// Make newform 1003.2.a.i in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1003_a();" // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();" // To make the newform (type ModFrm), type "MakeNewformModFrm_1003_2_a_i();". // 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_1003_2_a_i();". // 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 := [28, -100, -1952, 1189, 9201, -5604, -15697, 11182, 11533, -9826, -3825, 4304, 431, -984, 52, 112, -16, -5, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1040316564, 5242626976, 64993586656, -93338201887, -212478348524, 273476087608, 202318455723, -276891915821, -75963419650, 130824516472, 8453508555, -31655264755, 1553634958, 3840968938, -437835062, -195692906, 27603530, 1603205], [-1680834164, 2591601760, 16437157520, -29262232015, -30360885660, 63467758040, 23150614939, -54362020477, -11457941330, 23993616376, 4626655131, -6337233987, -1245763682, 1061307674, 172115546, -104332442, -9046678, 4468293], [3061678300, 19243183584, 33075248352, -176724978555, -99577814860, 440305730504, 12535086055, -411676152609, 72173081990, 180240532008, -50706085177, -39180669471, 14100829750, 3847479106, -1781428510, -83459586, 84688962, -6526647], [-599729124, -4274427680, 78337756256, -28420424579, -267468538284, 132442588328, 295042593455, -154717191481, -152435041482, 82189416168, 41448412623, -22957637767, -6013468890, 3455427122, 431026258, -262206418, -11556334, 7745729], [-736631868, 4552577048, 40558943600, -40703624959, -133664775808, 112629948452, 147120641455, -118650382749, -76407838686, 60917184580, 20921021015, -16838568259, -3055625518, 2543960726, 220312190, -196373670, -5938506, 6002425], [1835590924, 19664554208, -91595317216, -33137195863, 336609343060, -55429121448, -438843100125, 127822374123, 282941145998, -94156691400, -100266293789, 34840412021, 19790303422, -6987875910, -2030406822, 722209126, 84099226, -30052531], [-1088955028, -16222142464, -93309825920, 225691904993, 259080908628, -608004675912, -77018484181, 562653989427, -131568240834, -229270906824, 104573869707, 40801027885, -30205793170, -1510725526, 3887312362, -379778058, -186790710, 33160165], [-1527210676, 7531339520, 49161902080, -129377105239, -65264158156, 380890440888, -138565934845, -352535542229, 234252289166, 133661164920, -128256252797, -17567079211, 32718727454, -1494181414, -3965573350, 566863078, 184167610, -37154163], [-231904632, 2393923808, 5272197858, -4369624325, -25574527844, -3464216802, 50814226103, 2692357581, -44421256370, 3047670594, 19364549555, -2941782421, -4407747754, 913780746, 499734910, -122475798, -22234052, 6040615], [-64456832, -1770614060, -4229868864, 22352348333, 