/* 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![29, 8, -12, -1, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [5, 5, -w + 2], [5, 5, -3/5*w^3 - w^2 + 26/5*w + 42/5], [9, 3, 2/5*w^3 - 19/5*w - 3/5], [9, 3, -1/5*w^3 + 2/5*w + 4/5], [11, 11, -2/5*w^3 - w^2 + 14/5*w + 28/5], [11, 11, 1/5*w^3 - w^2 - 7/5*w + 36/5], [16, 2, 2], [29, 29, -w], [29, 29, 1/5*w^3 - 12/5*w + 1/5], [41, 41, -2/5*w^3 - w^2 + 14/5*w + 38/5], [41, 41, -w^2 + 10], [41, 41, 11/5*w^3 + 3*w^2 - 102/5*w - 149/5], [41, 41, -1/5*w^3 + w^2 + 7/5*w - 26/5], [49, 7, 1/5*w^3 + w^2 - 12/5*w - 19/5], [49, 7, 1/5*w^3 + w^2 - 12/5*w - 44/5], [59, 59, -4/5*w^3 - w^2 + 33/5*w + 36/5], [59, 59, -2*w^3 - 3*w^2 + 17*w + 26], [61, 61, -1/5*w^3 + w^2 + 12/5*w - 51/5], [61, 61, 4/5*w^3 + w^2 - 28/5*w - 41/5], [71, 71, 1/5*w^3 + w^2 - 7/5*w - 49/5], [71, 71, -2*w^3 - 3*w^2 + 18*w + 28], [89, 89, -3/5*w^3 + 16/5*w + 7/5], [89, 89, 3/5*w^3 + w^2 - 26/5*w - 57/5], [121, 11, -3/5*w^3 + 21/5*w + 2/5], [131, 131, w^2 - w - 8], [131, 131, 2/5*w^3 + w^2 - 19/5*w - 23/5], [149, 149, 2/5*w^3 - w^2 - 14/5*w + 22/5], [149, 149, 3/5*w^3 + w^2 - 21/5*w - 42/5], [179, 179, -6/5*w^3 - w^2 + 57/5*w + 69/5], [179, 179, -2/5*w^3 - 2*w^2 + 14/5*w + 73/5], [179, 179, 2*w^2 - 11], [179, 179, -3/5*w^3 + 2*w^2 + 16/5*w - 48/5], [181, 181, -7/5*w^3 - 2*w^2 + 59/5*w + 78/5], [181, 181, -2/5*w^3 + 24/5*w - 27/5], [191, 191, -3/5*w^3 + 2*w^2 + 21/5*w - 78/5], [191, 191, 2/5*w^3 - 2*w^2 - 9/5*w + 57/5], [199, 199, -1/5*w^3 + 17/5*w - 16/5], [199, 199, 1/5*w^3 - 17/5*w - 14/5], [211, 211, 2/5*w^3 - w^2 + 1/5*w + 7/5], [211, 211, -1/5*w^3 + w^2 - 3/5*w - 26/5], [211, 211, 4/5*w^3 + w^2 - 38/5*w - 36/5], [211, 211, -1/5*w^3 - 2*w^2 + 7/5*w + 44/5], [239, 239, 3/5*w^3 + w^2 - 16/5*w - 37/5], [239, 239, 2/5*w^3 + w^2 - 14/5*w - 53/5], [251, 251, 2/5*w^3 - 24/5*w - 13/5], [251, 251, 2*w - 3], [251, 251, w^2 + w - 10], [251, 251, -4/5*w^3 - w^2 + 28/5*w + 46/5], [269, 269, 1/5*w^3 - 12/5*w - 24/5], [269, 269, w - 5], [271, 271, -3/5*w^3 + 31/5*w - 33/5], [271, 271, -8/5*w^3 - 2*w^2 + 66/5*w + 77/5], [281, 281, 4/5*w^3 - w^2 - 38/5*w + 39/5], [281, 281, -14/5*w^3 - 3*w^2 + 128/5*w + 161/5], [311, 311, 1/5*w^3 - 2*w^2 - 2/5*w + 51/5], [311, 311, 1/5*w^3 - w^2 - 17/5*w + 21/5], [331, 331, 1/5*w^3 + 3/5*w - 4/5], [331, 331, 3/5*w^3 - 31/5*w - 2/5], [359, 359, -w^3 + 6*w - 1], [359, 359, -6/5*w^3 + 47/5*w - 6/5], [359, 359, 3/5*w^3 + w^2 - 31/5*w - 27/5], [359, 359, 3/5*w^3 + w^2 - 26/5*w - 22/5], [361, 19, 4/5*w^3 - 28/5*w - 11/5], [361, 19, 1/5*w^3 - 7/5*w - 24/5], [379, 379, 9/5*w^3 + 3*w^2 - 78/5*w - 121/5], [379, 379, -4/5*w^3 + 3*w^2 + 28/5*w - 99/5], [389, 389, -3/5*w^3 - 2*w^2 + 26/5*w + 42/5], [389, 389, -2*w^2 + w + 17], [401, 401, 4/5*w^3 - w^2 - 23/5*w + 24/5], [401, 401, 3/5*w^3 + w^2 - 26/5*w - 7/5], [401, 401, -4/5*w^3 - w^2 + 38/5*w + 66/5], [401, 401, 6/5*w^3 + w^2 - 52/5*w - 39/5], [409, 409, -6/5*w^3 + 52/5*w + 19/5], [409, 409, 1/5*w^3 + 2*w^2 - 7/5*w - 39/5], [421, 421, -1/5*w^3 - w^2 + 17/5*w + 44/5], [421, 421, 2/5*w^3 + w^2 - 24/5*w - 18/5], [431, 431, -3/5*w^3 - w^2 + 36/5*w + 47/5], [431, 431, -13/5*w^3 - 4*w^2 + 116/5*w + 182/5], [431, 431, -2/5*w^3 + w^2 + 24/5*w - 47/5], [431, 431, 2/5*w^3 - w^2 + 6/5*w - 8/5], [439, 439, 6/5*w^3 + w^2 - 47/5*w - 49/5], [439, 439, -2/5*w^3 + 2*w^2 + 9/5*w - 42/5], [439, 439, 4/5*w^3 - w^2 - 23/5*w + 14/5], [439, 439, -w^3 - 2*w^2 + 8*w + 17], [461, 461, w^3 + w^2 - 10*w - 9], [461, 461, -2*w^3 - 3*w^2 + 17*w + 27], [461, 461, -1/5*w^3 + w^2 - 8/5*w - 16/5], [461, 461, -3*w^3 - 4*w^2 + 26*w + 35], [479, 479, 2*w - 1], [479, 479, 2/5*w^3 - 24/5*w - 3/5], [491, 491, w^3 - 2*w^2 - 6*w + 11], [491, 491, w^3 + w^2 - 10*w - 15], [499, 499, -1/5*w^3 + 2*w^2 + 2/5*w - 36/5], [499, 499, 4/5*w^3 + 2*w^2 - 33/5*w - 91/5], [509, 509, 8/5*w^3 + w^2 - 66/5*w - 42/5], [509, 509, 1/5*w^3 - 2*w^2 + 8/5*w + 16/5], [521, 521, w^3 - 9*w - 4], [521, 521, 3/5*w^3 + w^2 - 16/5*w - 42/5], [521, 521, 3/5*w^3 - w^2 - 26/5*w + 23/5], [521, 521, -3/5*w^3 + 11/5*w + 22/5], [529, 23, 3/5*w^3 + w^2 - 36/5*w - 62/5], [529, 23, -4/5*w^3 + 2*w^2 + 23/5*w - 64/5], [541, 541, -w^3 - w^2 + 9*w + 7], [541, 541, 1/5*w^3 + 2*w^2 - 7/5*w - 64/5], [541, 541, 3/5*w^3 + w^2 - 26/5*w - 12/5], [571, 571, -w^3 - w^2 + 6*w + 9], [571, 571, -w^3 + w^2 + 8*w - 4], [619, 619, -w^3 - 2*w^2 + 8*w + 11], [619, 619, -2/5*w^3 + 2*w^2 + 9/5*w - 72/5], [619, 619, 2/5*w^3 - 2*w^2 - 4/5*w + 57/5], [619, 619, -6/5*w^3 - 2*w^2 + 52/5*w + 69/5], [631, 631, 1/5*w^3 + w^2 + 3/5*w - 24/5], [631, 631, 2/5*w^3 - w^2 - 24/5*w + 42/5], [659, 659, -13/5*w^3 - 4*w^2 + 111/5*w + 177/5], [659, 659, -11/5*w^3 - 3*w^2 + 92/5*w + 119/5], [659, 659, -8/5*w^3 - w^2 + 61/5*w + 27/5], [659, 659, -8/5*w^3 + 71/5*w + 2/5], [661, 661, -w^3 - w^2 + 7*w + 11], [661, 661, 4/5*w^3 - w^2 - 28/5*w + 9/5], [691, 691, 1/5*w^3 - w^2 - 17/5*w + 41/5], [691, 691, 3/5*w^3 + 2*w^2 - 26/5*w - 67/5], [701, 701, 12/5*w^3 + 4*w^2 - 99/5*w - 158/5], [701, 701, -17/5*w^3 - 4*w^2 + 149/5*w + 193/5], [709, 709, -1/5*w^3 + w^2 + 12/5*w - 6/5], [709, 709, 1/5*w^3 + w^2 - 2/5*w - 59/5], [719, 719, 2/5*w^3 + w^2 - 29/5*w - 8/5], [719, 719, -w^3 + w^2 + 7*w - 4], [719, 719, 6/5*w^3 + w^2 - 42/5*w - 44/5], [719, 719, -2/5*w^3 - w^2 + 29/5*w + 53/5], [739, 739, 14/5*w^3 + 4*w^2 - 123/5*w - 171/5], [739, 739, -1/5*w^3 + 2*w^2 + 12/5*w - 56/5], [739, 739, -6/5*w^3 + 3*w^2 + 37/5*w - 96/5], [739, 739, 2/5*w^3 + 2*w^2 - 9/5*w - 73/5], [751, 751, -9/5*w^3 - 2*w^2 + 73/5*w + 86/5], [751, 751, -w^3 + 2*w^2 + 5*w - 8], [761, 761, 1/5*w^3 + w^2 - 22/5*w + 16/5], [761, 761, -16/5*w^3 - 4*w^2 + 147/5*w + 214/5], [769, 769, 13/5*w^3 + 3*w^2 - 121/5*w - 172/5], [769, 769, 3/5*w^3 - w^2 - 41/5*w + 78/5], [769, 769, -12/5*w^3 - 4*w^2 + 99/5*w + 153/5], [769, 769, -6/5*w^3 + 3*w^2 + 27/5*w - 61/5], [809, 809, 1/5*w^3 + 3*w^2 - 22/5*w - 104/5], [809, 809, w^3 + 3*w^2 - 10*w - 17], [811, 811, -5*w + 11], [811, 811, -16/5*w^3 - 5*w^2 + 137/5*w + 214/5], [821, 821, -1/5*w^3 + w^2 + 17/5*w - 36/5], [821, 821, w^2 + 2*w - 6], [829, 829, 3*w^2 + w - 17], [829, 829, 2/5*w^3 + 3*w^2 - 9/5*w - 108/5], [841, 29, w^3 - 7*w - 3], [859, 859, 6/5*w^3 + w^2 - 37/5*w - 34/5], [859, 859, 2/5*w^3 + w^2 - 14/5*w - 68/5], [881, 881, 1/5*w^3 + 2*w^2 - 17/5*w - 79/5], [881, 881, 3/5*w^3 + 2*w^2 - 31/5*w - 47/5], [911, 911, 4/5*w^3 - 3*w^2 - 28/5*w + 89/5], [911, 911, 1/5*w^3 - w^2 - 7/5*w + 61/5], [911, 911, w^3 - 6*w - 3], [911, 911, -11/5*w^3 - 3*w^2 + 92/5*w + 139/5], [919, 919, 7/5*w^3 + 2*w^2 - 54/5*w - 88/5], [919, 919, 4/5*w^3 - 2*w^2 - 23/5*w + 39/5], [929, 929, -7/5*w^3 + w^2 + 54/5*w - 17/5], [929, 929, -1/5*w^3 + w^2 + 22/5*w - 71/5], [941, 941, 9/5*w^3 + w^2 - 73/5*w - 31/5], [941, 941, 1/5*w^3 - w^2 - 2/5*w + 66/5], [941, 941, 2/5*w^3 + 2*w^2 - 19/5*w - 103/5], [941, 941, -w^3 + w^2 + 5*w - 3], [961, 31, w^3 - 7*w - 2], [961, 31, 2/5*w^3 - 14/5*w - 33/5], [971, 971, -2/5*w^3 - 2*w^2 + 24/5*w + 93/5], [971, 971, 2/5*w^3 + 2*w^2 - 24/5*w - 33/5], [991, 991, -w^3 + 2*w^2 + 7*w - 10], [991, 991, -7/5*w^3 - 2*w^2 + 49/5*w + 78/5]]; primes := [ideal : I in primesArray]; heckePol := x^15 + 3*x^14 - 23*x^13 - 76*x^12 + 