/* 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![25, 5, -11, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, 1/5*w^3 + 4/5*w^2 - 11/5*w - 7], [5, 5, 1/5*w^3 - 6/5*w^2 - 1/5*w + 4], [11, 11, w + 1], [11, 11, 1/5*w^3 - 1/5*w^2 - 11/5*w + 2], [16, 2, 2], [19, 19, -1/5*w^3 + 1/5*w^2 + 1/5*w + 1], [19, 19, 1/5*w^3 - 1/5*w^2 - 11/5*w], [19, 19, -2/5*w^3 + 2/5*w^2 + 17/5*w], [19, 19, w - 1], [29, 29, w^2 - w - 8], [29, 29, 2/5*w^3 - 2/5*w^2 - 7/5*w + 2], [31, 31, 1/5*w^3 - 1/5*w^2 - 1/5*w + 2], [31, 31, 2/5*w^3 - 2/5*w^2 - 17/5*w + 3], [61, 61, -2/5*w^3 - 3/5*w^2 + 12/5*w + 5], [61, 61, -4/5*w^3 - 6/5*w^2 + 39/5*w + 15], [61, 61, 3/5*w^3 + 2/5*w^2 - 18/5*w - 3], [61, 61, 7/5*w^3 + 3/5*w^2 - 62/5*w - 13], [71, 71, 1/5*w^3 - 6/5*w^2 - 6/5*w + 4], [71, 71, w^2 - 8], [79, 79, 3/5*w^3 + 12/5*w^2 - 33/5*w - 22], [79, 79, 1/5*w^3 + 4/5*w^2 - 11/5*w - 9], [81, 3, -3], [101, 101, 3/5*w^3 + 7/5*w^2 - 28/5*w - 15], [101, 101, 1/5*w^3 + 9/5*w^2 - 16/5*w - 16], [121, 11, 3/5*w^3 - 3/5*w^2 - 18/5*w + 1], [131, 131, 1/5*w^3 + 4/5*w^2 - 16/5*w - 8], [131, 131, w^2 - 2*w - 2], [149, 149, 4/5*w^3 + 1/5*w^2 - 29/5*w - 5], [149, 149, 4/5*w^3 - 9/5*w^2 - 19/5*w + 6], [151, 151, 2/5*w^3 + 3/5*w^2 - 17/5*w - 3], [151, 151, -2/5*w^3 + 7/5*w^2 + 7/5*w - 8], [151, 151, 4/5*w^3 + 1/5*w^2 - 29/5*w - 2], [151, 151, -4/5*w^3 + 9/5*w^2 + 19/5*w - 9], [179, 179, -3/5*w^3 + 8/5*w^2 + 18/5*w - 6], [179, 179, 2/5*w^3 + 3/5*w^2 - 12/5*w - 6], [211, 211, 1/5*w^3 + 4/5*w^2 - 6/5*w - 10], [211, 211, 2/5*w^3 - 7/5*w^2 - 12/5*w + 2], [229, 229, 6/5*w^3 - 11/5*w^2 - 41/5*w + 11], [229, 229, 2*w^2 - 2*w - 9], [229, 229, -4/5*w^3 - 1/5*w^2 + 19/5*w + 2], [229, 229, -3/5*w^3 + 3/5*w^2 + 28/5*w - 3], [239, 239, 11/5*w^3 + 4/5*w^2 - 96/5*w - 19], [239, 239, -6/5*w^3 + 1/5*w^2 + 51/5*w + 3], [241, 241, 1/5*w^3 + 4/5*w^2 - 1/5*w - 6], [241, 241, 4/5*w^3 - 4/5*w^2 - 29/5*w - 1], [251, 251, -1/5*w^3 + 6/5*w^2 + 11/5*w - 12], [251, 251, -1/5*w^3 + 11/5*w^2 + 1/5*w - 15], [269, 269, 2/5*w^3 + 3/5*w^2 - 17/5*w - 2], [269, 269, -2/5*w^3 + 7/5*w^2 + 7/5*w - 9], [271, 271, 1/5*w^3 - 1/5*w^2 + 4/5*w], [271, 271, 3/5*w^3 - 3/5*w^2 - 28/5*w + 