// Make newform 507.4.b.k in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_507_b();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_Hecke();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_507_4_b_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_507_4_b_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function ConvertToHeckeField(input: pass_field := false, Kf := []) if not pass_field then poly := [729, 0, 137781, 0, 6703200, 0, 21033277, 0, 8545747, 0, 1397921, 0, 114847, 0, 5026, 0, 112, 0, 1]; Kf := NumberField(Polynomial([elt : elt in poly])); AssignNames(~Kf, ["nu"]); end if; Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [104051653875419246568, 0, 10467565459459412450652, 0, 33576707877832131502849, 0, 13667233608090341944987, 0, 2236951836672312074405, 0, 183822566061610352491, 0, 8045612589142543945, 0, 179305097519975323, 0, 1601044867583677, 0], [-86875928215438277826, 0, -11259288060469626490641, 0, -36322397623835941473329, 0, -14793416297435623057127, 0, -2421951006453304882537, 0, -199058877775579121987, 0, -8713432054116181529, 0, -194201793863294741, 0, -1734137237833382, 0], [-113523248333235433275, 0, -12862649157148374623424, 0, -42549134109079154978570, 0, -17366991252690389931998, 0, -2845151778625514796646, 0, -233902954714980508178, 0, -10239866564043144446, 0, -228234250530821033, 0, -2038071261700910, 0], [64519121555398648152, 0, 6869693527170214236975, 0, 22549257793644108016991, 0, 9198745336173102049853, 0, 1506817840103975516743, 0, 123874522923848204201, 0, 5423019545415382019, 0, 120873077291344265, 0, 1079366094781310, 0], [6033608228060496, 0, 629456453820527145, 0, 2050638752311140007, 0, 835899628653827689, 0, 136887404185381487, 0, 11251941816685501, 0, 492560731054663, 0, 10978375356709, 0, 98034109972, 0], [552995164470195684828, 0, 51169954896152686256301, 0, 166290168946269976351726, 0, 67776203941121748833254, 0, 11098876671253416854978, 0, 912311321808153371026, 0, 39937130751587450518, 0, 890136922280136022, 0, 7948726537666543, 0], [83114602924383017289, 0, 7814023679691706406211, 0, 25198186392175608099059, 0, 10262367969749678587517, 0, 1680072585616760848183, 0, 138079952831185821437, 0, 6044042621266104671, 0, 134704700598988610, 0, 1202833801010558, 0], [910221237339627426210, 0, 85994173135670544045135, 0, 277990265584479225693698, 0, 113242004905400576400818, 0, 18540464840465259889006, 0, 1523830694243708244014, 0, 66702099923845469930, 0, 1486610281078533560, 0, 13274576015701397, 0], [0, -24951778139040599323224, 0, -2279231692555537478384199, 0, -7358110153966373216362609, 0, -2997036285041392095830743, 0, -490667738706187857844865, 0, -40326974856989520487915, 0, -1765201224467671817689, 0, -39341354801072342995, 0, -351294310301276860], [0, 47261222379747912174174, 0, 6307068476565465585596775, 0, 20573829758624817568974100, 0, 8387406494434905735659440, 0, 1373574158454111751451624, 0, 112906948451625313607500, 0, 4942569832940758036828, 0, 110161189351206053410, 0, 983703339022442455], [0, -99536642989191731003697, 0, -10155196761284993270910147, 0, -32907476482060537655524259, 0, -13407972960160536941644949, 0, -2195364852078138476531287, 0, -180441933287432272993493, 0, -7898572866631670792183, 0, -176040047976219790778, 0, -1571948875555483478], [0, -44429073271077606612399, 0, -3496859371178284642113396, 0, -11217433878414679404604189, 0, -4566437600792833056717175, 0, -747466609147348852486949, 0, -61427319708209783608723, 0, -2688677052085443470617, 0, -59921178114515677318, 0, -535048544355919573], [0, -14126671221018042266649, 0, -1295716017016055651501761, 0, -4185171196258535895870177, 0, -1704744450192583427024935, 0, -279101082885132292202333, 0, -22938911069422955677215, 0, -1004091133388838528645, 0, -22378415796312586260, 0, -199825963018042202], [0, -117854530825576418762373, 0, -12250163265923155307477616, 0, -39669436928059992423501136, 0, -16162005282268721057812336, 0, -2646235448390013693745460, 0, -217497551368298832831436, 0, -9520568872239458397820, 0, -212189660851318789837, 0, -1894742639433313732], [0, -156186936516528005836725, 0, -14181026290366663812397572, 0, -45744322168224785295664520, 0, -18630808445116862226400760, 0, -3050117193501780762086452, 0, -250679917778232872609636, 0, -10972742991526896374708, 0, -244550446460167690205, 0, -2183680095349438856], [0, 63973440918298125700722, 0, 6143032929765195340399791, 0, 19873518927273929084242097, 