/* 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![4, 8, -7, -2, 1]; F := NumberField(g); ZF := Integers(F); NN := ideal; primesArray := [ [2, 2, 1/2*w^2 + 1/2*w - 1], [2, 2, -1/2*w^2 + 3/2*w], [9, 3, -1/2*w^2 + 1/2*w + 2], [13, 13, 1/2*w^3 - w^2 - 5/2*w + 2], [13, 13, -1/2*w^3 + 1/2*w^2 + 3*w - 1], [23, 23, 1/2*w^3 - w^2 - 5/2*w], [23, 23, -1/2*w^3 + 1/2*w^2 + 3*w - 3], [37, 37, 1/2*w^3 - 1/2*w^2 - 3*w - 3], [37, 37, 1/2*w^3 - w^2 - 5/2*w + 6], [59, 59, -1/2*w^3 + 7/2*w], [59, 59, -1/2*w^3 + 3/2*w^2 + 2*w - 3], [61, 61, -w^3 + 2*w^2 + 7*w - 7], [61, 61, w^3 - w^2 - 6*w + 3], [61, 61, w^3 - 2*w^2 - 5*w + 3], [61, 61, w^3 - w^2 - 8*w + 1], [71, 71, -1/2*w^3 + 1/2*w^2 + 5*w - 5], [71, 71, -w^3 + 3/2*w^2 + 15/2*w - 6], [83, 83, -1/2*w^3 + 3/2*w^2 + 2*w - 7], [83, 83, 1/2*w^3 - 7/2*w - 4], [97, 97, 2*w - 1], [107, 107, 1/2*w^3 + 1/2*w^2 - 4*w - 1], [107, 107, 3/2*w^3 - 7/2*w^2 - 9*w + 13], [109, 109, -3/2*w^2 + 7/2*w + 4], [109, 109, 1/2*w^3 + 1/2*w^2 - 4*w - 5], [109, 109, -1/2*w^3 + 2*w^2 + 3/2*w - 8], [109, 109, 3/2*w^2 + 1/2*w - 6], [121, 11, w^2 - w - 5], [121, 11, w^2 - w - 3], [131, 131, -w^3 + 11*w - 1], [131, 131, -w^3 + 3*w^2 + 8*w - 9], [157, 157, -w^3 + w^2 + 6*w + 1], [157, 157, 1/2*w^3 - 2*w^2 - 11/2*w + 4], [167, 167, w^2 - 3*w - 3], [167, 167, 1/2*w^3 - 3/2*w^2 - 2*w + 1], [167, 167, 3/2*w^3 - 5/2*w^2 - 10*w + 9], [167, 167, w^2 + w - 5], [169, 13, w^2 - w - 9], [179, 179, w^3 - 7*w - 5], [179, 179, 1/2*w^3 + 1/2*w^2 - 2*w - 3], [181, 181, -1/2*w^3 + 1/2*w^2 + 3*w - 5], [181, 181, 1/2*w^3 - w^2 - 5/2*w - 2], [191, 191, -1/2*w^3 + 3/2*w + 2], [191, 191, 2*w - 5], [191, 191, -2*w - 3], [191, 191, -1/2*w^3 + 2*w^2 + 3/2*w - 10], [193, 193, -1/2*w^3 + w^2 + 9/2*w - 4], [193, 193, 1/2*w^3 - 5/2*w^2 + w + 5], [193, 193, -1/2*w^3 - w^2 + 5/2*w + 4], [193, 193, 1/2*w^3 - 1/2*w^2 - 5*w + 1], [227, 227, -1/2*w^3 + 3/2*w^2 + 4*w - 9], [227, 227, w^3 - 2*w^2 - 7*w + 5], [227, 227, w^3 - w^2 - 8*w + 3], [227, 227, 1/2*w^3 - 11/2*w - 4], [239, 239, 1/2*w^3 - 3/2*w^2 - 4*w + 3], [239, 239, 1/2*w^3 - 11/2*w + 2], [251, 251, 1/2*w^3 - w^2 - 1/2*w - 2], [251, 251, -1/2*w^3 + 1/2*w^2 + w - 3], [263, 263, -1/2*w^3 - w^2 + 17/2*w], [263, 263, 1/2*w^3 - 5/2*w^2 - 5*w + 7], [277, 277, w^3 - w^2 - 6*w + 1], [277, 277, -w^3 + 2*w^2 + 5*w - 5], [311, 311, -w^3 + 5/2*w^2 + 13/2*w - 8], [311, 311, -w^3 + 1/2*w^2 + 17/2*w], [337, 337, 2*w^3 - 3*w^2 - 13*w + 9], [337, 337, 5/2*w^2 + 3/2*w - 8], [347, 347, 2*w^3 - 7/2*w^2 - 25/2*w + 14], [347, 347, 2*w^3 - 5/2*w^2 - 27/2*w], [349, 349, w^3 - 3/2*w^2 - 11/2*w + 4], [349, 349, -w^3 + 3/2*w^2 + 11/2*w - 2], [359, 359, -1/2*w^3 + w^2 + 1/2*w + 4], [359, 359, 1/2*w^3 - 1/2*w^2 - w + 5], [373, 373, 1/2*w^3 - 3/2*w + 4], [373, 373, 1/2*w^3 - 3/2*w^2 - 3], [383, 383, w^3 - 3/2*w^2 - 15/2*w + 8], [383, 383, -2*w^2 + 1], [409, 409, 1/2*w^3 + w^2 - 13/2*w - 4], [409, 409, -1/2*w^3 + 5/2*w^2 + 3*w - 9], [431, 431, 3/2*w^3 - 5/2*w^2 - 10*w + 5], [431, 431, 1/2*w^3 - 5/2*w^2 - 3*w + 13], [431, 431, 1/2*w^3 + w^2 - 13/2*w - 8], [431, 431, 3/2*w^3 - 2*w^2 - 21/2*w + 6], [433, 433, 2*w^3 - 4*w^2 - 14*w + 13], [433, 433, w^3 - 1/2*w^2 - 9/2*w + 6], [443, 443, 2*w^3 - 11/2*w^2 - 21/2*w + 24], [443, 443, w^3 - 5*w - 3], [457, 457, -2*w^3 + 4*w^2 + 12*w - 13], [457, 457, -w^3 + 1/2*w^2 + 9/2*w], [467, 467, -3/2*w^3 + 7/2*w^2 + 7*w - 7], [467, 467, 3/2*w^2 + 1/2*w - 8], [467, 467, w^3 - 7/2*w^2 - 15/2*w + 12], [467, 467, 3/2*w^3 - w^2 - 19/2*w + 2], [503, 503, -2*w^2 + 11], [503, 503, 2*w^3 - 2*w^2 - 12*w + 7], [503, 503, 2*w^3 - 4*w^2 - 10*w + 5], [503, 503, -2*w^2 + 4*w + 9], [529, 23, 3/2*w^2 - 3/2*w - 8], [541, 541, -1/2*w^3 + w^2 + 1/2*w + 6], [541, 541, 1/2*w^3 - 1/2*w^2 - w + 7], [563, 563, 1/2*w^2 + 3/2*w - 6], [563, 563, 1/2*w^2 - 5/2*w - 4], [577, 577, -1/2*w^3 + 5/2*w^2 + w - 11], [577, 577, -1/2*w^3 - w^2 + 9/2*w + 8], [587, 587, -w^3 + 1/2*w^2 + 13/2*w - 2], [587, 587, -w^3 + 5/2*w^2 + 9/2*w - 4], [599, 599, 1/2*w^3 + w^2 - 9/2*w - 12], [599, 599, -1/2*w^3 + 5/2*w^2 + w - 15], [601, 601, 1/2*w^3 - 1/2*w^2 - 3*w - 5], [601, 601, -1/2*w^3 + w^2 + 5/2*w - 8], [625, 5, -5], [659, 659, 3/2*w^3 - 1/2*w^2 - 8*w + 3], [659, 659, 3/2*w^3 - 4*w^2 - 9/2*w + 4], [673, 673, 2*w^3 - 2*w^2 - 16*w - 1], [673, 673, w^3 - 3/2*w^2 - 7/2*w - 8], [673, 673, w^3 - 3/2*w^2 - 7/2*w + 12], [673, 673, 2*w^3 - 4*w^2 - 14*w + 17], [683, 683, -1/2*w^3 - w^2 + 9/2*w - 2], [683, 683, -w^3 + 5/2*w^2 + 9/2*w - 12], [683, 683, -w^3 + 1/2*w^2 + 13/2*w + 6], [683, 683, 3/2*w^3 - w^2 - 11/2*w + 10], [709, 709, -1/2*w^3 + w^2 + 9/2*w - 10], [709, 709, 1/2*w^3 - 1/2*w^2 - 5*w - 5], [719, 719, -3/2*w^3 + 3/2*w^2 + 11*w - 3], [719, 719, -3/2*w^3 + 3*w^2 + 19/2*w - 8], [733, 733, w^3 - 3*w^2 - 8*w + 13], [733, 733, -w^3 + 7/2*w^2 + 7/2*w - 12], [733, 733, w^3 + 1/2*w^2 - 15/2*w - 6], [733, 733, -3/2*w^3 + 9/2*w^2 + 8*w - 19], [757, 757, w^3 - w^2 - 4*w + 1], [757, 757, -2*w^3 + 7/2*w^2 + 25/2*w - 10], [769, 769, -2*w^2 + 4*w + 7], [769, 769, -2*w^2 + 9], [827, 827, -1/2*w^3 - 1/2*w^2 + 6*w - 1], [827, 827, -1/2*w^3 + 2*w^2 + 7/2*w - 4], [829, 829, 1/2*w^3 + 