2], [281, 281, w^3 - 3*w^2 - 4*w + 13], [281, 281, -3/5*w^3 + 3/5*w^2 + 28/5*w - 6], [281, 281, -6/5*w^3 + 26/5*w^2 + 11/5*w - 16], [281, 281, w^3 + w^2 - 8*w - 9], [289, 17, 2/5*w^3 + 3/5*w^2 - 7/5*w - 5], [289, 17, 4/5*w^3 - 9/5*w^2 - 29/5*w + 8], [311, 311, -w^2 + 2*w + 11], [311, 311, -1/5*w^3 + 6/5*w^2 + 1/5*w - 1], [311, 311, 1/5*w^3 + 4/5*w^2 - 16/5*w + 1], [311, 311, -1/5*w^3 - 4/5*w^2 + 11/5*w + 10], [349, 349, -1/5*w^3 - 9/5*w^2 + 11/5*w + 16], [349, 349, 2/5*w^3 - 12/5*w^2 - 7/5*w + 7], [359, 359, -4/5*w^3 + 4/5*w^2 + 19/5*w - 1], [359, 359, -w^3 + 3*w^2 + 5*w - 12], [359, 359, w^3 - w^2 - 7*w + 2], [359, 359, 4/5*w^3 + 6/5*w^2 - 29/5*w - 11], [379, 379, 4/5*w^3 + 6/5*w^2 - 34/5*w - 9], [379, 379, 3/5*w^3 - 3/5*w^2 - 28/5*w + 7], [389, 389, w^2 - 3*w - 8], [389, 389, -4/5*w^3 + 14/5*w^2 + 19/5*w - 16], [401, 401, -3*w^2 + 2*w + 23], [401, 401, 1/5*w^3 - 16/5*w^2 + 4/5*w + 11], [409, 409, -w^3 + 2*w^2 + 3*w - 7], [409, 409, 7/5*w^3 - 2/5*w^2 - 57/5*w - 2], [419, 419, -1/5*w^3 + 1/5*w^2 + 16/5*w - 2], [419, 419, 1/5*w^3 - 1/5*w^2 - 16/5*w], [421, 421, 2/5*w^3 + 3/5*w^2 - 27/5*w - 3], [421, 421, 1/5*w^3 + 4/5*w^2 + 4/5*w - 3], [439, 439, -w^3 + 3*w^2 + 5*w - 16], [439, 439, -4/5*w^3 - 6/5*w^2 + 29/5*w + 7], [439, 439, -3/5*w^3 + 13/5*w^2 + 3/5*w - 11], [439, 439, 4/5*w^3 + 6/5*w^2 - 39/5*w - 10], [449, 449, 2/5*w^3 - 12/5*w^2 - 2/5*w + 5], [449, 449, 4/5*w^3 - 24/5*w^2 - 4/5*w + 15], [461, 461, -1/5*w^3 - 4/5*w^2 - 4/5*w + 4], [461, 461, 2*w^2 - 2*w - 11], [461, 461, 3/5*w^3 + 2/5*w^2 - 33/5*w - 4], [479, 479, -w^3 + 3*w^2 + 6*w - 14], [479, 479, -3/5*w^3 + 3/5*w^2 + 28/5*w + 3], [499, 499, 2/5*w^3 + 8/5*w^2 - 17/5*w - 6], [499, 499, -3/5*w^3 + 13/5*w^2 + 13/5*w - 17], [509, 509, -w^3 + 2*w^2 + 4*w - 6], [509, 509, 6/5*w^3 - 1/5*w^2 - 46/5*w - 4], [521, 521, -2/5*w^3 + 2/5*w^2 + 7/5*w - 7], [521, 521, -3/5*w^3 + 3/5*w^2 + 23/5*w - 8], [529, 23, -7/5*w^3 + 17/5*w^2 + 42/5*w - 15], [529, 23, 2/5*w^3 + 13/5*w^2 - 27/5*w - 21], [541, 541, 1/5*w^3 - 11/5*w^2 - 1/5*w + 10], [541, 541, 3/5*w^3 - 8/5*w^2 - 13/5*w + 13], [541, 541, -2*w^2 + w + 13], [541, 541, 3/5*w^3 + 2/5*w^2 - 23/5*w + 2], [569, 