 + 13/8*e^2 - 37/2*e + 9/2, -1/2*e^4 - e^3 + 13/2*e^2 + 6*e - 8, -1/4*e^4 + 1/2*e^3 + 15/4*e^2 - 5*e + 3, 1/4*e^4 - 1/2*e^3 - 23/4*e^2 + 9*e + 25, -5/8*e^4 - 11/4*e^3 + 27/8*e^2 + 47/2*e - 5/2, -1/2*e^4 - 2*e^3 + 9/2*e^2 + 19*e - 2, -e^4 - 3*e^3 + 13*e^2 + 26*e - 22, 5/8*e^4 + 7/4*e^3 - 31/8*e^2 - 12*e - 25/2, 1/2*e^4 + 5/2*e^3 - 7/2*e^2 - 29/2*e + 3, -3/4*e^4 - e^3 + 33/4*e^2 - 1/2*e - 14, 1/2*e^4 + e^3 - 13/2*e^2 - 10*e + 4, -1/2*e^4 - e^3 + 9/2*e^2 + 4*e - 4, 1/8*e^4 - 9/4*e^3 - 31/8*e^2 + 49/2*e + 9/2, -3/4*e^4 - 1/2*e^3 + 37/4*e^2 - e - 11, -1/2*e^4 - 5/2*e^3 + 9/2*e^2 + 55/2*e + 1, -1/2*e^4 + 1/2*e^3 + 23/2*e^2 - 5/2*e - 33, 3/8*e^4 - 5/4*e^3 - 77/8*e^2 + 15*e + 57/2, 1/8*e^4 + 3/4*e^3 - 3/8*e^2 - 8*e - 5/2, -1/4*e^4 - 1/2*e^3 + 7/4*e^2 + e + 5, -7/8*e^4 - 11/4*e^3 + 41/8*e^2 + 14*e + 23/2, -e + 2, 1/2*e^4 + 2*e^3 - 5/2*e^2 - 9*e - 2, 1/2*e^4 + 4*e^3 - 3/2*e^2 - 36*e, -1/2*e^4 - e^3 + 7/2*e^2 + 4*e + 4, -1/2*e^4 - e^3 + 19/2*e^2 + 9*e - 32, -1/2*e^4 - e^3 + 7/2*e^2 + 5*e + 12, 3/4*e^4 + 2*e^3 - 27/4*e^2 - 18*e - 5, -5/8*e^4 - 9/4*e^3 + 27/8*e^2 + 18*e + 5/2, 7/8*e^4 + 9/4*e^3 - 89/8*e^2 - 49/2*e + 23/2, 1/4*e^4 + 1/2*e^3 - 23/4*e^2 - 12*e + 19, 3/4*e^4 + 5/2*e^3 - 37/4*e^2 - 22*e + 27, 1/8*e^4 - 5/4*e^3 - 11/8*e^2 + 15*e - 33/2, -3/8*e^4 - 5/4*e^3 - 3/8*e^2 + 5/2*e + 77/2, -1/4*e^4 + 1/2*e^3 + 15/4*e^2 - 13*e + 7, e^3 - e^2 - 14*e, 1/2*e^4 + 1/2*e^3 - 9/2*e^2 + 9/2*e - 7, -3/8*e^4 - 1/4*e^3 + 29/8*e^2 - 1/2*e - 15/2, -3/4*e^4 - e^3 + 47/4*e^2 + 10*e - 15, 5/8*e^4 - 1/4*e^3 - 91/8*e^2 + 11/2*e + 65/2, -2*e^3 - 6*e^2 + 14*e + 24, -1/4*e^4 + 1/2*e^3 + 23/4*e^2 - 5*e - 25, 11/8*e^4 + 11/4*e^3 - 113/8*e^2 - 27/2*e + 43/2, 1/2*e^4 - 23/2*e^2 - e + 34, 3/4*e^4 + e^3 - 47/4*e^2 - 10*e + 39, 3/8*e^4 + 13/4*e^3 - 13/8*e^2 - 81/2*e - 1/2, -3/4*e^4 + 53/4*e^2 - 9/2*e - 22, -1/8*e^4 - 7/4*e^3 - 1/8*e^2 + 43/2*e - 9/2, -2*e^3 + e^2 + 29*e - 14, 1/2*e^4 + 3/2*e^3 - 7/2*e^2 - 23/2*e - 27, 7/4*e^4 + 11/2*e^3 - 63/4*e^2 - 69/2*e + 8, -3/4*e^4 - e^3 + 43/4*e^2 + 7*e - 13, -2*e^3 - 8*e^2 + 18*e + 40, 3/8*e^4 + 5/4*e^3 - 45/8*e^2 - 33/2*e - 9/2, -11/8*e^4 - 21/4*e^3 + 117/8*e^2 + 81/2*e - 43/2, -e^4 - 5*e^3 + 7*e^2 + 37*e - 4, 3*e^3 + 2*e^2 - 35*e + 18, -1/4*e^4 + 1/2*e^3 + 39/4*e^2 + 3*e - 37, 5/4*e^4 + 9/2*e^3 - 55/4*e^2 - 41*e + 21, -3/8*e^4 - 7/4*e^3 + 45/8*e^2 + 22*e - 53/2, 1/2*e^4 + 3*e^3 - 7/2*e^2 - 26*e + 10, 3*e^3 + 7*e^2 - 25*e - 32, 1/2*e^4 + 3*e^3 - 7*e^2 - 69/2*e + 13, -9/8*e^4 - 15/4*e^3 + 99/8*e^2 + 26*e - 35/2, 3/4*e^4 + 3/2*e^3 - 29/4*e^2 - 10*e - 7, -7/4*e^4 - 11/2*e^3 + 73/4*e^2 + 48*e - 29, -1/4*e^4 + 5/2*e^3 + 31/4*e^2 - 27*e - 5, 3/4*e^4 + 5/2*e^3 - 29/4*e^2 - 29*e - 5, -e^3 - 7/2*e^2 + 13/2*e + 21, -e^4 - 5*e^3 + 8*e^2 + 45*e - 20, 7/8*e^4 + 5/4*e^3 - 121/8*e^2 - 31/2*e + 75/2, -1/8*e^4 - 3/4*e^3 + 31/8*e^2 + 29/2*e - 29/2, e^4 + 3*e^3 - 11*e^2 - 23*e + 24, -1/2*e^3 + 15/2*e - 19, -1/4*e^4 - 7/2*e^3 - 5/4*e^2 + 32*e - 3, -7/8*e^4 - 9/4*e^3 + 57/8*e^2 + 13/2*e - 15/2, -1/2*e^4 + e^3 + 9/2*e^2 - 23*e, 1/2*e^4 - 1/2*e^3 - 19/2*e^2 + 25/2*e + 33, -1/8*e^4 - 11/4*e^3 - 1/8*e^2 + 57/2*e - 21/2, 3/4*e^4 - 43/4*e^2 + 4*e + 15, -3/4*e^4 - 1/2*e^3 + 45/4*e^2 - 21, 2*e^4 + 4*e^3 - 21*e^2 - 17*e + 30, -1/2*e^4 - 2*e^3 + 3/2*e^2 + 17*e + 28, -5/4*e^4 - 4*e^3 + 59/4*e^2 + 41/2*e - 44, -1/4*e^4 + 1/2*e^3 + 31/4*e^2 - 4*e - 43, -13/8*e^4 - 23/4*e^3 + 119/8*e^2 + 46*e - 71/2, e^4 + 9/2*e^3 - 15/2*e^2 - 33*e + 24, 5/4*e^4 + 13/2*e^3 - 35/4*e^2 - 60*e + 15, 1/4*e^4 - 3/2*e^3 - 3/4*e^2 + 20*e - 23, -e^4 - 9/2*e^3 + 13*e^2 + 87/2*e - 35, -7/2*e^3 - e^2 + 91/2*e + 5, 3/4*e^4 + 5/2*e^3 - 21/4*e^2 - 15*e - 25, 3/4*e^4 + 1/2*e^3 - 45/4*e^2 - 3*e + 39, 13/8*e^4 + 9/4*e^3 - 143/8*e^2 - 17/2*e + 13/2, 1/2*e^4 + 5*e^3 + 3/2*e^2 - 38*e, e^4 + 1/2*e^3 - 11*e^2 + 21/2*e + 7, -1/4*e^4 - 1/2*e^3 + 23/4*e^2 + 8*e - 7, -1/4*e^4 + e^3 + 15/4*e^2 - 37/2*e - 16, 1/2*e^4 + 5*e^3 + 1/2*e^2 - 44*e - 10, 5/8*e^4 - 5/4*e^3 - 139/8*e^2 + 29/2*e + 121/2, 3/4*e^4 + 1/2*e^3 - 45/4*e^2 + 5*e + 15, 1/4*e^4 - 5/2*e^3 - 31/4*e^2 + 19*e + 9, -e^4 - 1/2*e^3 + 14*e^2 - 7/2*e - 19, -2*e^4 - 9/2*e^3 + 19*e^2 + 37/2*e - 13, e^3 + 3/2*e^2 - 9/2*e - 25, -2*e^4 - 6*e^3 + 35/2*e^2 + 77/2*e - 15, -1/4*e^4 - 2*e^3 + 15/4*e^2 + 45/2*e - 14, -3/2*e^4 - 5*e^3 + 17/2*e^2 + 24*e + 30, 1/8*e^4 - 5/4*e^3 - 31/8*e^2 + 35/2*e - 3/2, 3/2*e^3 - 5/2*e^2 - 32*e + 18, -3/2*e^4 - 5*e^3 + 37/2*e^2 + 46*e - 30, e^3 + 3*e^2 - 6*e - 24, e^4 + 4*e^3 - 3*e^2 - 26*e - 8, 3/8*e^4 + 21/4*e^3 + 19/8*e^2 - 89/2*e - 9/2] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([7, 7, -w + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]