1840425726, -59382333630, 29244763413, 51907281475, -39689575104, -18409013782, 20581292633, 1941081393, -5128036560, 360506880, 613684104, -97976644, -28282200, 6053175], [-301716580, 8570974544, 4349495704, -113857594079, 47579502788, 303110008560, -198840292381, -269465073261, 216787275502, 100062459136, -104117203501, -13174676275, 24952854334, -986596822, -2917574854, 392850758, 132425074, -25731019], [656205004, 5349825328, -59079830624, -35961353735, 242495850636, 56580628080, -366506974869, -15371453797, 267515584966, -20090353488, -104218462469, 15817530869, 22083367798, -4512375294, -2389076942, 576714958, 102983154, -27616971], [-1512586372, 21482869568, 99470541936, -324651015307, -189752917180, 895533906920, -153858197321, -825445861681, 364158983542, 329590031528, -211764044169, -55249056879, 55256076454, 963704754, -6767846030, 711941006, 316018786, -55864183], [1077401004, 11706378208, -7738166112, -197884162319, 125519891236, 542752259080, -420831199701, -484568126045, 445032492334, 176131301896, -212861062133, -20012118563, 51151472350, -3034501318, -6007265254, 890033958, 273896826, -55930331], [555600236, -1751857152, 3188992512, 81114719985, -115388668836, -244328770576, 352769386931, 199306839139, -356503271578, -48552214560, 165748033427, -8103174819, -39016219162, 5763599234, 4510183266, -934344946, -203107614, 49926317]]; Rf_basisdens := [1, 1, 1, 568793824, 568793824, 568793824, 568793824, 142198456, 568793824, 568793824, 568793824, 71099228, 71099228, 284396912, 284396912, 568793824, 568793824, 284396912]; 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_1003_a();" function MakeCharacter_1003_a() N := 1003; order := 1; char_gens := [768, 120]; v := [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; function MakeCharacter_1003_a_Hecke(Kf) return MakeCharacter_1003_a(); 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, 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], [1, 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, -1, 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], [1, 0, 0, 0, 0, 0, 0, 1, 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], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1], [1, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1], [3, 0, 0, 1, 0, 1, 0, -1, 0, 0, 1, 1, 0, 0, 0, -1, 0, 0], [3, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, -1, 0, 0], [-1, 1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0], [2, -1, 1, -1, -2, 0, -1, 0, 0, -1, -1, 0, 0, 2, 0, 0, -2, 0], [3, -2, 1, -1, -1, -1, -1, 0, -1, 0, 0, 0, 0, 2, 0, -1, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 1, 2, 0, 1, 0, 1, 1, 0, -1, 1, -1, -1, 1, 