569, w^3 - 2*w^2 - 5*w + 7], [569, 569, -1/5*w^3 + 1/5*w^2 + 1/5*w - 6], [571, 571, w^2 + w - 9], [571, 571, -2/5*w^3 + 12/5*w^2 + 7/5*w - 12], [571, 571, 1/5*w^3 + 9/5*w^2 - 11/5*w - 11], [571, 571, -2/5*w^3 + 7/5*w^2 + 17/5*w - 4], [601, 601, -3/5*w^3 + 13/5*w^2 + 18/5*w - 11], [601, 601, -4/5*w^3 + 4/5*w^2 + 34/5*w + 1], [631, 631, 2/5*w^3 + 18/5*w^2 - 27/5*w - 30], [631, 631, 3/5*w^3 - 23/5*w^2 - 3/5*w + 15], [661, 661, -2*w^3 + 17*w + 12], [661, 661, -9/5*w^3 + 4/5*w^2 + 74/5*w + 1], [691, 691, -7/5*w^3 - 18/5*w^2 + 72/5*w + 35], [691, 691, 6/5*w^3 - 31/5*w^2 - 6/5*w + 19], [701, 701, -1/5*w^3 + 6/5*w^2 + 16/5*w - 6], [701, 701, 2/5*w^3 - 7/5*w^2 - 22/5*w + 8], [709, 709, -3/5*w^3 + 13/5*w^2 + 13/5*w - 18], [709, 709, 1/5*w^3 + 14/5*w^2 - 11/5*w - 20], [709, 709, -2/5*w^3 - 8/5*w^2 + 17/5*w + 5], [709, 709, 1/5*w^3 + 4/5*w^2 - 21/5*w - 10], [719, 719, -3/5*w^3 + 8/5*w^2 + 13/5*w - 12], [719, 719, 3/5*w^3 + 2/5*w^2 - 23/5*w + 1], [719, 719, -7/5*w^3 + 17/5*w^2 + 32/5*w - 14], [719, 719, 7/5*w^3 + 3/5*w^2 - 52/5*w - 8], [739, 739, 1/5*w^3 - 1/5*w^2 - 11/5*w - 5], [739, 739, w - 6], [761, 761, -4/5*w^3 + 9/5*w^2 + 19/5*w - 11], [761, 761, 4/5*w^3 + 1/5*w^2 - 29/5*w], [769, 769, 3/5*w^3 - 13/5*w^2 - 13/5*w + 8], [769, 769, -2/5*w^3 - 8/5*w^2 + 17/5*w + 15], [769, 769, 7/5*w^3 - 17/5*w^2 - 37/5*w + 18], [769, 769, -6/5*w^3 - 4/5*w^2 + 41/5*w + 5], [809, 809, -1/5*w^3 + 6/5*w^2 - 4/5*w - 14], [809, 809, -2/5*w^3 - 3/5*w^2 + 22/5*w - 4], [809, 809, -6/5*w^3 + 6/5*w^2 + 41/5*w - 5], [809, 809, w^3 - w^2 - 5*w + 4], [821, 821, -3/5*w^3 - 17/5*w^2 + 33/5*w + 30], [821, 821, -4/5*w^3 + 24/5*w^2 + 9/5*w - 15], [829, 829, 7/5*w^3 - 12/5*w^2 - 52/5*w + 10], [829, 829, -1/5*w^3 + 11/5*w^2 - 4/5*w - 19], [829, 829, -13/5*w^3 + 3/5*w^2 + 108/5*w + 10], [829, 829, -w^3 + 4*w^2 + 5*w - 23], [839, 839, 4/5*w^3 - 9/5*w^2 - 24/5*w + 4], [839, 839, 3/5*w^3 + 2/5*w^2 - 18/5*w - 8], [841, 29, -w^3 + w^2 + 6*w - 4], [859, 859, 6/5*w^3 - 11/5*w^2 - 36/5*w + 9], [859, 859, w^3 - 6*w - 3], [881, 881, 4/5*w^3 - 4/5*w^2 - 9/5*w - 1], [881, 881, 9/5*w^3 - 24/5*w^2 - 19/5*w + 14], [881, 881, -w^3 + 6*w + 2], [881, 881, 