0, 8096217599600815029990719, 0, 1325578663983348867970777, 0, 108949916379354111308963, 0, 4769061782192990779361, 0, 106290075697754159453, 0, 949112438926442588], [0, 214845478818258882970719, 0, 21138611739620106672143964, 0, 68400103572511201807276405, 0, 27865711783016489158493239, 0, 4562419570010159402832821, 0, 374988006387626869843771, 0, 16414355208445643925721, 0, 365833804227819620518, 0, 3266697595312124629]]; Rf_basisdens := [1, 3032421244108904232, 3032421244108904232, 3032421244108904232, 1516210622054452116, 50991629993928, 3032421244108904232, 336935693789878248, 3032421244108904232, 118264428520247265048, 118264428520247265048, 118264428520247265048, 39421476173415755016, 13140492057805251672, 118264428520247265048, 118264428520247265048, 39421476173415755016, 118264428520247265048]; Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf); Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]]; inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf); return Eltseq(inp_vec); end function; // To make the character of type GrpDrchElt, type "MakeCharacter_507_b();" function MakeCharacter_507_b() N := 507; order := 2; char_gens := [170, 340]; v := [2, 1]; // chi(gens[i]) = zeta^v[i] assert UnitGenerators(DirichletGroup(N)) eq char_gens; F := CyclotomicField(order); chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]); return MinimalBaseRingCharacter(chi); end function; // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_Hecke();" function MakeCharacter_507_b_Hecke(Kf) N := 507; order := 2; char_gens := [170, 340]; char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; assert UnitGenerators(DirichletGroup(N)) eq char_gens; values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field F := Universe(values);// the Hecke field chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values); return chi; end function; function ExtendMultiplicatively(weight, aps, character) prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec primes := PrimesUpTo(prec); prime_powers := primes; assert #primes eq #aps; log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1) F := Universe(aps); FXY := PolynomialRing(F, 2); // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ... R := PowerSeriesRing(FXY : Precision := log_prec + 1); recursion := Coefficients(1/(1 - X*T + Y*T^2)); coeffs := [F!0: i in [1..(prec+1)]]; coeffs[1] := 1; //a_1 for i := 1 to #primes do p := primes[i]; coeffs[p] := aps[i]; b := p^(weight - 1) * F!character(p); r := 2; p_power := p * p; //deals with powers of p while p_power le prec do Append(~prime_powers, p_power); coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]); p_power *:= p; r +:= 1; end while; end for; Sort(~prime_powers); for pp in prime_powers do for k := 1 to Floor(prec/pp) do if GCD(k, pp) eq 1 then coeffs[pp*k] := coeffs[pp]*coeffs[k]; end if; end for; end for; return coeffs; end function; function qexpCoeffs() // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" weight := 4; raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 1, 1, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-19, -4, 2, 1, -3, 3, 0, -3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 8, -2, 2, 2, -2, 6, 4], [-17, 2, -2, -5, 5, -3, -3, -2, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-14, -4, 18, 5, -6, 2, -8, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 1, 5, 2, -12, 12, 6, -6], [0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 2, 12, 9, 4, -13, -3, -12, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, -22, 12, -7, 7, -20, 18, -14, 6, 7], [-31, -7, -1, 5, -8, 4, -3, 5, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 5, 14, -20, 14, 8, 0, 6, -8], [-14, 6, 4, -10, 6, 9, 2, -8, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -3, 11, 1, 10, -8, -5, -7, -18], [-143, 0, -14, 9, 7, -15, 0, -11, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -32, 12, 9, 12, 34, 17, -3, 18, 37], [0, 0, 0, 0, 0, 0, 0, 0, 0, -31, 23, 19, -20, -26, 13, -7, 16, 5], [0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 10, 38, -14, -18, 8, -3, 0, -6], [-175, -27, -1, -3, -11, 36, -24, 13, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -50, -33, 7, -2, 12, 35, 30, 31, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 70, 26, 33, 14, 56, 37, 9, 22, 51], [0, 0, 0, 0, 0, 0, 0, 0, 0, 44, -10, -5, -4, 20, -75, 22, -10, 11], [224, 30, -40, 2, 53, -32, 25, 2, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16, -12, -16, 