/* 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([20, 0, -10, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [4, 2, 1/2*w^3 - 3*w + 2],\ [5, 5, w^2 - w - 5],\ [11, 11, -1/2*w^3 - 1/2*w^2 + 3*w + 3],\ [11, 11, -1/2*w^2 + w + 2],\ [11, 11, -1/2*w^2 - w + 2],\ [11, 11, 1/2*w^3 - 1/2*w^2 - 3*w + 3],\ [29, 29, -1/2*w^2 - w + 4],\ [29, 29, -1/2*w^3 + 1/2*w^2 + 3*w - 1],\ [29, 29, -1/2*w^3 - 1/2*w^2 + 3*w + 1],\ [29, 29, -1/2*w^2 + w + 4],\ [41, 41, w^3 - 1/2*w^2 - 6*w + 6],\ [41, 41, -1/2*w^3 + 5/2*w^2 + 5*w - 14],\ [41, 41, 1/2*w^3 - 1/2*w^2 - 3*w + 6],\ [41, 41, -3/2*w^2 - 2*w + 6],\ [79, 79, -w^3 - w^2 + 4*w - 1],\ [79, 79, -3/2*w^2 - w + 9],\ [79, 79, -w^3 + 7/2*w^2 + 9*w - 21],\ [79, 79, w^3 - w^2 - 6*w + 9],\ [81, 3, -3],\ [109, 109, w^2 - w - 7],\ [109, 109, -1/2*w^3 + 1/2*w^2 + 4*w - 1],\ [109, 109, 1/2*w^3 + 1/2*w^2 - 4*w - 1],\ [109, 109, w^2 + w - 7],\ [131, 131, -1/2*w^3 - 1/2*w^2 + 4*w + 7],\ [131, 131, -1/2*w^3 + 7/2*w^2 + 6*w - 18],\ [131, 131, 1/2*w^3 + 1/2*w^2 - 2*w + 2],\ [131, 131, 1/2*w^3 - 1/2*w^2 - 4*w + 7],\ [149, 149, w^3 + w^2 - 5*w - 7],\ [149, 149, 1/2*w^3 - w^2 - 5*w + 3],\ [149, 149, -1/2*w^3 - w^2 + 5*w + 3],\ [149, 149, -w^3 + w^2 + 5*w - 7],\ [199, 199, -w^3 - w^2 + 6*w + 3],\ [199, 199, -w^2 + 2*w + 7],\ [199, 199, -w^2 - 2*w + 7],\ [199, 199, w^3 - w^2 - 6*w + 3],\ [211, 211, 1/2*w^3 - 2*w^2 - 3*w + 11],\ [211, 211, -1/2*w^3 - 1/2*w^2 + 3*w + 7],\ [211, 211, 1/2*w^3 - 1/2*w^2 - 3*w + 7],\ [211, 211, 1/2*w^3 + 2*w^2 - 3*w - 11],\ [229, 229, -1/2*w^3 - 1/2*w^2 + 4*w - 2],\ [229, 229, 1/2*w^3 + 1/2*w^2 - 2*w - 7],\ [229, 229, 1/2*w^3 - 1/2*w^2 - 2*w + 7],\ [229, 229, 1/2*w^3 - 1/2*w^2 - 4*w - 2],\ [239, 239, 1/2*w^2 + 2*w - 6],\ [239, 239, -w^3 - 1/2*w^2 + 6*w - 1],\ [239, 239, w^3 - 1/2*w^2 - 6*w - 1],\ [239, 239, 1/2*w^2 - 2*w - 6],\ [241, 241, -w^3 + 1/2*w^2 + 6*w - 2],\ [241, 241, -w^3 - 1/2*w^2 + 6*w + 4],\ [241, 241, w^3 - 1/2*w^2 - 6*w + 4],\ [241, 241, w^3 + 1/2*w^2 - 6*w - 2],\ [251, 251, 