// Make newform 3024.2.k.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_3024_k();" // To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3024_k_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_3024_2_k_l();". // 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_3024_2_k_l();". // 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 := [256, -1920, 7200, -18000, 30948, -35292, 27090, -12522, 917, 1704, -378, 306, -45, -36, 18, -6, 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], [-90521992849087319232, 409906873135003778880, -1185368258007122055024, 1839630656591715546764, -1796800164972733552596, 1148647844033093893974, -280563829185536375770, -110340558648795826381, 30275770334978045064, -24300598042312983470, 8077218483857604690, 1516003722014862581, -1443574468406641356, 586463746974274502, -135922610320481566, 9625967735304871], [-128641878336, 608118237376, -1800396653968, 2995055019604, -3267629790492, 2381970760186, -909061030402, -24801774291, 117868348872, -39143035490, 23237736362, -1502246501, -3073658820, 1443496954, -438014494, 63679977], [8748872216183229304, 23390361071012248832, -76254427680148237152, 168246773779286763616, -242879832966233604796, 209888009648871349274, -126785162227774315461, 32986629793581824055, 20479970216212369304, -1657319496483625186, 2985818238940592013, -878506007414906579, -292350862854969940, 164374780549634918, -63670839763550169, 12896919611658579], [-9139436721701545248, 53009605304413211488, -170642159006615384472, 371056209408619765668, -490058282289866822840, 425247294085105447534, -232097741244840672690, 33529795552648846041, 31380199749231410196, -4823158399067954190, 5505988466081180250, -1170314294538647425, -565356612235773536, 303747509202031774, -109347424813408518, 19830533427325021], [98191886152776007488, -354073431298430664384, 875864394993863957520, -648330092363001340596, -577936068553120419876, 1406323934504147906966, -1501322878517262329270, 586657112311307922859, 232014507612254250840, 10087746234878269970, 32832904336136207694, -14580993899421032339, -2051377156717453308, 1522520126318531510, -718993239773640506, 166495347726429999], [-45979845120272208160, 265381678914536967264, -855237181615557275160, 1862112167002892038148, -2454209646766613232680, 2151136076115084440370, -1186408888646239693836, 186231486934515420959, 163419010013194163900, -23575723806692084026, 28122382173670077108, -6236140769180355679, -2866347090965366816, 1549632586804565914, -562758338401057656, 103445914180111363], [-860136715502795952, -25584805500528726272, 83409227248484142816, -184034753095094222816, 265675330140966521196, -229594268069864059746, 138694161338870264505, -36089568444875126467, -22399813683111549176, 1814833707151853938, -3266450045541282705, 960645259076306143, 319984785621266012, -179866152511286382, 69657074983267533, -14107663960345207], [-260758962053897874240, 1309729105070966195648, -3724989584265231860832, 7215792024074813908276, -8768944295897186435364, 7075088896396222306842, -3540699862484537086286, 338398304276928343289, 467665489437841643328, -92360559260180901442, 85078276272927929526, -14537311885819777697, -9392053185843428652, 4839202878549981066, -1663183207330582570, 285056008857833765], [-173395593155735587392, 990675147416249758400, -2927487349506788131872, 6579060527466609194276, -9170802974279233461588, 8208443510027441366546, -4796235394155472638766, 839647715163391821573, 662301871168381505280, -82678831114224238618, 112446815018011162998, -26936081144636182573, -11056175268959929980, 6097823575132527650, -2264341257780881178, 425315347188680289], [151122658785155736384, -800570606298078334400, 2314691850042689622464, -4799054844668716030132, 6239123689728757613220, -5313429374717505153882, 2923298596889134896290, -407759785279442730137, -393392250764883891936, 60745062947036955490, -69074359242044407882, 14487934719168682481, 7124583038670140172, -3815159516715513114, 1371551676913492126, -247812487370796629], [150396920982869709856, -835637634442003547424, 2685058836161561509704, -5826047463340864722956, 7675972396318969309952, -6588772098844568088486, 3532577508249729418284, -433930421707137283701, -461702437330773630492, 78843857616753617150, -83914631234874493860, 16523970977232350709, 8715983809605597256, -4636454425014531870, 1644440103187941264, -291135571090866401], [150233508340123593888, -916968500027046048032, 2950204768948573461960, -6411064036377998369420, 8520502969779199259536, -7318593594721039106686, 3973440416942672673336, -548638306586091491705, -532911706689001127612, 84608950357766551158, -94297253750326461456, 19578299564138657673, 