1, -1], [3, -2, 2, 0, 0, 0, 0, -1, -1, -1, 1, 2, 0, 3, 0, -1, -2, 1], [2, 0, 1, 1, 0, -1, -2, 0, -2, -2, 0, 2, -1, 1, 0, -1, -1, 1], [4, -2, -1, -1, 0, -1, -1, 1, 0, -1, 0, 1, -1, 1, 0, 1, -1, 1], [-3, 1, -2, 1, 3, 0, 3, -1, 3, 2, 1, -1, 1, -3, 1, 1, 3, 0], [0, -2, 1, 0, 1, -1, 2, 0, 0, 1, 0, 0, 1, -2, 1, 1, 2, 0], [2, -1, 0, 1, 1, 2, 1, 0, 0, 1, 1, 0, 1, -1, -1, 0, 1, -2], [0, 1, -1, 0, 0, 0, 2, -1, 2, 1, 0, -2, 0, 0, 0, 1, 2, 2], [4, -1, 2, -2, -3, -2, -2, 1, -2, -3, -1, 0, -1, 3, 1, 0, -1, 1], [0, 1, -1, 0, 0, 1, 0, -1, 0, 0, 2, 0, -2, -1, 0, -1, 0, 1], [2, -2, 2, -4, 0, 0, -1, 1, 0, -1, -3, 0, 0, 1, 1, -1, -1, 2], [2, 1, 1, -3, -1, 1, -1, 0, 0, 0, -1, 0, -2, 0, 0, -2, -1, 0], [4, 2, -1, 0, -3, 0, -2, -2, -1, -1, 0, 1, -1, 4, -1, -2, -1, 1], [-1, -1, 2, -1, 0, 0, -2, 0, 0, -2, 0, 0, 1, 1, 2, -1, -1, 0], [-1, 1, -1, 0, 0, 0, 0, 0, -1, 2, -1, 1, 2, -2, 2, 1, 3, -1], [1, -2, -1, 1, 1, 0, 1, 0, 2, 1, 1, -2, 0, -1, 0, 2, 2, -1], [-1, 1, 1, 0, 1, 1, -1, 1, 1, -2, -3, -1, 0, -1, -1, 1, 0, -1], [6, -3, -3, 0, -1, -1, -1, 0, -1, -1, 2, 0, 0, 0, 2, 1, 0, 0], [-1, 0, 1, -2, -1, 0, 1, 0, 1, 0, -2, -2, 3, 0, 0, 1, 0, -1], [0, 1, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, -1, 1, 0, -2, -2], [0, 1, 0, 1, -2, 0, -2, 1, 0, -2, 2, 1, -1, 5, -2, -2, -3, 1], [2, 0, 2, 1, 1, -1, -2, 0, 1, 1, 0, -1, 0, 1, -2, -1, -1, 1], [5, -1, 1, 0, 2, 0, 0, -2, 2, 2, 2, -2, 0, -2, 0, -1, 1, 0], [0, -2, 0, -2, -1, 0, 0, 2, -2, 0, -3, 2, 0, -3, 3, 2, 0, -1], [0, 5, 0, 4, -1, 1, 0, -3, 0, 1, -1, 1, -1, 1, -2, -1, 0, 1], [-2, 0, 0, 0, 3, 0, 1, 1, 2, 2, 3, 0, 0, -3, 1, 0, 1, 2], [1, 0, 2, -4, -3, -1, -2, 3, 0, -1, -3, 0, 0, 0, 1, 1, -2, -1], [-2, -3, -3, -2, 0, 0, -1, 0, 1, 1, -5, 0, 2, -5, 2, 5, 3, -4], [0, -1, 0, -1, -1, -2, -1, 2, -1, -3, 0, -1, 1, 6, -1, 0, -2, 0], [0, -1, 4, 1, 1, 0, 3, -2, 0, 0, 3, 0, 1, 4, -2, 0, -2, 2], [-3, 0, -1, 0, 0, 0, 2, -1, 2, 3, 0, -3, 0, -1, -2, 2, 1, 2], [-2, 3, -4, 0, 2, -1, 1, -2, 1, 2, 3, -1, 1, -1, -1, 0, 2, 2], [-4, 0, 2, -1, 1, -1, 0, -2, 2, 1, -1, -1, 0, 2, -1, -2, -1, 0], [2, 1, -2, -1, 1, 1, -2, 1, 0, 1, -2, 2, -1, -6, 1, 0, 4, -1], [-5, 0, -1, -2, 0, 2, 0, -1, 3, 1, -1, -1, -2, -2, -3, -1, -1, -1], [0, -1, 5, 0, -1, 0, 2, 1, 1, 0, -5, 1, 2, -1, 4, 1, -1, -2], [3, 1, -3, 2, -1, 1, 0, -2, -3, 1, 5, 3, -1, 3, -2, -1, -1, 2], [3, 5, 2, -3, -1, 1, 0, 0, 2, 0, -3, 1, -2, -1, 2, -3, 0, 1], [4, -3, 0, -1, -2, -1, -2, 3, 0, -1, -4, -2, -2, 3, -1, 1, -1, 0], [5, 4, -2, 1, 2, 2, -2, -1, 1, 1, 4, 0, 0, 1, -2, 0, 1, 0], [-1, 0, 4, -2, -2, 2, -3, 2, 0, -2, 0, 0, 1, 6, -1, -2, -4, 1], [5, -2, 0, 0, 0, 0, -1, 0, -1, -1, 1, 2, 0, 3, -3, -1, -2, 1], [1, 2, -1, 3, 0, 1, -1, 2, -1, -2, 1, 1, -2, 2, -2, -1, -1, 0], [2, 3, 3, 2, 1, 3, -4, -1, 