1/2*w^3 + w^2 - 5*w - 7],\ [251, 251, w^3 + w^2 - 5*w - 3],\ [251, 251, -w^3 + w^2 + 5*w - 3],\ [251, 251, -1/2*w^3 + w^2 + 5*w - 7],\ [269, 269, 1/2*w^3 + 1/2*w^2 - 2*w - 6],\ [269, 269, 1/2*w^3 - 1/2*w^2 - 4*w - 1],\ [269, 269, -1/2*w^3 - 1/2*w^2 + 4*w - 1],\ [269, 269, -1/2*w^3 + 1/2*w^2 + 2*w - 6],\ [281, 281, w^3 - 6*w + 1],\ [281, 281, w^3 - 1/2*w^2 - 6*w + 3],\ [281, 281, -w^3 - 1/2*w^2 + 6*w + 3],\ [281, 281, -w^3 + 6*w + 1],\ [331, 331, 3/2*w^3 + w^2 - 7*w + 1],\ [331, 331, -1/2*w^3 + 3*w^2 + 5*w - 17],\ [331, 331, -3/2*w^3 + w^2 + 9*w - 13],\ [331, 331, 1/2*w^3 + 3*w^2 + w - 11],\ [349, 349, 1/2*w^3 + 3/2*w^2 - 2*w - 9],\ [349, 349, -1/2*w^3 + 3/2*w^2 + 4*w - 6],\ [349, 349, 1/2*w^3 + 3/2*w^2 - 4*w - 6],\ [349, 349, -1/2*w^3 + 3/2*w^2 + 2*w - 9],\ [359, 359, -w^3 + 7/2*w^2 + 9*w - 19],\ [359, 359, w^3 - 2*w^2 - 8*w + 11],\ [359, 359, w^3 - 1/2*w^2 - 6*w - 2],\ [359, 359, -5/2*w^2 - 3*w + 9],\ [361, 19, 2*w^2 - 11],\ [361, 19, -1/2*w^2 + 7],\ [389, 389, -1/2*w^3 + 5/2*w^2 + 6*w - 11],\ [389, 389, 5/2*w^3 - 3/2*w^2 - 16*w + 17],\ [389, 389, 3/2*w^3 - 1/2*w^2 - 10*w + 7],\ [389, 389, -3/2*w^3 + 7/2*w^2 + 12*w - 21],\ [401, 401, -w^3 + 1/2*w^2 + 5*w - 1],\ [401, 401, 1/2*w^3 + 1/2*w^2 - 5*w - 4],\ [401, 401, -1/2*w^3 + 1/2*w^2 + 5*w - 4],\ [401, 401, w^3 + 1/2*w^2 - 5*w - 1],\ [439, 439, -2*w^3 + 3/2*w^2 + 12*w - 16],\ [439, 439, -w^3 + 1/2*w^2 + 5*w - 7],\ [439, 439, w^3 + 1/2*w^2 - 5*w - 7],\ [439, 439, w^3 + 1/2*w^2 - 4*w + 4],\ [479, 479, -w^3 + 9/2*w^2 + 10*w - 26],\ [479, 479, -3/2*w^3 + 3/2*w^2 + 9*w - 14],\ [479, 479, -3/2*w^3 - 1/2*w^2 + 7*w - 6],\ [479, 479, -5/2*w^2 - 2*w + 14],\ [491, 491, 1/2*w^3 + 5/2*w^2 - 4*w - 14],\ [491, 491, 1/2*w^3 + 5/2*w^2 - 2*w - 11],\ [491, 491, -1/2*w^3 + 5/2*w^2 + 2*w - 11],\ [491, 491, -1/2*w^3 + 5/2*w^2 + 4*w - 14],\ [509, 509, 1/2*w^3 - 1/2*w^2 - 5*w - 3],\ [509, 509, -w^3 + 1/2*w^2 + 5*w - 8],\ [509, 509, w^3 + 1/2*w^2 - 5*w - 8],\ [509, 509, -7/2*w^2 - 3*w + 18],\ [521, 521, -9/2*w^2 - 4*w + 21],\ [521, 521, -5/2*w^3 + 2*w^2 + 16*w - 19],\ [521, 521, -1/2*w^3 + 2*w^2 + 6*w - 9],\ [521, 521, w^3 - 7/2*w^2 - 8*w + 22],\ [571, 571, 1/2*w^3 - 1/2*w^2 - 4*w - 4],\ [571, 571, 3/2*w^3 - 3/2*w^2 - 9*w + 7],\ [571, 571, -3/2*w^3 - 3/2*w^2 + 9*w + 7],\ [571, 571, -1/2*w^3 - 1/2*w^2 + 4*w - 4],\ [599, 599, 1/2*w^3 + w^2 - 2*w - 9],\ [599, 599, -1/2*w^3 + w^2 + 4*w - 1],\ [599, 599, 1/2*w^3 + w^2 - 4*w - 1],\ [599, 599, -1/2*w^3 + w^2 + 2*w - 9],\ [601, 601, 5/2*w^2 + w - 11],\ [601, 601, 1/2*w^3 - 5/2*w^2 - 3*w + 14],\ [601, 601, -1/2*w^3 - 5/2*w^2 + 3*w + 14],\ [601, 601, 5/2*w^2 - w - 11],\ [641, 641, 5/2*w^2 + w - 13],\ [641, 641, -1/2*w^3 + 5/2*w^2 + 3*w - 12],\ [641, 641, 1/2*w^3 + 5/2*w^2 - 3*w - 12],\ [641, 641, 5/2*w^2 - w - 13],\ [691, 691, 1/2*w^3 + 5/2*w^2 - 3*w - 13],\ [691, 691, -1/2*w^3 - w^2 + 3*w + 11],\ [691, 691, 1/2*w^3 - w^2 - 3*w + 11],\ [691, 691, -1/2*w^3 + 5/2*w^2 + 3*w - 13],\ [709, 709, 3/2*w^3 - 3/2*w^2 - 8*w + 2],\ [709, 709, -1/2*w^3 + 3/2*w^2 + 6*w - 13],\ [709, 709, 1/2*w^3 + 3/2*w^2 - 6*w - 13],\ [709, 709, -3/2*w^3 - 3/2*w^2 + 8*w + 2],\ [719, 719, w^2 + 2*w - 9],\ [719, 719, w^3 + w^2 - 6*w - 1],\ [719, 719, -w^3 + w^2 + 6*w - 1],\ [719, 719, w^2 - 2*w - 9],\ [761, 761, 1/2*w^3 + w^2 - 6*w - 9],\ [761, 761, 3/2*w^3 + w^2 - 8*w - 1],\ [761, 761, 3/2*w^3 - 5*w^2 - 14*w + 29],\ [761, 761, -1/2*w^3 + w^2 + 6*w - 9],\ [811, 811, 3/2*w^3 - 1/2*w^2 - 8*w + 12],\ [811, 811, w^3 - 2*w^2 - 9*w + 11],\ [811, 811, -w^3 - 2*w^2 + 9*w + 11],\ [811, 811, -3/2*w^3 + 9/2*w^2 + 12*w - 28],\ [829, 829, w^3 + w^2 - 5*w - 9],\ [829, 829, 1/2*w^3 - w^2 - 5*w + 1],\ [829, 829, -1/2*w^3 - w^2 + 5*w + 1],\ [829, 829, -w^3 + w^2 + 5*w - 9],\ [839, 839, w^3 - 5/2*w^2 - 8*w + 13],\ [839, 839, 2*w^3 - 1/2*w^2 - 13*w - 1],\ [839, 839, -2*w^3 - 1/2*w^2 + 13*w - 1],\ [839, 839, w^3 + 5/2*w^2 - 8*w - 13],\ [881, 881, -w^3 + 9/2*w^2 + 9*w - 27],\ [881, 881, -2*w^3 + 1/2*w^2 + 12*w - 9],\ [881, 881, w^3 - 7/2*w^2 - 10*w + 19],\ [881, 881, 2*w^3 - 5/2*w^2 - 13*w + 23],\ [919, 919, 1/2*w^3 + 2*w^2 - 4*w - 7],\ [919, 919, 1/2*w^3 + 2*w^2 - 2*w - 13],\ [919, 919, -1/2*w^3 + 2*w^2 + 2*w - 13],\ [919, 919, -1/2*w^3 + 2*w^2 + 4*w - 7],\ [961, 31, -w^2 + 11],\ [961, 31, 5/2*w^2 - 13],\ [971, 971, 1/2*w^3 + 7/2*w^2 - 5*w - 17],\ [971, 971, 2*w^3 - 13*w - 3],\ [971, 971, -2*w^3 + 13*w - 3],\ [971, 971, -1/2*w^3 + 7/2*w^2 + 5*w - 17]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 4*x^3 - 44*x^2 - 176*x + 16 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, 0, e, -1/44*e^3 - 3/11*e^2 + 9/11*e + 72/11, -3/44*e^3 + 2/11*e^2 + 27/11*e - 48/11, 1/11*e^3 + 1/11*e^2 - 47/11*e - 68/11, 0, 0, 0, 0, 1/44*e^3 + 3/11*e^2 + 2/11*e - 138/11, -5/44*e^3 - 4/11*e^2 + 56/11*e + 74/11, -3/44*e^3 + 2/11*e^2 + 16/11*e - 114/11, 7/44*e^3 - 1/11*e^2 - 74/11*e - 86/11, 0, 0, 0, 0, -6, 0, 0, 0, 0, 1/22*e^3 + 6/11*e^2 - 7/11*e - 188/11, -9/44*e^3 - 5/11*e^2 + 103/11*e + 164/11, -3/44*e^3 + 2/11*e^2 + 5/11*e - 92/11, 5/22*e^3 - 3/11*e^2 - 101/11*e - 16/11, 0, 0, 0, 0, 0, 0, 0, 0, -3/22*e^3 + 4/11*e^2 + 43/11*e - 316/11, 1/4*e^3 - 11*e - 28, 1/44*e^3 + 3/11*e^2 + 13/11*e - 292/11, -3/22*e^3 - 7/11*e^2 + 65/11*e - 8/11, 0, 0, 0, 0, 0, 0, 0, 0, 3/44*e^3 + 9/11*e^2 - 38/11*e - 106/11, 3/44*e^3 - 2/11*e^2 - 60/11*e + 158/11, -1/4*e^3 + 12*e + 22, 5/44*e^3 - 7/11*e^2 - 34/11*e + 322/11, -1/44*e^3 + 8/11*e^2 + 53/11*e - 280/11, -2/11*e^3 + 9/11*e^2 + 39/11*e - 348/11, -3/11*e^3 - 14/11*e^2 + 141/11*e + 336/11, 21/44*e^3 - 3/11*e^2 - 233/11*e - 192/11, 0, 0, 0, 0, -5/44*e^3 + 7/11*e^2 + 34/11*e - 146/11, 1/4*e^3 - 12*e - 6, -3/44*e^3 + 2/11*e^2 + 60/11*e + 18/11, -3/44*e^3 - 9/11*e^2 + 38/11*e + 282/11, -21/44*e^3 + 3/11*e^2 + 167/11*e - 116/11, 15/22*e^3 + 2/11*e^2 - 303/11*e - 356/11, 7/22*e^3 + 9/11*e^2 - 93/11*e - 392/11, -23/44*e^3 - 14/11*e^2 + 229/11*e + 380/11, 0, 0, 0, 0, 0, 0, 0, 0, -6/11*e^3 - 6/11*e^2 + 216/11*e + 210/11, 6/11*e^3 + 6/11*e^2 - 216/11*e - 342/11, 0, 0, 0, 