2*w^2 - 7/2*w - 12], [829, 829, 7/2*w^3 - 8*w^2 - 41/2*w + 32], [829, 829, 3/2*w^3 - w^2 - 15/2*w - 2], [829, 829, -w^3 + w^2 + 8*w + 5], [839, 839, 3/2*w^3 - w^2 - 23/2*w], [839, 839, -3/2*w^3 + 7/2*w^2 + 9*w - 11], [853, 853, 1/2*w^3 - 3/2*w^2 - 2*w + 11], [853, 853, -1/2*w^3 + 7/2*w + 8], [863, 863, 2*w^3 - 11/2*w^2 - 25/2*w + 26], [863, 863, 3/2*w^3 - 5/2*w^2 - 10*w + 13], [887, 887, 2*w^3 - 7/2*w^2 - 29/2*w + 8], [887, 887, 2*w^3 - 5/2*w^2 - 31/2*w + 8], [911, 911, 2*w^3 - w^2 - 15*w - 3], [911, 911, 2*w^3 - 5*w^2 - 11*w + 17], [947, 947, -w^3 + 3*w^2 + 6*w - 9], [947, 947, w^3 - 9*w - 1], [961, 31, -3/2*w^3 + 3*w^2 + 23/2*w - 8], [961, 31, 3/2*w^3 - 3/2*w^2 - 13*w + 5], [983, 983, -1/2*w^3 + 2*w^2 - 1/2*w - 6], [983, 983, 1/2*w^3 + 1/2*w^2 - 2*w - 5], [997, 997, w^3 - 13*w + 7], [997, 997, 1/2*w^3 - 5/2*w^2 - w + 9], [997, 997, 1/2*w^3 + w^2 - 9/2*w - 6], [997, 997, -w^3 + 3*w^2 + 10*w - 5]]; primes := [ideal : I in primesArray]; heckePol := x^22 - 38*x^20 + 621*x^18 - 5710*x^16 + 32467*x^14 - 118294*x^12 + 277439*x^10 - 411354*x^8 + 370104*x^6 - 187524*x^4 + 46720*x^2 - 4400; K := NumberField(heckePol); heckeEigenvaluesArray := [e, 411/322400*e^21 - 4357/80600*e^19 + 315411/322400*e^17 - 19957/2015*e^15 + 19898137/322400*e^13 - 2471322/10075*e^11 + 15441413/24800*e^9 - 39561023/40300*e^7 + 36297323/40300*e^5 - 33875451/80600*e^3 + 562239/8060*e, -15/1612*e^20 + 639/1612*e^18 - 11529/1612*e^16 + 57695/806*e^14 - 704265/1612*e^12 + 2712899/1612*e^10 - 507743/124*e^8 + 2466728/403*e^6 - 2122801/403*e^4 + 914098/403*e^2 - 135010/403, -1, -287/8060*e^20 + 2714/2015*e^18 - 176097/8060*e^16 + 80017/403*e^14 - 8934739/8060*e^12 + 7908767/2015*e^10 - 5452841/620*e^8 + 24377472/2015*e^6 - 19260107/2015*e^4 + 7617397/2015*e^2 - 212024/403, 817/40300*e^21 - 7754/10075*e^19 + 504467/40300*e^17 - 229602/2015*e^15 + 25653989/40300*e^13 - 45420469/20150*e^11 + 15674011/3100*e^9 - 140958199/20150*e^7 + 56713262/10075*e^5 - 23335972/10075*e^3 + 682226/2015*e, 3899/80600*e^21 - 37313/20150*e^19 + 2452399/80600*e^17 - 1130309/4030*e^15 + 128265233/80600*e^13 - 57848442/10075*e^11 + 81587917/6200*e^9 - 374864189/20150*e^7 + 306112889/20150*e^5 - 125676959/20150*e^3 + 1803246/2015*e, -165/1612*e^20 + 6223/1612*e^18 - 25156/403*e^16 + 455713/806*e^14 - 5071801/1612*e^12 + 17904223/1612*e^10 - 1539861/62*e^8 + 13771334/403*e^6 - 10938411/403*e^4 + 4383659/403*e^2 - 619466/403, -287/8060*e^20 + 2714/2015*e^18 - 176097/8060*e^16 + 80017/403*e^14 - 8934739/8060*e^12 + 7908767/2015*e^10 - 