9732783733294230104, -5208086892130408390, 1865845786412506188, -335980506902322613], [13175314775391168, -69828260949961536, 201983454004697152, -418925624772450876, 545001311541427884, -464511509659530078, 250693834742223922, -35996761740085251, -34284127789537632, 5331726217263510, -5960180317929130, 1253112709022811, 619813991867076, -331040469619086, 118714615744902, -21499912175367], [-759002784629311903296, 4021991012936718836416, -11632056514056489451456, 24122384513063254660324, -31374365399252977991988, 26733031893110706133426, -14529482345426408592374, 2064376738135099866693, 1975303852222086794016, -306400368700789644986, 344643919038718439934, -72396225461829865421, -35733762908496343068, 19103269614095631970, -6856827026809858242, 1240747642399240641], [347623021430852021600, -2021950457437052554016, 6504653598592364971464, -14133556987400189799500, 18688582627295661656696, -16122917299412198591318, 8745715316358408665652, -1198325909011618735165, -1169554041448852139572, 186530862938561573742, -207566655142818235692, 42981643978667404477, 21409340134734606944, -11459302404703427278, 4103720694856443000, -738158905111162601]]; Rf_basisdens := [1, 7684522302142264288, 24698853056, 1921130575535566072, 1921130575535566072, 15369044604284528576, 7684522302142264288, 960565287767783036, 15369044604284528576, 15369044604284528576, 7684522302142264288, 7684522302142264288, 7684522302142264288, 434694100132496, 15369044604284528576, 7684522302142264288]; 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_3024_k();" function MakeCharacter_3024_k() N := 3024; order := 2; char_gens := [1135, 757, 785, 2593]; v := [2, 2, 1, 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_3024_k_Hecke();" function MakeCharacter_3024_k_Hecke(Kf) N := 3024; order := 2; char_gens := [1135, 757, 785, 2593]; char_values := [[1, 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], [-1, 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]]; 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 := 2; raw_aps := [[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], [0, 1, 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, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 1, -1, 0], [0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0], [0, -1, 0, 0, 0, 2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0], [0, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -2, 0, 0, -1, 1, 0], [0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0], [-2, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 1, 1, 0], [0, 0, 0, 0, 2, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 3], [-3, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0], [0, -1, 3, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, -3, 2, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, -1], [0, 0, -2, 0, 0, 2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 3, -2, 0, 0, 1, -2, 0], [3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, -2, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 1, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 3, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 2, 2, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -1, 0, 0, 2, 1, 0], [5, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0], [0, 0, 0, 0, -1, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -2], [-5, 0, 0, -2, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0], [0, 1, 5, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, -2, 0, 0, 2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, -3, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -2, 0, 0, 2, 0, 0], [-5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -1, 0, 0, -2, 0, 0], [0, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, -2], [0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 2, -2, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, -4, -4, 0, 0, 1], [0, 1, -2, 0, 0, -3, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 3, 0, 0, 0, 0, 2, 2, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, -3, -3, -3, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 3, 0, 0, 1, 0, 0], [0, 0, 0, 0, -3, 0, -2, 0, 0, 0, 0, -2, -2, 0, 0, 2], [0, 5, 3, 0, 0, 3, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 5, 0, 0, 1, -1, 0], [0, -3, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0, 1, 1, 0, 0, 1], [3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -5, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -6, 0, 0, 3, 2, 0], [0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, -1, 8, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 5], [0, -1, -3, 0, 