149*x^11 + 658*x^10 - 88*x^9 - 2176*x^8 - 1476*x^7 + 2192*x^6 + 2629*x^5 - 210*x^4 - 1182*x^3 - 395*x^2 - 10*x + 1; K := NumberField(heckePol); heckeEigenvaluesArray := [95752307/63244487*e^14 + 128558854/63244487*e^13 - 2422800864/63244487*e^12 - 3269218181/63244487*e^11 + 19883532072/63244487*e^10 + 30255211335/63244487*e^9 - 60360404141/63244487*e^8 - 110467158749/63244487*e^7 + 48386335734/63244487*e^6 + 139309969292/63244487*e^5 + 11988778965/63244487*e^4 - 57190450693/63244487*e^3 - 18027790555/63244487*e^2 + 1397170238/63244487*e + 126255219/63244487, e, 300616356/63244487*e^14 + 467502655/63244487*e^13 - 7548767757/63244487*e^12 - 11880800234/63244487*e^11 + 60965200525/63244487*e^10 + 108278098048/63244487*e^9 - 175423270456/63244487*e^8 - 388410589205/63244487*e^7 + 99447307949/63244487*e^6 + 478426501757/63244487*e^5 + 100918780462/63244487*e^4 - 186831612768/63244487*e^3 - 79916397810/63244487*e^2 - 29453745/63244487*e + 188068850/63244487, -235568438/63244487*e^14 - 370526337/63244487*e^13 + 5905730022/63244487*e^12 + 9424617584/63244487*e^11 - 47518147589/63244487*e^10 - 85928531819/63244487*e^9 + 135037448065/63244487*e^8 + 308433686601/63244487*e^7 - 67985026068/63244487*e^6 - 379776513005/63244487*e^5 - 96006774215/63244487*e^4 + 144526782957/63244487*e^3 + 71796228431/63244487*e^2 + 3556035506/63244487*e - 413206600/63244487, 130543335/63244487*e^14 + 207463673/63244487*e^13 - 3299269461/63244487*e^12 - 5294282384/63244487*e^11 + 26995224224/63244487*e^10 + 48496503914/63244487*e^9 - 80059019627/63244487*e^8 - 176652929439/63244487*e^7 + 52985403250/63244487*e^6 + 227561446443/63244487*e^5 + 39904335064/63244487*e^4 - 95135401993/63244487*e^3 - 37162354085/63244487*e^2 + 946111559/63244487*e + 153356551/63244487, -320438407/63244487*e^14 - 542097751/63244487*e^13 + 8024140123/63244487*e^12 + 13824592917/63244487*e^11 - 64399273768/63244487*e^10 - 125809045217/63244487*e^9 + 180422692588/63244487*e^8 + 452907522300/63244487*e^7 - 76136649344/63244487*e^6 - 568408043070/63244487*e^5 - 157753950354/63244487*e^4 + 225509240907/63244487*e^3 + 110614812304/63244487*e^2 + 2490137279/63244487*e - 219896193/63244487, 165548125/63244487*e^14 + 253829572/63244487*e^13 - 4125858013/63244487*e^12 - 6440609663/63244487*e^11 + 32786433696/63244487*e^10 + 58608479002/63244487*e^9 - 90306766744/63244487*e^8 - 