56, -40, 35, 25, -29, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [319, -26, 32, 7, 30, -29, -3, 29, -39, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -110, 4, -9, -3, -66, -12, -30, 52, -37], [-642, -15, 67, 2, -5, 21, -39, 32, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0], [411, 25, -29, 1, -12, -33, 84, 26, -17, 0, 0, 0, 0, 0, 0, 0, 0, 0], [227, 19, -9, -15, -10, 83, -1, -33, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 123, 5, 40, -53, -98, -3, 11, 44, -72], [-328, 78, -20, -42, 66, -10, 3, -75, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 235, -95, 11, 1, 102, 8, 48, 38, -54], [0, 0, 0, 0, 0, 0, 0, 0, 0, -204, -18, 65, -32, 72, -45, 103, -18, -2], [149, 3, 81, -73, 16, -92, 19, -9, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 142, -30, -157, 16, -58, 15, -75, -82, -69], [0, 0, 0, 0, 0, 0, 0, 0, 0, 49, -89, -26, 119, -44, 65, -85, -4, 106], [1434, -2, -184, -55, 36, -4, 10, -96, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [183, 32, 52, -7, 55, 36, -34, 171, -51, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-406, 17, -171, -106, 121, -101, 6, 59, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-688, 1, 49, -26, 215, -75, 38, 161, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -5, -93, 30, -99, -66, -19, 51, 24, -82], [0, 0, 0, 0, 0, 0, 0, 0, 0, -448, -17, 143, -70, 14, 49, 52, 75, 8], [453, 10, 194, -30, 61, -53, -146, 113, 162, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1193, -23, -175, -49, 66, 28, 38, -96, 125, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 106, -194, -169, -53, 96, -98, 106, -54, -107], [0, 0, 0, 0, 0, 0, 0, 0, 0, 114, -88, 17, -38, 110, 107, 10, 12, -41], [0, 0, 0, 0, 0, 0, 0, 0, 0, -117, -41, -69, 6, -16, 219, 1, -10, 173], [-809, 80, 14, 1, 217, -75, -38, 79, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -52, -136, 176, -51, 138, -185, 329, -8, -134], [0, 0, 0, 0, 0, 0, 0, 0, 0, -539, 31, -20, 137, -116, -13, -56, -22, 168], [-258, 0, 350, 20, -149, 0, -28, 153, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0], [850, 41, 49, -90, 81, 159, -92, 11, 250, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1612, 55, -9, -100, 115, -225, 116, 123, -112, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-577, 170, -180, -137, 169, 46, -131, 96, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 997, -313, -137, 152, 116, -149, 14, -226, -37], [1384, -68, 228, -56, 192, -212, 21, 179, -20, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -7, -125, 115, -13, 82, -138, 112, 214, -75], [-21, 46, -102, -11, -13, 103, -162, -5, -83, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -412, 92, 89, -182, 140, 61, 247, 190, 80], [0, 0, 0, 0, 0, 0, 0, 0, 0, -379, 255, 257, -212, 96, 185, 115, 104, 33], [-344, 157, -127, -42, 57, -49, -62, 99, -62, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-375, -56, -182, 127, -101, -176, 255, 174, -119, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 218, 199, 128, -285, 10, 365, -20, 293, -91], [0, 0, 0, 0, 0, 0, 0, 0, 0, -874, -112, -25, 157, 48, -328, 64, 12, 103], [-3060, -10, 80, -163, 172, -218, 198, 128, -25, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2395, 23, -475, -147, 121, -208, 75, 150, -29, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1720, -52, -102, -48, -24, -134, 224, -256, 78], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1002, 106, 203, 6, 148, -285, 219, -62, 127], [0, 0, 0, 0, 0, 0, 0, 0, 0, 150, -194, -525, -8, -572, 143, 47, -28, -477], [2037, -55, 35, 269, -179, 40, -162, -49, -247, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3046, -142, -10, 100, -100, -56, 157, 71, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -24, 420, 298, -103, 330, -201, 559, 288, 222], [0, 0, 0, 0, 0, 0, 0, 0, 0, -168, 4, -433, -138, -296, 81, -291, -140, -65], [-3516, -8, 144, 41, -54, 72, 283, 65, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 425, -139, -432, 186, -74, 162, -592, -514, -386], [0, 0, 0, 0, 0, 0, 0, 0, 0, -97, -165, -247, 326, -102, -169, -177, -278, -261], [0, 0, 0, 0, 0, 0, 0, 0, 0, 489, -62, -262, 358, -306, 4, -515, -113, -269], [3062, -140, 220, 199, -44, -118, 207, -249, 161, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -121, -309, -178, -80, 38, -134, 16, 246, 12], [0, 0, 0, 0, 0, 0, 0, 0, 0, 310, 80, -405, 410, -568, 169, -451, 178, -141], [530, -287, -485, -157, -49, 