0, -5, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, -3, -4, 0], [5, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -3, 3, 0, 0, 0], [-5, 0, 0, -2, 0, 0, 0, 2, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, -1], [0, 2, -10, 0, 0, -1, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 6, 0, 0, -3, 3, 0], [0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, -1, -1, 0, 0, -5], [1, 0, 0, -5, 0, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 0], [5, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 3, -3, 0, 0, 0], [0, 7, 5, 0, 0, -3, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 7, 0, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 4, 4, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 2, 0, 0, 3, -2, 0], [0, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0, 7, 7, 0, 0, -2], [0, -8, 11, 0, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [4, 0, 0, 4, 0, 0, 0, -2, 0, 0, 0, -4, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -3, 0, 0, 0, 0, -3, -3, 0, 0, -3], [0, 0, 0, 0, 4, 0, -3, 0, 0, 0, 0, -5, -5, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 5, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 5, 0, 0, -3, 2, 0], [2, 0, 0, 5, 0, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, 0], [0, -5, 8, 0, 0, 6, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [-5, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, -2, 0, 0, 0], [0, -8, 0, 0, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 3, -7, 0, 0, 5, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [6, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 2, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -5, 0, 0, 4, 0, 0], [7, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 3, -3, 0, 0, 0], [0, 3, 11, 0, 0, -1, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, -1, -2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -16, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -2, 0, 0, 0, 0, 2, 2, 0, 0, 0], [-13, 0, 0, -5, 0, 0, 0, 1, 0, 0, 0, 7, -7, 0, 0, 0], [1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, -5, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -8, 0, 0, 1, 3, 0], [0, 9, -8, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -7, -2, 0], [-21, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, -4, -4, 0, 0, 1], [0, 1, 8, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 6, 0, 0, 4, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 5, 0, 0, 1, 1, 0], [0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 0, 0, 0, -5, 0, -2, 0, 0, 0, 0, 2, 2, 0, 0, -7], [8, 0, 0, -4, 0, 0, 0, -4, 0, 0, 0, 6, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 2, 0, 0, -5, -2, 0], [0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, -2, -2, 0, 0, 0], [-7, 0, 0, -3, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -5, -1, 0, 0, -3, 0, 0], [0, 0, 0, 0, -6, 0, 2, 0, 0, 0, 0, 4, 4, 0, 0, -4], [0, 2, -2, 0, 0, -8, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 1, 0, 0, 5, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, -1, -2, 0], [0, 0, 0, 0, 5, 0, -3, 0, 0, 0, 0, -4, -4, 0, 0, 7], [2, 0, 0, -3, 0, 0, 0, -1, 0, 0, 0, 3, -3, 0, 0, 0], [0, 4, -6, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -5, 1, 0, 0, -2, -2, 0], [0, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, -6], [0, 0, 0, 0, 0, 0, 0, 0, 6, 6, -1, 0, 0, 2, -4, 0], [-5, 0, 0, -1, 0, 0, 0, -3, 0, 0, 0, -5, 5, 0, 0, 0], [0, -9, 7, 0, 0, 1, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -7], [0, 0, 0, 0, 6, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -4], [14, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, -5, 7, 0, 0, 2, 1, 0], [20, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, -1, 1, 0, 0, 0], [13, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 7, -7, 0, 0, 0], [0, 0, 10, 0, 0, -1, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -3, 0, 0, 0, 0, 1, 1, 0, 0, -9], [0, 3, 4, 0, 0, -4, 0, 0, -5, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 4, 4, 0, 0, -3], [0, -2, 12, 0, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 6, 0, 0, 0, 4, 0], [0, 0, 0, 0, -1, 0, 2, 0, 0, 0, 0, 8, 8, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, -9, 3, 0], [-23, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, -6, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, -1, 4, 0], [0, 0, 0, 0, -5, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -5], [0, -7, 11, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, -3, 0, 0, 0, 0, -4, -4, 0, 0, 2], [0, -6, -14, 0, 0, 1, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0, -8, -8, 2, 0, 0, -7, -1, 0], [23, 0, 0, 1, 0, 0, 0, -3, 0, 0, 0, -3, 3, 0, 0, 