/* This code can be loaded, or copied and pasted, into Magma. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. At the *bottom* of the file, there is code to recreate the Hilbert modular form in Magma, by creating the HMF space and cutting out the corresponding Hecke irreducible subspace. From there, you can ask for more eigenvalues or modify as desired. It is commented out, as this computation may be lengthy. */ P := PolynomialRing(Rationals()); g := P![11, 0, -7, 0, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [4, 2, -w^2 + w + 3], [5, 5, w + 1], [5, 5, -w^3 + w^2 + 4*w - 4], [11, 11, w], [29, 29, w^3 - 2*w^2 - 3*w + 7], [29, 29, -w^3 - 2*w^2 + 3*w + 7], [31, 31, -w^3 + w^2 + 4*w - 2], [31, 31, -w^3 - w^2 + 4*w + 2], [41, 41, w^3 + 2*w^2 - 4*w - 6], [41, 41, w^3 - 5*w + 2], [49, 7, -w^3 + w^2 + 4*w - 1], [49, 7, w^3 + w^2 - 4*w - 1], [59, 59, -3*w^2 - w + 10], [59, 59, -3*w^2 + w + 10], [61, 61, -2*w^3 + 2*w^2 + 7*w - 5], [61, 61, 2*w^3 + w^2 - 8*w - 6], [71, 71, 2*w^2 + w - 9], [71, 71, 2*w^2 - w - 9], [81, 3, -3], [101, 101, 2*w^2 - w - 10], [101, 101, 2*w^2 + w - 10], [109, 109, 2*w^3 + w^2 - 8*w - 3], [109, 109, -2*w^3 + w^2 + 8*w - 3], [121, 11, -w^2 + 7], [131, 131, 4*w^2 - 2*w - 15], [131, 131, 2*w^2 + 2*w - 3], [131, 131, 2*w^2 - 2*w - 3], [131, 131, -w^3 - w^2 + 3*w + 7], [139, 139, w^3 + 4*w^2 - 5*w - 16], [139, 139, w^3 + 3*w^2 - 6*w - 10], [139, 139, 2*w^3 - w^2 - 8*w + 1], [139, 139, -4*w^2 - w + 12], [149, 149, 2*w^3 - w^2 - 8*w + 4], [149, 149, 2*w^3 + w^2 - 8*w - 4], [151, 151, 3*w^2 - 3*w - 8], [151, 151, 2*w^3 - 4*w^2 - 8*w + 17], [151, 151, w^2 - 2*w - 7], [151, 151, w^3 + 2*w^2 - 6*w - 8], [179, 179, w^3 - w^2 - 5*w + 1], [179, 179, -w^3 - w^2 + 5*w + 1], [191, 191, 2*w^3 - w^2 - 7*w + 2], [191, 191, 3*w^3 - 2*w^2 - 10*w + 2], [199, 199, w^3 - 4*w^2 - 3*w + 13], [199, 199, -w^3 - 4*w^2 + 3*w + 13], [211, 211, -w^3 + w^2 + 6*w - 5], [211, 211, w^3 - 5*w - 5], [211, 211, -w^3 + 5*w - 5], [211, 211, w^3 + w^2 - 6*w - 5], [229, 229, w^3 - 4*w^2 - 2*w + 14], [229, 229, 2*w^3 + 3*w^2 - 7*w - 12], [229, 229, 2*w^3 - 3*w^2 - 7*w + 12], [229, 229, 2*w^3 + w^2 - 7*w - 8], [241, 241, w^3 + w^2 - 6*w - 4], [241, 241, -w^3 + w^2 + 6*w - 4], [251, 251, 2*w^2 - w - 2], [251, 251, 2*w^2 + w - 2], [271, 271, 2*w^3 - 2*w^2 - 8*w + 5], [271, 271, -4*w^2 - w + 14], [271, 271, -4*w^2 + w + 14], [271, 271, 2*w^3 + 2*w^2 - 8*w - 5], [281, 