5452841/620*e^8 + 24377472/2015*e^6 - 19258092/2015*e^4 + 7603292/2015*e^2 - 209606/403, 626/10075*e^21 - 95467/40300*e^19 + 1565179/40300*e^17 - 1442993/4030*e^15 + 20534192/10075*e^13 - 298476531/40300*e^11 + 53259807/3100*e^9 - 249121369/10075*e^7 + 416590513/20150*e^5 - 87866414/10075*e^3 + 2577137/2015*e, 7857/80600*e^21 - 149743/40300*e^19 + 4900657/80600*e^17 - 2250037/4030*e^15 + 254483919/80600*e^13 - 457992599/40300*e^11 + 161335731/6200*e^9 - 742194827/20150*e^7 + 304169076/10075*e^5 - 250931487/20150*e^3 + 3594498/2015*e, 57/806*e^20 - 4131/1612*e^18 + 64327/1612*e^16 - 281715/806*e^14 + 762821/403*e^12 - 10577771/1612*e^10 + 1807583/124*e^8 - 8135095/403*e^6 + 6579286/403*e^4 - 2702104/403*e^2 + 389336/403, 491/8060*e^20 - 9209/4030*e^18 + 73974/2015*e^16 - 132932/403*e^14 + 14639107/8060*e^12 - 25466387/4030*e^10 + 4290029/310*e^8 - 37208516/2015*e^6 + 28293251/2015*e^4 - 10760141/2015*e^2 + 296184/403, -51/1612*e^20 + 523/403*e^18 - 36297/1612*e^16 + 86999/403*e^14 - 2020517/1612*e^12 + 1831613/403*e^10 - 1272139/124*e^8 + 5636239/403*e^6 - 4359528/403*e^4 + 1683570/403*e^2 - 230936/403, -227/2015*e^20 + 34409/8060*e^18 - 279709/4030*e^16 + 509917/806*e^14 - 7147344/2015*e^12 + 101858477/8060*e^10 - 8862117/310*e^8 + 80405113/2015*e^6 - 64941183/2015*e^4 + 26453398/2015*e^2 - 755390/403, 3899/80600*e^21 - 37313/20150*e^19 + 2452399/80600*e^17 - 1130309/4030*e^15 + 128265233/80600*e^13 - 57848442/10075*e^11 + 81587917/6200*e^9 - 374864189/20150*e^7 + 306133039/20150*e^5 - 125858309/20150*e^3 + 1835486/2015*e, 6091/80600*e^21 - 28596/10075*e^19 + 3691291/80600*e^17 - 1672841/4030*e^15 + 186982397/80600*e^13 - 166503231/20150*e^11 + 116221753/6200*e^9 - 530049001/20150*e^7 + 215383388/10075*e^5 - 176479281/20150*e^3 + 2528934/2015*e, -1727/8060*e^21 + 32473/4030*e^19 - 261983/2015*e^17 + 948199/806*e^15 - 52787329/8060*e^13 + 46699622/2015*e^11 - 8072404/155*e^9 + 145485687/2015*e^7 - 116638062/2015*e^5 + 47127272/2015*e^3 - 1334878/403*e, 1687/20150*e^21 - 64451/20150*e^19 + 2111349/40300*e^17 - 1937353/4030*e^15 + 27309452/10075*e^13 - 97727459/10075*e^11 + 68202017/3100*e^9 - 309408139/10075*e^7 + 248930264/10075*e^5 - 100447884/10075*e^3 + 2822622/2015*e, 727/8060*e^20 - 28391/8060*e^18 + 474787/8060*e^16 - 444375/806*e^14 + 25543029/8060*e^12 - 93187383/8060*e^10 + 16590221/620*e^8 - 76946452/2015*e^6 + 63458282/2015*e^4 - 26310757/2015*e^2 + 759596/403, 9287/80600*e^21 - 174963/40300*e^19 + 5654787/80600*e^17 - 1280206/2015*e^15 + 285015329/80600*e^13 - 