0], [0, 7, 10, 0, 0, -3, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0], [7, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, -5, 5, 0, 0, 0], [0, -1, -9, 0, 0, 9, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [9, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 5, 5, 0, 0, -4, 4, 0], [-7, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 4, -4, 0, 0, 0], [0, -3, 14, 0, 0, -4, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 3, 0, 0, 0, 0, -5, -5, 0, 0, 5], [0, 5, 2, 0, 0, 2, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -3, -4, 0, 0, 8, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 8, -4, 0, 0, 3, -2, 0], [-20, 0, 0, -2, 0, 0, 0, -4, 0, 0, 0, -1, 1, 0, 0, 0], [0, 0, 11, 0, 0, -2, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, -8, 2, 0, 0, -7, 8, 0], [0, 7, -11, 0, 0, 5, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [9, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0], [0, 0, 0, 0, 5, 0, 1, 0, 0, 0, 0, 4, 4, 0, 0, 0], [-9, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, -6, 6, 0, 0, 0], [0, 7, -6, 0, 0, 4, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 11, -4, 0, 0, 8, -4, 0], [0, 0, 0, 0, 5, 0, -1, 0, 0, 0, 0, -2, -2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 0, -4, -7, 0], [9, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 3, -3, 0, 0, 0], [0, 0, 0, 0, 5, 0, -2, 0, 0, 0, 0, 1, 1, 0, 0, -6], [0, -2, 8, 0, 0, -1, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [-24, 0, 0, -7, 0, 0, 0, -1, 0, 0, 0, 6, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7, 7, 2, 0, 0, 4, -3, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 3, 0, 0, -5], [0, 0, 0, 0, -8, 0, 2, 0, 0, 0, 0, -2, -2, 0, 0, 5], [-17, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0], [0, 4, 9, 0, 0, 2, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0], [0, -8, 10, 0, 0, -3, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 9, 0, 0, -1, -3, 0], [0, 1, 10, 0, 0, -2, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [20, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 2, -2, 0, 0, 0], [0, 0, 0, 0, 3, 0, -1, 0, 0, 0, 0, 2, 2, 0, 0, 5], [14, 0, 0, 1, 0, 0, 0, 7, 0, 0, 0, -1, 1, 0, 0, 0], [0, 7, -7, 0, 0, 1, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, -6, -6, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 8, 0, 0, -9, 2, 0], [-15, 0, 0, 6, 0, 0, 0, -1, 0, 0, 0, -3, 3, 0, 0, 0], [0, 5, 2, 0, 0, -8, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, -4, 0, 0, -3, 4, 0], [0, 4, -14, 0, 0, -7, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [-30, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -5, 5, 0, 0, 0], [-9, 0, 0, -3, 0, 0, 0, 3, 0, 0, 0, 1, -1, 0, 0, 0], [0, -1, 0, 0, 0, 13, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [0, 3, 5, 0, 0, -15, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 9, 9, 0, 0, 0, 8, -3, 0], [0, 0, 0, 0, 6, 0, -3, 0, 0, 0, 0, 5, 5, 0, 0, 2], [0, 0, 0, 0, -6, 0, -6, 0, 0, 0, 0, -8, -8, 0, 0, -6], [0, 0, 0, 0, -12, 0, 3, 0, 0, 0, 0, 3, 3, 0, 0, -3], [0, 6, 1, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 3, 6, 0, 0, 8, 0, 0, 5, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, 0, -1, 0, 0, 0, 0, -6, -6, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 0, -3, -3, -10, 0, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, -6, -4, 0, 0, 3, -6, 0], [0, 0, 0, 0, 3, 0, -4, 0, 0, 0, 0, -6, -6, 0, 0, 0], [-15, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, -6, 6, 0, 0, 0], [0, -4, 10, 0, 0, 2, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [17, 0, 0, -7, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -9, 0, 0, 1, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -5, 0, 0, 0, 0, 7, 7, 0, 0, -11], [-29, 0, 0, -3, 0, 0, 0, -2, 0, 0, 0, 5, -5, 0, 0, 0], [0, 14, -22, 0, 0, -6, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11, 11, -7, 0, 0, 6, -7, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, -9, -1, 0], [11, 0, 0, -7, 0, 0, 0, 3, 0, 0, 0, -3, 3, 0, 0, 0], [0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 1, 0, 0, 10, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 6, 0, 0, 0, -7, 7, 0, 0, 0], [0, -9, 7, 0, 0, 7, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [-17, 0, 0, -1, 0, 0, 0, 9, 0, 0, 0, -1, 1, 0, 0, 0], [0, 2, 4, 0, 0, 9, 0, 0, 6, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, -1, -8, 0], [0, 0, 0, 0, 8, 0, -2, 0, 0, 0, 0, 4, 4, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, 7, 7, -3, 0, 0, 12, 2, 0], [-30, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -4, 4, 0, 0, 0], [0, 0, 0, 0, 11, 0, -3, 0, 0, 0, 0, -2, -2, 0, 0, 7], [-9, 0, 0, 7, 0, 0, 0, -2, 0, 0, 