1, -1, -2, 0, -2, 0, 1, -3, -3, -2], [3, 4, 1, -1, 0, 1, 1, 0, 2, -2, -6, -2, 0, 0, 0, 0, 0, -1], [1, 1, 1, 2, 2, -1, 0, 0, 2, 1, -3, 1, 1, -3, 1, 0, 0, -3], [4, 1, 1, 1, 3, 1, 0, -1, 3, 0, 0, -1, 0, 1, -5, -1, 0, 0], [2, -4, 0, -2, 1, -2, 1, 3, -2, 0, -4, 0, 0, -3, 3, 4, 0, -1], [8, 0, 2, -1, 0, 1, 1, 1, 1, -2, -3, 0, 2, 5, 0, -1, -4, 1], [-6, 1, -3, 3, 1, 1, 1, 0, 1, 3, 1, -2, 2, -3, -2, 2, 6, -3], [2, 1, 1, -2, 0, -1, -1, 0, 0, -2, 1, 0, 0, 2, 1, -3, 4, 1], [2, 3, 3, -3, -4, 0, -5, 0, -1, -3, 0, 1, -4, 9, -1, -5, -5, 3], [-2, 1, -3, -1, -1, 0, -4, -1, 1, 2, -1, 2, 0, -5, 4, 2, 4, 0], [5, -3, -1, 2, 1, 2, 4, 0, 1, 3, -2, -1, 4, -7, 1, 1, 2, -6], [6, -2, 1, 0, 1, 2, -4, 2, -1, -2, -4, 3, 2, -3, 3, 1, -2, -4], [1, -2, 5, 0, -1, 1, 1, 0, 1, 0, -4, -2, -1, -3, -1, -1, -1, -2], [1, -4, 1, -2, -3, 1, -3, 2, -5, -2, -2, 0, 0, 3, 2, 0, 0, -1], [0, 2, 0, -4, -6, 0, -5, 3, -2, -3, -6, 2, 0, -2, 3, 0, -1, -2], [-3, -2, -3, 1, 5, -1, 1, -2, 2, 2, 2, 1, 2, -5, 2, 3, 7, 0], [-2, 2, -4, -3, -2, -2, -1, -1, 0, 2, 1, 1, 0, -3, 1, 0, 1, 2], [4, 5, 1, -2, -3, -1, -3, 4, 0, -1, -1, -2, 2, 2, -1, 0, -1, 0], [4, 2, 2, 2, 2, -1, 2, -2, 0, 0, 9, -4, 0, 7, -3, -4, 0, 3], [0, -4, 4, -2, -1, -1, -4, 2, -3, -3, 0, 3, -2, 5, -2, -1, -3, 2], [3, -2, -1, 2, 1, 1, 3, 0, -1, 0, 0, -2, 3, 3, 0, 1, -3, -4], [0, 3, -5, -1, 0, -1, -1, 0, 0, 2, 4, 0, -2, -3, 1, 1, 4, -2], [-7, 3, -1, -2, -3, 4, -1, -3, 0, 1, 3, 1, -2, 4, -3, -3, -3, -1], [9, -5, 1, 1, -1, -2, -1, 1, -1, -2, 0, -2, -1, 2, 2, 1, -2, -1], [-4, 2, 3, -2, 3, -1, -2, 4, 1, -2, 1, -2, -1, 4, -2, -1, -1, 2], [1, -2, -1, -2, -4, -1, -1, 2, 0, -2, -1, 0, -1, 4, -4, 0, -1, 3], [0, -2, 0, -1, -2, -2, 1, 3, 1, -1, 1, 1, 2, 3, 1, 0, -2, -2], [-7, 3, -7, 5, 5, -1, 2, -4, 1, 3, 1, 1, 2, -6, -3, 2, 8, -1], [3, 4, -3, 2, 0, -1, 2, -1, -1, 1, -1, 0, 2, 0, -1, 2, 0, 1], [-2, 1, 1, 0, -2, 4, 3, -2, 2, 2, 5, 1, 2, 3, -1, -4, 0, -1], [2, 3, -1, 0, 2, 1, 3, 0, 4, 0, -6, -2, 2, -2, 0, 2, 0, 0], [4, -3, 3, -2, -3, 1, -1, 0, -3, -4, 3, 5, -4, 7, 0, -3, -8, 3], [-1, 2, -4, -2, 0, 1, -5, 4, 2, 1, -4, 4, -2, -9, 4, 3, -1, -3], [-3, 5, 2, 2, -2, -1, 0, 0, -1, 1, 2, -1, -2, 2, -1, -3, -1, -1], [0, -1, 0, 3, 5, 5, 6, -5, 6, 5, 4, -1, 3, -2, -2, 0, 1, -1], [1, -9, 0, -4, 0, -2, 1, 3, 2, -2, -9, 3, 1, -3, 2, 3, 1, 0], [1, 1, -1, 1, 3, -2, 2, 2, -1, -1, 6, -2, 2, 6, 0, -1, 1, 1], [9, -5, 1, -2, -2, -1, -4, 2, 0, -3, -5, 3, 0, -2, 7, -1, 0, 0], [1, 0, 2, -5, 1, -1, 0, 1, 0, -3, 1, 0, 1, 3, 1, -4, -2, 1], [2, 1, -4, -3, -2, -1, -4, 3, -1, -5, -4, 0, -5, 2, -3, -2, -4, 1], [-4, -1, 2, -6, -2, 1, 3, 2, 1, 1, -7, 1, 1, -5, 3, 2, 1, -1], [4, -1, 3, 1, 0, 2, -3, 0, 0, -1, 2, -1, 1, 6, -4, 1, -3, 0], [1, 5, -4, 7, 2, -1, 3, -5, -3, 1, 4, 2, 