253, -70, -1, -193, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1022, 373, -341, -490, 660, -427, 68, 414, -46, 0, 0, 0, 0, 0, 0, 0, 0, 0], [588, 251, -457, -115, -27, 129, -36, -15, 71, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 291, 559, 372, -28, 318, -18, -62, 52, 162], [0, 0, 0, 0, 0, 0, 0, 0, 0, -300, -384, -245, 134, 46, -185, -559, -46, -346], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1153, 24, -207, 232, -232, -329, -372, -175, 246], [0, 0, 0, 0, 0, 0, 0, 0, 0, 764, 68, 388, -232, 276, -128, -212, -20, 104], [-1459, 97, 421, 219, -420, 503, -514, -412, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 616, -22, -406, 299, 166, -373, -225, -438, 158], [0, 0, 0, 0, 0, 0, 0, 0, 0, 976, 176, 47, -290, 26, 17, -24, 436, -249], [3661, -557, 391, 54, 72, 90, -471, 405, -208, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -148, -380, -628, 166, -156, -222, 434, -156, -748], [-3623, -250, 118, 591, -281, 105, 23, -318, -285, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1876, -458, -446, 192, 204, -110, -194, -330, 390], [-3733, -271, 269, 703, -370, 60, -67, -361, -39, 0, 0, 0, 0, 0, 0, 0, 0, 0], [893, -104, 38, 501, -751, 395, 31, -282, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 866, -76, -333, -231, -86, 188, 254, -180, -723], [5788, -149, -307, -78, 475, 445, -551, -22, 432, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1901, 405, 517, -422, 294, -13, -219, -440, 365], [-4168, -341, 531, -376, -343, 300, -420, 283, 420, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3730, 14, -56, 204, -1214, 508, 261, -1403, -172, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4747, 600, 258, -655, 759, -615, -137, 908, 161, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 538, -73, -767, 194, -422, -3, 830, -147, -274], [0, 0, 0, 0, 0, 0, 0, 0, 0, -543, -141, -70, 251, -282, -273, 37, -408, -145], [0, 0, 0, 0, 0, 0, 0, 0, 0, -390, 472, 310, 158, 964, -424, 128, -454, 154], [1466, 157, 365, 340, -687, 15, 512, -1247, 168, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-960, 18, -352, -237, 334, -362, 669, 103, 105, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1038, 439, -11, -882, 868, -225, -62, 1208, -102, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 966, 18, -452, -37, -12, 253, 7, -154, 490], [0, 0, 0, 0, 0, 0, 0, 0, 0, -842, 208, -566, 352, -304, 1242, -1156, 118, 260], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1053, -65, 400, 482, 790, 54, 236, 878, 528], [0, 0, 0, 0, 0, 0, 0, 0, 0, -48, -601, -107, -550, 600, -573, 1552, -215, -446], [-1155, 314, -198, 279, 427, -897, -69, 80, -459, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2461, -333, 1194, -60, 206, -382, 1516, 894, -976], [5180, -374, 126, 532, -250, 312, 201, -1073, 644, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6983, -211, -1129, -299, -171, 274, 46, -821, 575, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2541, -224, 74, -523, -638, 531, -153, -793, 179, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 282, -710, -107, -706, 394, -1183, 985, -468, -417], [-6391, 188, -874, 287, -282, -63, 231, -395, 137, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-7339, 51, 835, -145, 145, -1056, 87, 266, -723, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -240, -44, 932, 164, 266, -544, 969, 1190, -110], [0, 0, 0, 0, 0, 0, 0, 0, 0, 542, -372, -639, -561, -312, -164, 408, -624, -545], [-3758, 338, 304, 146, -204, 346, -211, 807, 494, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2317, 327, -645, 552, -1464, 533, -355, 576, 213], [-696, -26, 146, 142, 556, -648, 378, -118, -686, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3839, 49, 225, 1019, 743, -562, 209, 788, -73, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1552, 538, 1039, 824, 870, -507, 115, 414, 257], [0, 0, 0, 0, 0, 0, 0, 0, 0, 656, 392, -616, 235, 326, -21, 363, 644, 22], [0, 0, 0, 0, 0, 0, 0, 0, 0, 21, -179, -614, -16, -194, 1356, -668, -38, -402], [-1566, 401, 179, -229, 713, -219, 855, 882, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1158, -140, -108, -836, -144, 410, 9, -97, -764, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1438, -34, -415, 746, 580, -627, -709, -694, 175], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1114, -316, 404, -998, 2056, -314, 1302, 38, -300], [0, 0, 0, 0, 0, 0, 0, 0, 0, 391, 838, 1478, -90, 364, -700, 441, 449, 347], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1968, -784, -99, -153, -892, 