1/5*w^3 - 1/5*w^2 - 1/5*w - 6], [911, 911, w^3 - w^2 - 5*w + 1], [911, 911, 6/5*w^3 - 6/5*w^2 - 41/5*w + 2], [919, 919, -7/5*w^3 + 12/5*w^2 + 42/5*w - 7], [919, 919, -6/5*w^3 + 1/5*w^2 + 36/5*w + 5], [929, 929, -2/5*w^3 - 8/5*w^2 + 17/5*w + 16], [929, 929, -3/5*w^3 + 13/5*w^2 + 13/5*w - 7], [961, 31, w^3 - w^2 - 6*w + 2], [971, 971, 8/5*w^3 - 13/5*w^2 - 48/5*w + 6], [971, 971, -2/5*w^3 - 3/5*w^2 + 22/5*w], [971, 971, 7/5*w^3 - 7/5*w^2 - 57/5*w + 7], [971, 971, -1/5*w^3 + 6/5*w^2 - 4/5*w - 10], [991, 991, 2/5*w^3 + 8/5*w^2 - 22/5*w - 7], [991, 991, -2/5*w^3 + 12/5*w^2 + 2/5*w - 15]]; primes := [ideal : I in primesArray]; heckePol := x^10 + x^9 - 17*x^8 - 19*x^7 + 92*x^6 + 119*x^5 - 154*x^4 - 255*x^3 - 23*x^2 + 64*x + 12; K := NumberField(heckePol); heckeEigenvaluesArray := [e, -3471/9323*e^9 - 723/9323*e^8 + 58967/9323*e^7 + 19450/9323*e^6 - 324626/9323*e^5 - 156219/9323*e^4 + 607980/9323*e^3 + 388375/9323*e^2 - 161336/9323*e - 61038/9323, 626/9323*e^9 + 493/9323*e^8 - 11685/9323*e^7 - 7937/9323*e^6 + 71982/9323*e^5 + 42461/9323*e^4 - 159174/9323*e^3 - 75980/9323*e^2 + 85473/9323*e - 7122/9323, 5003/9323*e^9 + 470/9323*e^8 - 85270/9323*e^7 - 17041/9323*e^6 + 473711/9323*e^5 + 157878/9323*e^4 - 908613/9323*e^3 - 431645/9323*e^2 + 265279/9323*e + 69969/9323, -2725/9323*e^9 - 761/9323*e^8 + 46025/9323*e^7 + 15353/9323*e^6 - 253113/9323*e^5 - 104159/9323*e^4 + 485699/9323*e^3 + 241773/9323*e^2 - 175167/9323*e - 47573/9323, 5793/9323*e^9 - 695/9323*e^8 - 95876/9323*e^7 - 1084/9323*e^6 + 511800/9323*e^5 + 94434/9323*e^4 - 936878/9323*e^3 - 366716/9323*e^2 + 288999/9323*e + 85078/9323, -5013/9323*e^9 + 1905/9323*e^8 + 83044/9323*e^7 - 20571/9323*e^6 - 444926/9323*e^5 + 28559/9323*e^4 + 827070/9323*e^3 + 131425/9323*e^2 - 291188/9323*e - 21185/9323, 3528/9323*e^9 + 1170/9323*e^8 - 59331/9323*e^7 - 26949/9323*e^6 + 323704/9323*e^5 + 196439/9323*e^4 - 597215/9323*e^3 - 457814/9323*e^2 + 121625/9323*e + 58930/9323, -2019/9323*e^9 - 622/9323*e^8 + 33502/9323*e^7 + 9976/9323*e^6 - 178336/9323*e^5 - 52191/9323*e^4 + 316936/9323*e^3 + 96958/9323*e^2 - 78205/9323*e - 16109/9323, -3075/9323*e^9 - 1543/9323*e^8 + 