281, -2*w^3 - w^2 + 9*w + 2], [281, 281, 2*w^3 - w^2 - 9*w + 2], [311, 311, -w^3 + 3*w - 5], [311, 311, w^3 - 3*w - 5], [331, 331, -3*w^3 + 3*w^2 + 12*w - 13], [331, 331, 3*w^3 + 3*w^2 - 12*w - 13], [349, 349, 2*w^3 + 2*w^2 - 7*w - 10], [349, 349, -2*w^3 + 2*w^2 + 7*w - 10], [361, 19, 4*w^2 - 15], [361, 19, -4*w^2 + 13], [379, 379, -w^3 + 3*w^2 + 3*w - 13], [379, 379, w^3 + 3*w^2 - 3*w - 13], [409, 409, 3*w^3 - 2*w^2 - 11*w + 2], [409, 409, w^3 + 3*w^2 - 6*w - 9], [419, 419, -w^3 + w^2 + 6*w + 2], [419, 419, -3*w^3 + 5*w^2 + 13*w - 19], [421, 421, -w^3 + 3*w^2 + 7*w - 13], [421, 421, 2*w^3 - 2*w^2 - 9*w + 6], [421, 421, -2*w^3 - 2*w^2 + 9*w + 6], [421, 421, w^3 + 3*w^2 - 7*w - 13], [439, 439, 3*w^2 + 2*w - 13], [439, 439, 3*w^3 + 2*w^2 - 12*w - 10], [439, 439, -3*w^3 + 2*w^2 + 12*w - 10], [439, 439, 3*w^2 - 2*w - 13], [449, 449, 2*w^3 - 4*w^2 - 9*w + 14], [449, 449, -3*w^3 + 3*w^2 + 11*w - 9], [449, 449, 3*w^3 + 3*w^2 - 11*w - 9], [449, 449, w^3 - 2*w^2 - 5*w + 2], [461, 461, -w - 5], [461, 461, w - 5], [499, 499, w^3 - 5*w^2 - w + 17], [499, 499, -w^3 - 5*w^2 + w + 17], [509, 509, w^3 + 5*w^2 - 4*w - 16], [509, 509, w^2 - 2*w - 10], [509, 509, w^2 + 2*w - 10], [509, 509, w^3 - w^2 - 5*w + 9], [541, 541, 3*w^3 - 10*w - 6], [541, 541, -2*w^3 + 5*w^2 + 5*w - 16], [569, 569, 3*w^3 + 2*w^2 - 13*w - 6], [569, 569, -2*w^3 + 4*w^2 + 7*w - 10], [599, 599, w^3 + 2*w^2 - 7*w - 10], [599, 599, -w^3 + 2*w^2 + 7*w - 10], [601, 601, -3*w^3 - w^2 + 12*w + 6], [601, 601, 3*w^3 - w^2 - 12*w + 6], [619, 619, 3*w^3 - 12*w + 2], [619, 619, -3*w^3 + 12*w + 2], [631, 631, -w^3 + w^2 + 2*w - 8], [631, 631, w^3 + w^2 - 2*w - 8], [641, 641, w^3 + w^2 - 7*w - 1], [641, 641, -3*w^3 - 2*w^2 + 12*w + 4], [641, 641, 3*w^3 - 2*w^2 - 12*w + 4], [641, 641, w^3 - w^2 - 7*w + 1], [659, 659, -2*w^3 + w^2 + 10*w - 1], [659, 659, -w^3 + 5*w^2 + 4*w - 19], [659, 659, w^3 + 5*w^2 - 4*w - 19], [659, 659, 2*w^3 + w^2 - 10*w - 1], [691, 691, 3*w^2 - 2*w - 14], [691, 691, 3*w^2 + 2*w - 14], [701, 701, w^3 + 4*w^2 - 7*w - 12], [701, 701, 3*w^3 - w^2 - 12*w - 1], [719, 719, -w^3 + 7*w - 3], [719, 719, w^3 - 7*w - 3], [739, 739, 3*w^3 + w^2 - 12*w - 3], [739, 739, w^3 + 2*w^2 - 3*w - 1], [739, 739, -w^3 + 2*w^2 + 3*w - 1], [739, 739, -3*w^3 + w^2 + 12*w - 3], [751, 751, -2*w^3 + 6*w^2 + 7*w - 20], [751, 751, 2*w^3 + 6*w^2 - 7*w - 20], [761, 761, 2*w^3 + 6*w^2 - 8*w - 21], [761, 761, -2*w^3 + 6*w^2 + 8*w - 21], [769, 769, -2*w^3 + 6*w^2 + 8*w - 23], [769, 769, w^2 + 2*w + 3], [809, 809, 2*w^3 + 3*w^2 - 7*w - 14], [809, 809, -2*w^3 + 3*w^2 + 7*w - 14], [821, 821, -3*w^3 - w^2 + 11*w + 1], [821, 821, 3*w^3 - w^2 - 11*w + 1], [829, 829, 3*w^3 - w^2 - 12*w + 4], [829, 829, w^3 - 4*w^2 - 6*w + 12], [829, 829, -w^3 - 4*w^2 + 6*w + 12], [829, 829, -3*w^3 - w^2 + 12*w + 4], [839, 839, -w^3 + 3*w - 6], [839, 839, w^3 - 3*w - 6], [841, 29, 5*w^2 - 19], [859, 859, w^3 + 2*w^2 - 7*w - 9], [859, 859, -w^3 + 2*w^2 + 7*w - 9], [911, 911, -2*w^3 + 4*w^2 + 9*w - 13], [911, 911, 2*w^3 + 4*w^2 - 9*w - 13], [919, 919, 2*w^3 + 4*w^2 - 7*w - 17], [919, 919, 3*w^3 + 2*w^2 - 11*w - 6], [919, 919, 3*w^3 - 2*w^2 - 11*w + 6], [919, 919, 2*w^3 - 4*w^2 - 7*w + 17], [941, 941, -2*w^3 + 2*w^2 + 7*w - 15], [941, 941, 2*w^3 + 2*w^2 - 7*w - 15], [961, 31, -2*w^2 + 13], [971, 971, w^3 + 2*w^2 - 7*w - 7], [971, 971, -w^3 + 2*w^2 + 7*w - 7], [991, 991, 3*w^3 + 2*w^2 - 11*w - 7], [991, 991, -3*w^3 + 2*w^2 + 11*w - 7]]; primes := [ideal : I in primesArray]; heckePol := x^4 + x^3 - 9*x^2 + 2*x + 4; K := NumberField(heckePol); heckeEigenvaluesArray := [-1, e, -1/2*e^3 - 1/2*e^2 + 9/2*e - 1, -1, -1/2*e^3 - 1/2*e^2 + 9/2*e + 1, e + 2, 1/2*e^3 + 5/2*e^2 - 3/2*e - 6, -1/2*e^3 - 5/2*e^2 + 3/2*e + 10, 1/2*e^3 - 1/2*e^2 - 9/2*e + 6, -1/2*e^3 + 1/2*e^2 + 9/2*e - 5, 2*e^3 + 4*e^2 - 12*e - 4, -2*e^3 - 4*e^2 + 12*e + 6, -3/2*e^3 - 5/2*e^2 + 21/2*e + 8, e^3 + 2*e^2 - 5*e + 4, e^3 + 2*e^2 - 3*e - 4, -5/2*e^3 - 7/2*e^2 + 39/2*e - 2, 2*e^3 + 4*e^2 - 10*e - 6, -3*e^3 - 5*e^2 + 21*e + 2, -e^3 - e^2 + 11*e + 2, -2*e^2 - 5*e + 8, 3/2*e^3 + 7/2*e^2 - 23/2*e - 7, -e^2 - 4*e + 4, 3/2*e^3 + 5/2*e^2 - 25/2*e - 2, 3/2*e^3 + 3/2*e^2 - 33/2*e + 4, 1/2*e^3 + 7/2*e^2 + 9/2*e - 12, 3*e^3 + 3*e^2 - 22*e + 4, -5/2*e^3 - 5/2*e^2 + 33/2*e - 7, -3*e^3 - 6*e^2 + 23*e + 8, 3*e^3 + 6*e^2 - 17*e - 6, 1/2*e^3 - 1/2*e^2 - 23/2*e + 2, 3*e^3 + 4*e^2 - 27*e - 2, -7/2*e^3 - 13/2*e^2 + 45/2*e + 8, -e^3 - 2*e^2 + 3*e - 4, 5/2*e^3 + 7/2*e^2 - 39/2*e - 6, 2*e^3 + 2*e^2 - 16*e + 10, -e^3 - 5*e^2 + e + 20, 2*e^3 + 6*e^2 - 12*e - 10, -e^3 - e^2 + 5*e + 4, -3*e^3 - 3*e^2 + 28*e - 8, -1/2*e^3 - 1/2*e^2 + 21/2*e - 3, -1/2*e^3 - 1/2*e^2 + 5/2*e + 5, e^3 + e^2 - 8*e + 8, -1/2*e^3 - 1/2*e^2 + 23/2*e + 9, -7/2*e^3 - 7/2*e^2 + 65/2*e + 3, -2*e^2 - 3*e + 12, -5/2*e^3 - 5/2*e^2 + 45/2*e - 1, 5*e + 4, 1/2*e^3 + 5/2*e^2 - 5/2*e - 5, -e^3 - 3*e^2 + 2*e - 2, 5/2*e^3 + 11/2*e^2 - 39/2*e - 4, -3*e^2 - 8*e + 18, 5/2*e^3 + 9/2*e^2 - 37/2*e - 13, 7/2*e^3 + 7/2*e^2 - 59/2*e + 7, -e^3 - e^2 + 2*e - 2, 3/2*e^3 + 3/2*e^2 - 27/2*e - 1, -3*e - 4, 3/2*e^3 - 5/2*e^2 - 41/2*e + 19, e^3 + 7*e^2 + 2*e - 24, -5/2*e^3 - 17/2*e^2 + 29/2*e + 23, 3/2*e^3 + 11/2*e^2 - 25/2*e - 17, -2*e^2 - 4*e + 4, e^3 + 3*e^2 - 7*e - 12, -e^2 + 2*e + 20, -3/2*e^3 - 1/2*e^2 + 29/2*e + 8, 2*e^3 + e^2 - 16*e + 2, -3/2*e^3 - 1/2*e^2 + 21/2*e - 14, 7/2*e^3 + 11/2*e^2 - 43/2*e - 3, -4*e^3 - 6*e^2 + 27*e, 5/2*e^3 + 5/2*e^2 - 55/2*e + 1, -1/2*e^3 - 1/2*e^2 + 11/2*e - 1, -3*e^3 - 4*e^2 + 15*e, 11/2*e^3 + 13/2*e^2 - 85/2*e + 8, -2*e^3 - 10*e^2 + 6*e + 30, 2*e^3 + 10*e^2 - 6*e - 34, 5*e^3 + 9*e^2 - 36*e - 16, -5/2*e^3 - 13/2*e^2 + 17/2*e + 5, -4*e^3 - 10*e^2 + 25*e + 6, -1/2*e^3 - 1/2*e^2 - 11/2*e - 1, 5*e^3 + 5*e^2 - 44*e + 10, 5/2*e^3 + 17/2*e^2 - 17/2*e - 35, 7/2*e^3 + 5/2*e^2 - 67/2*e + 4, 6*e^3 + 10*e^2 - 42*e + 6, -4*e^3 - 8*e^2 + 20*e + 22, 1/2*e^3 + 3/2*e^2 - 21/2*e - 11, -11/2*e^3 - 11/2*e^2 + 97/2*e + 7, -2*e^3 - 2*e^2 + 20*e + 24, -e^3 - e^2 + 13*e + 26, 1/2*e^3 + 1/2*e^2 + 13/2*e + 19, -13/2*e^3 - 19/2*e^2 + 95/2*e + 8, 4*e^3 + 7*e^2 - 20*e + 2, -1/2*e^3 + 5/2*e^2 + 23/2*e - 20, -2*e^3 - 5*e^2 + 16*e + 4, 4*e^3 + 9*e^2 - 28*e - 2, 3/2*e^3 + 13/2*e^2 + 3/2*e - 20, -5*e^3 - 10*e^2 + 37*e + 12, -3/2*e^3 - 13/2*e^2 + 1/2*e + 32, -1/2*e^3 + 9/2*e^2 + 19/2*e - 28, -5*e^2 - 4*e + 18, -3*e^3 - 7*e^2 + 11*e + 18, 6*e^3 + 10*e^2 - 44*e, 2*e^3 + 2*e^2 - 12*e + 28, -3*e^3 - 3*e^2 + 23*e + 18, -3*e^3 - 11*e^2 + 15*e + 18, 2*e^3 + 10*e^2 - 4*e - 44, 3*e^3 + 5*e^2 - 21*e + 28, -2*e^3 - 4*e^2 + 10*e + 36, 6*e^3 + 6*e^2 - 52*e + 6, -e^3 - e^2 - 3*e - 8, -3*e^3 - 5*e^2 + 13*e - 12, -e^3 - 3*e^2 + 9*e + 18, -e^3 + e^2 + 13*e, 6*e^3 + 8*e^2 - 46*e - 12, -e^3 + 5*e - 30, 7*e^3 + 8*e^2 - 53*e + 2, -9/2*e^3 - 11/2*e^2 + 51/2*e - 12, 5/2*e^3 + 3/2*e^2 - 43/2*e - 14, 5/2*e^3 + 1/2*e^2 - 53/2*e + 1, e^3 + 3*e^2 - 12*e - 20, 9/2*e^3 + 13/2*e^2 - 81/2*e - 3, e^3 - e^2 - 20*e + 8, -2*e^3 - 2*e^2 + 18*e + 14, 4*e + 18, 4*e^3 + 10*e^2 - 29*e - 36, 5/2*e^3 + 23/2*e^2 - 23/2*e - 36, -e^3 - 10*e^2 - 5*e + 38, -1/2*e^3 - 13/2*e^2 - 19/2*e + 9, -e^3 + 5*e^2 + 21*e - 28, -3*e^3 - 9*e^2 + 23*e + 22, 6*e^3 + 12*e^2 - 46*e - 10, -e^3 - 7*e^2 - 9*e + 30, -1/2*e^3 + 7/2*e^2 + 3/2*e - 33, 7/2*e^3 - 1/2*e^2 - 69/2*e + 11, 9/2*e^3 + 15/2*e^2 - 41/2*e - 5, -17/2*e^3 - 23/2*e^2 + 129/2*e - 4, -2*e^3 - 4*e^2 + 20*e + 30, -2*e^3 + 24*e + 12, -3/2*e^3 - 9/2*e^2 + 1/2*e + 14, -2*e^3 - 2*e^2 + 12*e - 30, 3*e^3 + 3*e^2 - 23*e - 20, 5*e^3 + 8*e^2 - 39*e, -9/2*e^3 - 27/2*e^2 + 21/2*e + 43, 21/2*e^3 + 39/2*e^2 - 153/2*e - 8, -7/2*e^3 - 7/2*e^2 + 77/2*e - 16, -3/2*e^3 - 5/2*e^2 + 41/2*e - 2, -4*e^3 - 3*e^2 + 40*e - 16, -17/2*e^3 - 15/2*e^2 + 141/2*e - 15, 7/2*e^3 + 5/2*e^2 - 31/2*e + 18, -13/2*e^3 - 31/2*e^2 + 77/2*e + 16, e^3 + 9*e^2 + 17*e - 42, -9*e^3 - 17*e^2 + 71*e + 10, 11/2*e^3 + 29/2*e^2 - 55/2*e - 41, -5/2*e^3 - 7/2*e^2 + 15/2*e + 12, 7*e^3 + 8*e^2 - 57*e + 22, -4*e^3 - 4*e^2 + 44*e - 4, -5/2*e^3 - 9/2*e^2 + 21/2*e + 5, 5*e^3 + 7*e^2 - 38*e + 2, -e^3 + e^2 + 25*e - 6, -7*e^3 - 9*e^2 + 63*e]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); ALEigenvalues[ideal] := 1; ALEigenvalues[ideal] := 1; // EXAMPLE: // pp := Factorization(2*ZF)[1][1]; // heckeEigenvalues[pp]; print "To reconstruct the Hilbert newform f, type f, iso := Explode(make_newform());"; function make_newform(); M := HilbertCuspForms(F, NN); S := NewSubspace(M); // SetVerbose("ModFrmHil", 1); NFD := NewformDecomposition(S); newforms := [* Eigenform(U) : U in NFD *]; if #newforms eq 0 then; print "No Hilbert newforms at this level"; return 0; end if; print "Testing ", #newforms, " possible newforms"; newforms := [* f: f in newforms | IsIsomorphic(BaseField(f), K) *]; print #newforms, " newforms have the correct Hecke field"; if #newforms eq 0 then; print "No Hilbert newform found with the correct Hecke field"; return 0; end if; autos := Automorphisms(K); xnewforms := [* *]; for f in newforms do; if K eq RationalField() then; Append(~xnewforms, [* f, autos[1] *]); else; flag, iso := IsIsomorphic(K,BaseField(f)); for a in autos do; Append(~xnewforms, [* f, a*iso *]); end for; end if; end for; newforms := xnewforms; for P in primes do; xnewforms := [* *]; for f_iso in newforms do; f, iso := Explode(f_iso); if HeckeEigenvalue(f,P) eq iso(heckeEigenvalues[P]) then; Append(~xnewforms, f_iso); end if; end for; newforms := xnewforms; if #newforms eq 0 then; print "No Hilbert newform found which matches the Hecke eigenvalues"; return 0; else if #newforms eq 1 then; print "success: unique match"; return newforms[1]; end if; end if; end for; print #newforms, "Hilbert newforms found which match the Hecke eigenvalues"; return newforms[1]; end function;