503428509/40300*e^11 + 173425221/6200*e^9 - 777218907/20150*e^7 + 619447107/20150*e^5 - 249671267/20150*e^3 + 3585008/2015*e, -2673/40300*e^21 + 100329/40300*e^19 - 810299/20150*e^17 + 737038/2015*e^15 - 82965791/40300*e^13 + 299077547/40300*e^11 - 26587967/1550*e^9 + 249241203/10075*e^7 - 210158228/10075*e^5 + 89967318/10075*e^3 - 2701729/2015*e, -53/806*e^20 + 1008/403*e^18 - 32837/806*e^16 + 149585/403*e^14 - 835761/403*e^12 + 5910915/806*e^10 - 507633/31*e^8 + 18048935/806*e^6 - 7076347/403*e^4 + 2776582/403*e^2 - 381542/403, -287/8060*e^20 + 2714/2015*e^18 - 176097/8060*e^16 + 80017/403*e^14 - 8934739/8060*e^12 + 7908767/2015*e^10 - 5452841/620*e^8 + 24379487/2015*e^6 - 19282272/2015*e^4 + 7677847/2015*e^2 - 216860/403, 1399/8060*e^20 - 52827/8060*e^18 + 427657/4030*e^16 - 775781/806*e^14 + 43228483/8060*e^12 - 152791251/8060*e^10 + 6576073/155*e^8 - 117611614/2015*e^6 + 93212269/2015*e^4 - 37149059/2015*e^2 + 1045126/403, -289/8060*e^20 + 10377/8060*e^18 - 40036/2015*e^16 + 69556/403*e^14 - 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1709373/806*e^14 + 18827311/1612*e^12 - 32945683/806*e^10 + 2814001/31*e^8 - 100048869/806*e^6 + 39424257/403*e^4 - 15618560/403*e^2 + 2194818/403, -55353/80600*e^21 + 260268/10075*e^19 - 33594653/80600*e^17 + 15191083/4030*e^15 - 1689435051/80600*e^13 + 745710474/10075*e^11 - 1027653099/6200*e^9 + 2303232554/10075*e^7 - 1833919479/10075*e^5 + 1471120123/20150*e^3 - 20789512/2015*e, -26601/80600*e^21 + 504499/40300*e^19 - 16456151/80600*e^17 + 3772798/2015*e^15 - 854395267/80600*e^13 + 1543820007/40300*e^11 - 547710733/6200*e^9 + 1272398143/10075*e^7 - 2110534011/20150*e^5 + 881174591/20150*e^3 - 12699054/2015*e, -333/2015*e^20 + 49731/8060*e^18 - 794307/8060*e^16 + 354275/403*e^14 - 19335327/4030*e^12 + 133127353/8060*e^10 - 22159191/620*e^8 + 94945707/2015*e^6 - 71507542/2015*e^4 + 26987827/2015*e^2 - 731292/403, -3/130*e^20 + 123/260*e^18 - 153/130*e^16 - 1263/26*e^14 + 78769/130*e^12 - 873761/260*e^10 + 103771/10*e^8 - 1199264/65*e^6 + 1178019/65*e^4 - 559914/65*e^2 + 17430/13, -639/8060*e^20 + 5818/2015*e^18 - 366689/8060*e^16 + 164012/403*e^14 - 18362243/8060*e^12 + 16686129/2015*e^10 - 12137317/620*e^8 + 58810314/2015*e^6 - 51376109/2015*e^4 + 22579339/2015*e^2 - 684256/403, 2229/8060*e^20 - 81737/8060*e^18 + 321326/2015*e^16 - 566422/403*e^14 + 61398983/8060*e^12 - 211338291/8060*e^10 + 8868068/155*e^8 - 154690379/2015*e^6 + 119443859/2015*e^4 - 46304644/2015*e^2 + 1275966/403]; 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;