52022/9323*e^7 + 32122/9323*e^6 - 280491/9323*e^5 - 216840/9323*e^4 + 484532/9323*e^3 + 492815/9323*e^2 - 47129/9323*e - 102180/9323, 1, 2891/9323*e^9 - 1372/9323*e^8 - 48230/9323*e^7 + 17928/9323*e^6 + 260078/9323*e^5 - 59544/9323*e^4 - 489514/9323*e^3 + 24441/9323*e^2 + 206232/9323*e - 46235/9323, -386/9323*e^9 - 1555/9323*e^8 + 9171/9323*e^7 + 24940/9323*e^6 - 72920/9323*e^5 - 125816/9323*e^4 + 214314/9323*e^3 + 205103/9323*e^2 - 162882/9323*e - 54257/9323, 15359/9323*e^9 + 2192/9323*e^8 - 261807/9323*e^7 - 67892/9323*e^6 + 1448511/9323*e^5 + 586315/9323*e^4 - 2743318/9323*e^3 - 1545977/9323*e^2 + 779675/9323*e + 292483/9323, 1360/9323*e^9 + 3305/9323*e^8 - 23569/9323*e^7 - 49710/9323*e^6 + 131422/9323*e^5 + 226880/9323*e^4 - 237597/9323*e^3 - 321802/9323*e^2 + 36822/9323*e + 59290/9323, 5203/9323*e^9 - 415/9323*e^8 - 87365/9323*e^7 - 1318/9323*e^6 + 476037/9323*e^5 + 74431/9323*e^4 - 890632/9323*e^3 - 300735/9323*e^2 + 252048/9323*e + 17266/9323, 5433/9323*e^9 + 898/9323*e^8 - 92105/9323*e^7 - 21927/9323*e^6 + 503884/9323*e^5 + 168190/9323*e^4 - 935681/9323*e^3 - 406571/9323*e^2 + 279252/9323*e - 4652/9323, 1859/9323*e^9 + 1330/9323*e^8 - 31826/9323*e^7 - 24419/9323*e^6 + 172746/9323*e^5 + 154376/9323*e^4 - 288435/9323*e^3 - 350854/9323*e^2 - 28680/9323*e + 97428/9323, -189/9323*e^9 + 2934/9323*e^8 + 2679/9323*e^7 - 52663/9323*e^6 - 20005/9323*e^5 + 306292/9323*e^4 + 108076/9323*e^3 - 593123/9323*e^2 - 240756/9323*e + 99729/9323, -5127/9323*e^9 + 1011/9323*e^8 + 83772/9323*e^7 - 5573/9323*e^6 - 433759/9323*e^5 - 51881/9323*e^4 + 721633/9323*e^3 + 270303/9323*e^2 - 71921/9323*e - 35615/9323, 3620/9323*e^9 - 2034/9323*e^8 - 61227/9323*e^7 + 22610/9323*e^6 + 338572/9323*e^5 - 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527064/9323, 33483/9323*e^9 + 4968/9323*e^8 - 570058/9323*e^7 - 149857/9323*e^6 + 3137548/9323*e^5 + 1276881/9323*e^4 - 5804654/9323*e^3 - 3288899/9323*e^2 + 1166897/9323*e + 437403/9323, 20422/9323*e^9 + 16381/9323*e^8 - 361690/9323*e^7 - 316088/9323*e^6 + 2122432/9323*e^5 + 1993613/9323*e^4 - 4365340/9323*e^3 - 4241130/9323*e^2 + 1256346/9323*e + 890172/9323, 13388/9323*e^9 - 507/9323*e^8 - 232537/9323*e^7 - 19088/9323*e^6 + 1337113/9323*e^5 + 284378/9323*e^4 - 2778951/9323*e^3 - 865679/9323*e^2 + 1204347/9323*e - 65936/9323, 40473/9323*e^9 - 4986/9323*e^8 - 692224/9323*e^7 - 761/9323*e^6 + 3872384/9323*e^5 + 584410/9323*e^4 - 7596935/9323*e^3 - 2260996/9323*e^2 + 2794224/9323*e + 226492/9323, 34089/9323*e^9 + 888/9323*e^8 - 586195/9323*e^7 - 102496/9323*e^6 + 3285746/9323*e^5 + 1179451/9323*e^4 - 6304051/9323*e^3 - 3517815/9323*e^2 + 1703994/9323*e + 546495/9323, -32420/9323*e^9 - 1048/9323*e^8 + 540044/9323*e^7 + 90643/9323*e^6 - 2883067/9323*e^5 - 1006866/9323*e^4 + 5128232/9323*e^3 + 2954028/9323*e^2 - 1152800/9323*e - 575670/9323, -39969/9323*e^9 - 2838/9323*e^8 + 666434/9323*e^7 + 128765/9323*e^6 - 3570424/9323*e^5 - 1267559/9323*e^4 + 6426155/9323*e^3 + 3547429/9323*e^2 - 1527567/9323*e - 594989/9323, -3557/9323*e^9 + 10379/9323*e^8 + 51011/9323*e^7 - 141793/9323*e^6 - 207597/9323*e^5 + 539079/9323*e^4 + 201317/9323*e^3 - 487408/9323*e^2 + 61486/9323*e - 59166/9323, -25337/9323*e^9 - 459/9323*e^8 + 432986/9323*e^7 + 46932/9323*e^6 - 2411270/9323*e^5 - 527222/9323*e^4 + 4660207/9323*e^3 + 1478513/9323*e^2 - 1567722/9323*e - 201690/9323, -11005/9323*e^9 - 1414/9323*e^8 + 188696/9323*e^7 + 30083/9323*e^6 - 1069145/9323*e^5 - 207110/9323*e^4 + 2256631/9323*e^3 + 499806/9323*e^2 - 1359617/9323*e - 144210/9323, -13265/9323*e^9 - 5398/9323*e^8 + 217031/9323*e^7 + 116254/9323*e^6 - 1127127/9323*e^5 - 836949/9323*e^4 + 1886564/9323*e^3 + 2059821/9323*e^2 - 204155/9323*e - 549024/9323, 44639/9323*e^9 + 3150/9323*e^8 - 754975/9323*e^7 - 147139/9323*e^6 + 4149621/9323*e^5 + 1446114/9323*e^4 - 7819156/9323*e^3 - 3944852/9323*e^2 + 2169103/9323*e + 402490/9323, -26469/9323*e^9 + 16670/9323*e^8 + 455099/9323*e^7 - 203721/9323*e^6 - 2565026/9323*e^5 + 553507/9323*e^4 + 5127324/9323*e^3 + 291923/9323*e^2 - 2154954/9323*e - 138890/9323]; heckeEigenvalues := AssociativeArray(); for i := 1 to #heckeEigenvaluesArray do heckeEigenvalues[primes[i]] := heckeEigenvaluesArray[i]; end for; ALEigenvalues := AssociativeArray(); 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;