0, 6/11*e^3 - 5/11*e^2 - 216/11*e + 54/11, -7/11*e^3 - 7/11*e^2 + 296/11*e + 410/11, -3/11*e^3 - 3/11*e^2 + 64/11*e + 138/11, 4/11*e^3 + 15/11*e^2 - 144/11*e - 426/11, 0, 0, 0, 0, 0, 0, 0, 0, 29/44*e^3 + 10/11*e^2 - 283/11*e - 460/11, -1/2*e^3 - e^2 + 17*e + 32, -15/22*e^3 - 2/11*e^2 + 281/11*e + 180/11, 23/44*e^3 + 3/11*e^2 - 185/11*e - 204/11, 0, 0, 0, 0, 2/11*e^3 - 9/11*e^2 - 28/11*e + 502/11, -6/11*e^3 + 5/11*e^2 + 260/11*e + 298/11, -e^2 - 4*e + 46, 4/11*e^3 + 15/11*e^2 - 188/11*e - 250/11, -5/44*e^3 - 15/11*e^2 + 45/11*e + 272/11, 5*e - 8, 5/11*e^3 + 5/11*e^2 - 235/11*e - 428/11, -15/44*e^3 + 10/11*e^2 + 135/11*e - 328/11, 0, 0, 0, 0, 19/44*e^3 - 20/11*e^2 - 204/11*e + 326/11, -19/44*e^3 - 13/11*e^2 + 270/11*e + 466/11, 1/4*e^3 + e^2 - 18*e - 46, -1/4*e^3 + 2*e^2 + 12*e - 50, -25/44*e^3 - 20/11*e^2 + 280/11*e + 502/11, 5/44*e^3 + 15/11*e^2 + 10/11*e - 558/11, 35/44*e^3 - 5/11*e^2 - 370/11*e - 298/11, -15/44*e^3 + 10/11*e^2 + 80/11*e - 438/11, 15/44*e^3 + 12/11*e^2 - 245/11*e - 508/11, -7/22*e^3 + 24/11*e^2 + 159/11*e - 532/11, 9/22*e^3 - 23/11*e^2 - 195/11*e + 464/11, -19/44*e^3 - 13/11*e^2 + 281/11*e + 532/11, 0, 0, 0, 0, 0, 0, 0, 0, 4/11*e^3 - 7/11*e^2 - 100/11*e + 410/11, -8/11*e^3 + 3/11*e^2 + 332/11*e + 390/11, -2/11*e^3 - 13/11*e^2 + 28/11*e + 598/11, 6/11*e^3 + 17/11*e^2 - 260/11*e - 342/11, 1/4*e^3 - 17*e - 16, 2*e^2 - e - 56, 3/11*e^3 - 19/11*e^2 - 97/11*e + 412/11, -23/44*e^3 - 3/11*e^2 + 295/11*e + 248/11, 0, 0, 0, 0, 0, 0, 0, 0, 9/44*e^3 + 5/11*e^2 - 158/11*e - 450/11, -5/44*e^3 + 18/11*e^2 + 56/11*e - 630/11, 13/44*e^3 - 16/11*e^2 - 128/11*e + 142/11, -17/44*e^3 - 7/11*e^2 + 230/11*e + 146/11, 0, 0, 0, 0, e^3 + e^2 - 36*e - 54, -e^3 - e^2 + 36*e + 38, 1/4*e^3 - 7*e + 36, -9/22*e^3 + 1/11*e^2 + 173/11*e + 504/11, -5/22*e^3 - 8/11*e^2 + 79/11*e + 676/11, 17/44*e^3 + 7/11*e^2 - 175/11*e + 140/11] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, 1/2*w^3 - 3*w + 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]