208183504735/63244487*e^7 + 35189107440/63244487*e^6 + 246535811742/63244487*e^5 + 76928957701/63244487*e^4 - 84759879819/63244487*e^3 - 50880704383/63244487*e^2 - 6047980991/63244487*e - 81133254/63244487, 1, -99538099/63244487*e^14 - 234213106/63244487*e^13 + 2416212753/63244487*e^12 + 5951849654/63244487*e^11 - 18092002233/63244487*e^10 - 52562297806/63244487*e^9 + 38496741963/63244487*e^8 + 181286331884/63244487*e^7 + 38019677911/63244487*e^6 - 210675404836/63244487*e^5 - 116698684313/63244487*e^4 + 71373709671/63244487*e^3 + 55761353235/63244487*e^2 + 5716501940/63244487*e - 70517287/63244487, 278989471/63244487*e^14 + 530844866/63244487*e^13 - 6895059621/63244487*e^12 - 13485958701/63244487*e^11 + 53786200690/63244487*e^10 + 120786039200/63244487*e^9 - 136790381810/63244487*e^8 - 423633566984/63244487*e^7 - 2136697247/63244487*e^6 + 502939267133/63244487*e^5 + 207336995437/63244487*e^4 - 178112912833/63244487*e^3 - 116550691686/63244487*e^2 - 9738616660/63244487*e + 210177557/63244487, -62938151/63244487*e^14 - 109143675/63244487*e^13 + 1584055706/63244487*e^12 + 2795969436/63244487*e^11 - 12831942074/63244487*e^10 - 25582845369/63244487*e^9 + 36535301129/63244487*e^8 + 93408597673/63244487*e^7 - 15738196392/63244487*e^6 - 121109746063/63244487*e^5 - 37482506616/63244487*e^4 + 46944517859/63244487*e^3 + 27439548853/63244487*e^2 + 3305101107/63244487*e + 275701630/63244487, 195229694/63244487*e^14 + 463567501/63244487*e^13 - 4804897745/63244487*e^12 - 11901108913/63244487*e^11 + 37078884288/63244487*e^10 + 106939876020/63244487*e^9 - 87277435346/63244487*e^8 - 383469764472/63244487*e^7 - 48544106332/63244487*e^6 + 492847720888/63244487*e^5 + 237564549188/63244487*e^4 - 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2289605897128/63244487*e^5 - 974243930824/63244487*e^4 + 983963615574/63244487*e^3 + 552609540979/63244487*e^2 - 7000643686/63244487*e - 3921655993/63244487, -733460430/63244487*e^14 - 441730636/63244487*e^13 + 18849454343/63244487*e^12 + 10982396146/63244487*e^11 - 159460259300/63244487*e^10 - 111006223017/63244487*e^9 + 535548201614/63244487*e^8 + 425541341726/63244487*e^7 - 654136125191/63244487*e^6 - 492090564741/63244487*e^5 + 243050353531/63244487*e^4 + 121297919007/63244487*e^3 + 24121458095/63244487*e^2 + 38314096657/63244487*e + 314962361/63244487]; 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;