864, -172, 1378, 393], [1161, -247, 1153, 32, -276, -328, -72, 736, 278, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-14457, 255, 1231, 323, -222, -224, 411, 1133, 125, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 279, -1167, -1478, 1238, -330, 496, -1008, -586, -502], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1871, -265, -1058, 374, -210, -166, 1619, 204, -772], [9352, 178, -1042, -594, 467, 462, 55, -330, -516, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2094, -62, 37, 323, -440, 416, 100, 1640, -914], [2087, 34, -788, 139, 181, -695, 489, -822, -971, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1557, 19, -668, 247, -904, -1173, 123, 490, -1018], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3705, 1715, 867, -579, -168, 1416, -1206, -608, 1297], [-242, 180, -436, 630, -656, 726, 230, -1302, 990, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4171, -178, 492, 447, 733, -527, 335, 1014, -129, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1575, -341, -430, 344, 146, -304, -372, -958, -1310], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1595, 123, 164, -113, 810, 581, -1825, -390, 854], [-4145, 352, -38, 787, -857, 953, -996, -1809, 369, 0, 0, 0, 0, 0, 0, 0, 0, 0], [14277, -440, 1824, 15, -751, 31, -32, -1185, 915, 0, 0, 0, 0, 0, 0, 0, 0, 0], [10791, -1451, 1029, 287, -990, 414, -242, 692, 535, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-15412, -846, 1004, 542, -242, -12, -737, 777, 350, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2077, 249, 649, -917, 98, 364, 86, 436, -285, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-15402, -296, 318, 862, -240, -641, 714, -688, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3783, 1483, -186, 431, 330, -293, -195, -642, -952], [-8659, -108, -436, -702, -665, 877, -1548, -773, 762, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 689, 587, 1008, -1470, -317, -175, 550, 268], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2069, 543, -478, 204, 1304, -136, -1931, 208, 970], [-3444, 645, -1051, -996, -347, 247, -558, -1199, 792, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 4543, -385, -806, 1250, -1396, 2094, -1890, 486, 177], [2928, 738, -1282, -337, 732, -1094, 461, -195, 393, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -8130, 868, -814, 1293, -632, 363, -3737, -1400, 1006], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2774, 164, 1335, -752, 1212, -1045, 571, 966, 229], [7240, -1210, 408, 1404, -1352, 161, 500, -1110, -180, 0, 0, 0, 0, 0, 0, 0, 0, 0], [5852, -1375, 429, 396, -1113, 593, -1090, -55, -30, 0, 0, 0, 0, 0, 0, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_507_b_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_507_4_b_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose lines below. // The precision argument determines an initial guess on how many Fourier coefficients to use. // This guess is increased enough to uniquely determine the newform. function MakeNewformModFrm_507_4_b_k(:prec:=18) chi := MakeCharacter_507_b(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); S := BaseChange(S, Kf); maxprec := NextPrime(997) - 1; while true do trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]); B := Basis(S, prec + 1); S_basismat := Matrix([AbsEltseq(g): g in B]); if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then S_basismat := ChangeRing(S_basismat,Kf); f_lincom := Solution(S_basismat,trunc_vec); f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]]; return f; end if; error if prec eq maxprec, "Unable to distinguish newform within newspace"; prec := Min(Ceiling(1.25 * prec), maxprec); end while; end function; // To make the Hecke irreducible modular symbols subspace (type ModSym) // containing the newform, type "MakeNewformModSym_507_4_b_k();". // This may take a long time! To see verbose output, uncomment the SetVerbose line below. // The default sign is -1. You can change this with the optional parameter "sign". function MakeNewformModSym_507_4_b_k( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_507_b(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![1032256, 0, 28371296, 0, 129478420, 0, 70428381, 0, 16267053, 0, 1992090, 0, 138863, 0, 5516, 0, 116, 0, 1]>,<5,R![2391268778161201, 0, 1145975316696272, 0, 174948957367794, 0, 12132347525883, 0, 433913382945, 0, 8372407387, 0, 86264149, 0, 449620, 0, 1105, 0, 1]>],Snew); return Vf; end function;