0, -3, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -14, 0, 0, 1, 0, 0], [0, 0, 0, 0, -6, 0, -2, 0, 0, 0, 0, 6, 6, 0, 0, -12], [0, -6, 6, 0, 0, -6, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -1, 0, 0, 0, 0, 12, 12, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 13, 0, 0, 4, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, -9, -9, -10, 0, 0, -7, 8, 0], [0, -19, 11, 0, 0, -3, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0], [0, 0, 0, 0, 6, 0, 7, 0, 0, 0, 0, 2, 2, 0, 0, 1], [0, 0, 0, 0, -9, 0, -4, 0, 0, 0, 0, -7, -7, 0, 0, -3], [-28, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, -4, 0, 0, 5, 0, 0, -3, 3, 0, 0, 0, 0, 0, 0], [0, -5, -15, 0, 0, 7, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [0, 1, -13, 0, 0, -5, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [-27, 0, 0, 4, 0, 0, 0, -5, 0, 0, 0, -3, 3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 6, 0, 0, 0, 3, 0], [-1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, -5, 5, 0, 0, 0], [0, 5, 16, 0, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -10, 0, 0, -5, -1, 0], [0, 0, 0, 0, -10, 0, 1, 0, 0, 0, 0, 11, 11, 0, 0, -9], [0, -13, 16, 0, 0, -6, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 1, 0, 0, -11, 6, 0], [7, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 11, -11, 0, 0, 0], [0, 0, 0, 0, 4, 0, -3, 0, 0, 0, 0, 8, 8, 0, 0, -3], [0, -2, 18, 0, 0, -8, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -9, -9, 0, 0, 6], [0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 7, -7, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 2, 0, 0, -8, -2, 0], [0, 0, 0, 0, -2, 0, 9, 0, 0, 0, 0, 6, 6, 0, 0, -1], [0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 8], [0, 1, -10, 0, 0, -9, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 8, 0, 0, 0, 0, 4, 4, 0, 0, 10], [0, 0, 0, 0, 0, 0, 0, 0, 6, 6, -12, 0, 0, 0, -2, 0], [0, 0, -6, 0, 0, -2, 0, 0, 8, -8, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -13, 13, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 2, 0], [0, 0, 0, 0, 3, 0, -2, 0, 0, 0, 0, -8, -8, 0, 0, 13], [14, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, -6, 6, 0, 0, 0], [0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 7, 7, 0, 0, -2], [39, 0, 0, -3, 0, 0, 0, -4, 0, 0, 0, 10, -10, 0, 0, 0], [0, 0, 0, 0, -2, 0, 4, 0, 0, 0, 0, -1, -1, 0, 0, -4], [0, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0, 11, 11, 0, 0, -11], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 15, 0, 0, -7, 4, 0], [-3, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, -3, 3, 0, 0, 0], [-17, 0, 0, -5, 0, 0, 0, -7, 0, 0, 0, -4, 4, 0, 0, 0], [0, 0, -1, 0, 0, -8, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, -11, 13, 0, 0, -13, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [-17, 0, 0, 9, 0, 0, 0, 6, 0, 0, 0, -12, 12, 0, 0, 0], [0, 5, 9, 0, 0, -9, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, 4, 0, 0, 0, 0, 4, 4, 0, 0, 7], [0, 0, 0, 0, -4, 0, 9, 0, 0, 0, 0, 6, 6, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, -10, 0, 0, -4, 4, 0], [39, 0, 0, -1, 0, 0, 0, -3, 0, 0, 0, -2, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 6, 6, -3, 0, 0, 3, -5, 0], [0, 0, 0, 0, -2, 0, -6, 0, 0, 0, 0, 8, 8, 0, 0, -3], [0, -12, -2, 0, 0, -11, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 2, 0, 0, -8, -2, 0], [0, 0, -9, 0, 0, 10, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -13, 0, 0, 3, 2, 0], [0, -18, 1, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [19, 0, 0, 1, 0, 0, 0, -9, 0, 0, 0, 5, -5, 0, 0, 0], [0, -1, 16, 0, 0, 2, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 5, 0, 0, 0, 0, 1, 1, 0, 0, -9], [0, 11, 2, 0, 0, -4, 0, 0, -8, 8, 0, 0, 0, 0, 0, 0], [0, 2, 5, 0, 0, 8, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -11, 0, 1, 0, 0, 0, 0, 4, 4, 0, 0, 1], [0, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, -6, -6, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -6, 0, 0, 5, -7, 0], [28, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_3024_k_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_3024_2_k_l();". // 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_3024_2_k_l(:prec:=16) chi := MakeCharacter_3024_k(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(1999) - 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_3024_2_k_l();". // 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_3024_2_k_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_3024_k(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![196, 0, -391, 0, 215, 0, -29, 0, 1]>,<11,R![2704, 0, 1764, 0, 380, 0, 33, 0, 1]>,<13,R![7744, 0, 5036, 0, 920, 0, 55, 0, 1]>],Snew); return Vf; end function;