1, -1, -1, 1, 2, 1], [2, -2, -5, 2, 1, 1, 2, 0, -2, 1, 2, 1, 1, 0, 1, -2, -1, -1], [-5, -7, 1, -4, -1, -4, -1, 5, 1, 0, -8, -1, 2, -3, 1, 7, 2, -2], [4, 1, 1, 4, 0, -2, 1, -2, -2, -3, 2, 2, 1, 2, -2, 0, 1, 4], [6, -1, 1, 3, 5, -1, 4, -1, 3, 0, 4, 0, 2, -3, 3, -2, 0, 1], [4, 0, 3, -1, 0, 0, 3, 1, 1, -2, -1, -4, -2, 5, -2, 1, -4, 3], [-10, 7, -5, 4, 2, 2, 1, -2, 4, 6, 8, -1, 0, -4, -2, 3, 5, -4], [-2, -3, 5, -2, -2, 4, -8, -1, 0, -2, 3, 2, -2, 2, -2, -2, -5, -2], [-4, 4, -10, 3, 5, 2, -2, -6, 3, 3, 5, 0, 0, -4, -3, 1, 7, 1], [0, -5, 2, 2, -2, -4, 1, 2, -2, 0, 3, 2, 1, 1, 0, 5, -1, 2], [9, 4, -5, 5, 2, -3, 1, -4, -1, 4, 8, 0, 1, -4, 0, 0, 5, 1], [1, -7, 1, -6, -5, 0, -6, 4, -4, -3, -6, 3, -2, -1, 5, 5, 0, -2], [-9, -1, -2, 3, 4, -1, 8, 0, 0, 3, 10, -3, 2, 0, 1, -1, 4, 2], [-1, 1, 1, -1, 1, 0, 3, -1, 3, 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[35, -13, 9, -6, -8, 4, -13, 7, -6, -4, -7, 5, -4, 4, 3, -1, -7, -3], [-6, -1, -10, 2, -2, 0, 0, -7, 0, 2, 3, 9, -1, -9, 9, 6, 5, 2], [-9, 5, 5, 3, -4, 9, 2, -4, 1, 1, 4, -2, -6, 3, -13, -7, -3, -1], [-7, 2, 0, -8, -4, 2, -13, 6, -4, -8, -22, 7, 1, -11, 5, 2, 4, -6], [16, -11, -7, -4, 6, -7, -2, -1, 1, -1, -12, -9, 4, -16, 11, 7, 11, -3], [7, 5, -2, 5, -7, -12, -5, 3, -14, -10, -2, 5, 1, 10, 1, 4, -3, -7], [3, -1, -7, 8, 5, 2, 11, -5, -6, 4, 6, -2, 4, 11, -4, -2, -4, 1], [9, -8, 13, -3, -5, 1, -13, 10, -1, -4, -9, 2, -1, 6, -4, -4, -12, -2], [-6, -5, -5, 12, 7, 3, 16, -5, 12, 9, 19, -3, 0, -3, -5, 1, 8, 0], [0, 4, -14, 2, 7, -1, 10, -2, 6, 1, -6, -2, 0, -22, 5, 2, 17, -5], [-1, -15, 3, 1, -4, 6, 3, 7, 3, -1, 4, 1, 1, 9, -4, 6, -3, 0], [-5, -2, 19, -12, 2, -3, 1, 6, -1, -4, 1, -1, 2, -5, 6, -4, -2, 2], [13, -3, 6, -5, -4, -2, 1, 5, 2, -3, 15, -5, -1, 19, -3, -6, -2, 4], [14, 20, -9, 9, 10, 6, -2, -8, 6, 2, 0, -4, 1, -12, -2, 2, 11, -3], [-12, 7, 1, -6, 2, 0, -1, 0, 7, 3, 1, -2, 2, 11, -2, -5, -11, 0], [24, -5, 5, 0, -3, 1, -5, -8, -8, -4, 10, 7, -4, 16, -1, -11, -16, 5], [-5, -3, 2, -8, -5, -5, 0, 4, -8, -5, -3, -5, -5, -1, -2, 1, -7, 7], [8, -3, -10, 6, 14, 4, 13, -2, 4, 6, 9, -1, -1, -11, -10, -2, 10, 0], [22, -3, -7, 2, 0, 4, -4, -15, -3, 1, 16, -1, 0, 9, -2, -8, 0, 1], [6, 16, 0, 4, 2, 3, 5, 3, 0, -3, 4, 2, 0, -5, -5, 7, -2, -4], [6, 7, -7, -5, 1, 5, -11, 4, -3, -1, -4, 7, -1, -12, 5, -4, 5, -8], [4, -17, 10, 0, 5, 0, 5, -4, 6, 4, -4, -11, 8, -5, -2, -3, 6, -5], [21, 2, -10, 0, -2, 0, 0, 1, 2, -2, -5, -5, 2, 5, -10, 6, 4, -1], [-6, 4, -11, 9, -10, 3, -2, -8, 2, 3, 12, -5, 0, 10, -11, -3, -3, 0], [1, 7, -9, 5, 0, -1, 5, 5, 4, 3, 6, -1, -2, 3, -5, 6, 4, 2], [2, 8, 6, -8, 1, -5, -5, 1, -3, -8, -18, 6, 0, -4, -1, -5, -3, 4], [8, -4, 4, -9, -4, 4, 0, 0, -6, -2, 2, 5, -4, 3, 5, -2, 2, 