// Make newform 700.4.a.e in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_700_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_700_4_a_e();". // 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_700_4_a_e();". // 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 Kf := Rationals(); end if; return [Kf!elt[1] : elt in input]; end function; // To make the character of type GrpDrchElt, type "MakeCharacter_700_a();" function MakeCharacter_700_a() N := 700; order := 1; char_gens := [351, 477, 101]; v := [1, 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_700_a_Hecke(Kf) return MakeCharacter_700_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 := 4; raw_aps := [[0], [-4], [0], [-7], [-12], [82], [30], [68], [-216], [246], [-112], [-110], [-246], [172], [-192], [-558], [540], [110], [-140], [-840], [550], [-208], [-516], [-1398], [-1586], [-1242], [-680], [-996], [1382], [750], [-176], [-1548], [-378], [-2500], [846], [-2536], [1186], [-2108], [1944], [1362], [1596], [-1690], [3552], [2686], [1410], [-2968], [-1348], [-3872], [-5364], [-874], [-378], [1920], [4322], [5292], [5118], [-3768], [3918], [4880], [3538], [-5430], [6436], [-1350], [-3332], [-4728], [-5114], [-7206], [6260], [5326], [-36], [3134], [-1218], [-10008], [1072], [274], [7652], [-2160], [-1074], [-6926], [1938], [-9574], [-5052], [3422], [2208], [6814], [12584], [-6996], [9474], [-5786], [3438], [-9392], [4956], [-20592], [13432], [-14172], [-5956], [-16968], [5214], [-1398], [18580], [-18970], [16036], [-8310], [-7092], [-7158], [6500], [-21794], [-9756], [-5586], [-24], [4298], [-8480], [1906], [-7482], [-7348], [4520], [-19806], [5020], [-28392], [17562], [4716], [-22762], [-4802], [-21558], [-3780], [-5500], [10230], [10190], [9408], [33064], [6322], [-20740], [-32040], [-12832], [19906], [10842], [28274], [32346], [-30116], [6594], [-43014], [-14164], [34830], [-31016], [-9876], [-3154], [36936], [-9638], [-10266], [-4084], [192], [-19910], [14802], [32548], [-1464], [49564], [8448], [14600], [-21102], [20806], [24510], [44148], [-27114], [10264], [51468], [23790], [-26424], [39488], [-30854], [-19246], [-43782], [62676], [42302], [-19272], [52198], [-38080], [-62742], [22964], [-25650], [48904], [-21250], [37168], [-1500], [-11630], [5766], [5568], [-42042], [1114], [62476], [39194], [-16944], [-38498], [15516], [34940], [-13218], [17556], [7782], [12434], [-25238], [35838], [83496], [-43338], [-53632], [-27542], [-2590], [-15924], [3474], [48080], [61956], [-8742], [-53764], [26926], [-61770], [27880], [-17820], [42168], [14954], [-47504], [68706], [-50664], [75498], [-80146], [63272], [39810], [-55712], [-66756], [63614], [-78282], [42000], [45160], [-64812], [10618], [103412], [-83008], [104730], [34612], [-61392], [-83614], [-70350], [14004], [-1800], [28884], [114812], [32392], [7094], [-90354], [-62376], [5536], [12948], [-38284], [88656], [-112646], [-60702], [-49176], [-85414], [-19614], [60828], [-122242], [60916], [36762], [63430], [-23792], [3588], [-48442], [-34382], [107406], [80420], [-16362], [-108822], [-49556], [55314], [11918], [104884], [18454], [39056], [-13970], [65176], [15324], [88358], [150362], [-31164], [-88032], [-328], [18504], [12902], [118132], [-112320], [106894], [101586], [-37480], [65154], [72726], [140700], [122454], [31692], [39850], [110334], [27584], [45786], [80364], [-63044], [-30314], [100026], [2864], [-61548], [-144124], [90238], [-120084], [48734], [159384], [-134078], [32976], [-81474], [66690], [-74972], [-80376], [127610], [95604], [-100800], [-58562], [-16206], [36692], [72022], [-98610], [60256], [70086], [-46366], [63164], [3652], [-71664], [-32622], [34934], [-60150], [5168], [98940], [66116], [-53724], [-172618], [-74178], [-176806], [-153632], [24562], [-146442], [-52458], [148376], [164010], [-22212], [98678], [-45236], [-227376], [-125046], [-122236], [61990], [-123018], [22336], [112622], [168918], [227856], [-64596], [113646], [-28536], [-105374], [142938], [86112], [64020], [-40436], [-44282], [-131886], [-65960], [-138166], [-220116], [26588], [124368], [38766], [-130360], [-157382], [-89148], [-82992], [-187490], [-40734], [-91322], [-169122], [212262], [113848], [14046], [22916], [-195192], [-19888], [-13142], [5548], [106416], [-56062], [-113286], [240900], [10348], [-14760], [39382], [160976], [229866], [189284], [-14274], [74414], [257022], [49552], [118758], [109398], [120680], [187594], [-200814], [-260852], [178404], [31246], [-51390], [63036], [-99700], [-51194], [66894], [11328], [87448], [98046], [-109032], [117174], [236602], [-212400], [-281076], [72790], [-134718], [-27852], [242826], [156332], [227352]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_700_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_700_4_a_e();". // 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_700_4_a_e(:prec:=1) chi := MakeCharacter_700_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 4)); 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_700_4_a_e();". // 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_700_4_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_700_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<3,R![4, 1]>,<11,R![12, 1]>,<13,R![-82, 1]>],Snew); return Vf; end function;