5], [-20, 6, -8, -5, 10, -3, 1, 3, 2, 2, 2, -4, 0, -21, 1, 3, 16, -4], [-3, -6, 5, 1, -7, -1, -13, -6, 0, -4, -6, 5, -6, 7, -5, -7, 0, 5], [3, -5, -11, 2, 1, -3, 3, -1, -12, 1, 16, 0, 6, -2, 4, 4, 12, 4], [-5, 2, -7, -7, 5, 4, -5, -2, 4, 2, 4, -4, 3, -1, -5, 2, 5, 0], [-11, -9, 2, -7, -2, 1, -2, 6, -1, 0, -5, -1, 0, 6, -1, 3, -8, -3], [11, -1, -1, -8, -5, 3, 3, 3, -4, 2, -8, 5, 2, -8, 4, 4, 17, 1], [4, 13, 0, -1, -10, 6, -10, -3, -3, -5, 6, 1, -5, 6, -13, -5, -6, 5], [5, -1, 11, 1, -1, -3, -1, 7, 6, -4, -9, -11, 0, 3, -3, -3, -3, -3], [3, 5, 5, -3, -3, -4, -19, -1, -3, -3, -11, 1, 1, -3, 2, -1, -1, -3], [-14, -10, -7, 3, 1, -1, 9, -5, 6, 10, 5, -8, 1, -5, 1, 12, 15, 6], [18, 4, -4, -3, -1, -2, 3, 7, -9, 0, 3, 7, -6, 0, -9, 0, 2, 4], [-10, -13, 10, 2, 2, -1, 4, 3, 8, 4, -8, -10, 4, -5, 8, 1, 1, -6], [6, 3, 11, 1, 1, 0, -17, -2, -4, -9, -14, 8, -4, -7, 0, -2, -11, -4], [6, -3, -3, 2, 3, 10, 8, -7, 0, 7, 10, 2, 0, -10, 2, -4, 4, -2], [0, -3, 7, -6, 2, 7, 6, 9, -1, -6, -8, -1, 0, -5, 7, -1, -3, -13], [9, 7, 10, -2, -6, -3, -10, 4, -19, -10, -11, 8, -7, 14, -2, -8, -15, 3], [-11, -9, 10, -11, -6, -8, -3, 9, -4, -4, -13, -1, -6, 5, -4, 0, -6, 4], [20, -11, 3, 1, -1, 4, 6, -1, -2, -1, -6, 0, -2, 7, 8, -2, -1, 6], [0, -12, 1, -3, 13, -3, 12, 2, 6, 9, 5, -5, 7, -14, 6, 8, 5, 2], [3, 3, 0, 0, -8, -4, 5, -11, -8, 1, 13, 2, -5, 0, 4, -6, -1, 6], [-4, 13, -3, -2, -3, 8, 5, -2, 0, 1, -5, 2, 4, -8, 0, 7, 0, -6], [-14, -4, 1, -5, -2, -11, -3, 7, -8, -3, -5, 0, 1, -5, 3, 13, 11, -3], [21, 4, -2, 11, 7, -3, 14, -9, 2, 1, 16, -2, 8, 8, 1, -3, 0, 3], [2, -9, 1, 1, -6, -2, -6, 1, -4, 3, 3, 10, 2, 5, 4, -2, -1, -2], [-17, 11, -8, 3, -5, 6, -5, -3, 8, 7, -7, -4, -3, -4, -4, -3, 12, -4], [-7, 19, -11, 4, 7, 1, 0, 6, 9, 5, 9, 3, -3, -15, 3, 3, 9, -5], [9, -1, -3, -1, -4, 7, -7, -9, -4, -3, -9, 7, -4, -9, -1, -8, -1, -3]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1003_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1003_2_a_i();". // 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_1003_2_a_i(:prec:=18) chi := MakeCharacter_1003_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); S := BaseChange(S, Kf); maxprec := NextPrime(2999) - 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_1003_2_a_i();". // 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_1003_2_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1003_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![28, -100, -1952, 1189, 9201, -5604, -15697, 11182, 11533, -9826, -3825, 4304, 431, -984, 52, 112, -16, -5, 1]>,<3,R![128, 3760, 11672, -5572, -42652, -3090, 60913, 8592, -43240, -5784, 16995, 1901, -3862, -331, 504, 29, -35, -1, 1]>],Snew); return Vf; end function;