// Make newform 560.6.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_560_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_560_6_a_g();". // 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_560_6_a_g();". // 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_560_a();" function MakeCharacter_560_a() N := 560; order := 1; char_gens := [351, 421, 337, 241]; v := [1, 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_560_a_Hecke(Kf) return MakeCharacter_560_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 := 6; raw_aps := [[0], [12], [25], [49], [-556], [-354], [770], [2684], [1528], [-2418], [-7840], [-314], [-17878], [-16476], [-5376], [1654], [29492], [27630], [-57716], [-70648], [74202], [-74336], [-44068], [129306], [-137646], [52086], [-128216], [-113340], [-138946], [-28046], [26704], [-7028], [140202], [95876], [-347002], [270328], [-151186], [-588148], [-61912], [4862], [17964], [228422], [357840], [-494174], [595302], [261208], [-788564], [1358416], [461068], [1255094], [503914], [-1509024], [-63022], [964820], [-495150], [-2180920], [-473346], [2192336], [388502], [-75462], [-2172716], [142902], [-1052772], [3112136], [-1956758], [2747998], [-2230188], [3134386], [-2430028], [3460622], [-1256270], [-878712], [-234880], [97430], [1997220], [-119664], [327670], [4466526], [-1832494], [-5593926], [-562772], [679158], [-5894656], [-1644990], [4201384], [565684], [648386], [-1747126], [-317378], [5846784], [3672572], [1663328], [-4345048], [6650164], [-742996], [5571256], [-5912466], [6340650], [-118300], [5362478], [579116], [6618798], [2659068], [-11264998], [4052420], [-5075662], [9070788], [14779266], [-9062120], [-2654822], [-15293040], [14906950], [5120298], [4831396], [-5782120], [-16569086], [9284908], [13150472], [-6282066], [21600716], [-5792346], [14507106], [16677462], [-3355548], [-80036], [12640302], [10320342], [-13232976], [-1894472], [-3250226], [-25636612], [11033160], [-17597664], [-9254442], [13663962], [-29849822], [23287830], [10169148], [2431918], [14195978], [-1755484], [16971462], [8852408], [-24841772], [3556910], [-25140104], [-10124218], [-10552182], [-33775756], [31265168], [-28992114], [-6432110], [24116380], [-4163592], [41824772], [2963232], [-22235496], [-9650462], [-44511590], [10898334], [13737020], [-27886726], [-7364440], [8715076], [-16003182], [38174616], [37697872], [44025142], [-36433166], [-8936794], [-26121180], [-52821650], [-48241336], [45295370], [8000496], [51978714], [21519588], [59382102], [-2628376], [40954974], [39113072], [-25378228], [40786278], [-68126694], [-23449824], [35283110], [-71077346], [-29579284], [-3510902], [3851040], [24189778], [-38019804], [-8343924], [-61455026], [15824172], [33271482], [-318030], [-68732066], [46268434], [-13896760], [17869758], [-27838256], [-32766362], [-53166782], [-4027484], [-14475666], [-50542368], [-51776500], [110151210], [-85708348], [-13443438], [79461222], [-98679944], [-14338508], [-47311880], [-56321302], [-102708352], [46791250], [-46790536], [780030], [-12857674], [-136759640], [-88417470], [12000224], [35283548], [-99106490], [124894986], [-80958256], [-123333016], [-45295980], [-123981874], [-123209220], [-36447120], [-101408918], [-11908444], [35266752], [27469650], [-149550570], [47572580], [-107813352], [-65139108], [-436204], [-59476024], [4425214], [-113481022], [58499016], [-32466608], [-22766388], [65144020], [158282624], [37041118], [138114690], [156161640], [108481578], [9327998], [29214060], [-66029050], [82215220], [41623926], [-118454294], [158934896], [-144244948], [-41895338], [23785678], [66585522], [44274668], [-66134370], [-218574950], [-155972076], [-170596154], [36811646], [-83458052], [-248703862], [-332784], [-86320382], [-55169544], [-9838092], [62957550], [-173574614], [-43936612], [148026960], [57196840], [76903288], [-237189546], [-120786140], [-241788672], [-147431502], [219221814], [13939880], [74934050], [110063134], [-30826052], [270901434], [114979236], [107613422], [301635342], [-22414112], [-242699802], [57830996], [232959148], [75836954], [-53789394], [-104556064], [210508380], [266740292], [-161151838], [-260299612], [201530846], [315769464], [291543462], [32970304], [226121126], [247032994], [-81192628], [-85257144], [-196358166], [-177298084], [72687408], [244093010], [240715666], [204797596], [300862202], [-220516914], [-221932112], [71244138], [-108725422], [-135170660], [92748500], [237031440], [29462086], [102667614], [311342878], [323170336], [-142014804], [93449076], [76362644], [-329258450], [-232547886], [-215662230], [55194400], [341944982], [-430461030], [331116438], [-203999000], [258925790], [142321468], [270114358], [252342916], [446218512], [231501382], [-74830836], [189756362], [-515161410], [-222689152], [-87655706], [-440377782], [-77376064], [-127888956], [-98990766], [-319234216], [189590694], [92881002], [187851584], [536141252], [-170878196], [-245858454], [3675006], [-383740504], [-233455750], [-148572548], [-279508060], [-543082624], [223694694], [380107272], [-447357986], [427996764], [-522411504], [-461983598], [480953522], [-445105558], [68150830], [-317775030], [505971032], [455844898], [130403212], [-415963512], [696529312], [380693830], [-547458924], [-193631664], [-561011710], [-413185194], [-489421964], [-698359236], [-149923080], [-337876102], [-600184928], [22871978], [542068276], [23039430], [-258448274], [240917410], [-282183008], [-536662806], [207767286], [-264871432], [-124869618], [-752005966], [709386204], [-495590676], [564698210], [296911542], [-795416396], [365178460], [-599684566], [681672030], [55752816], [-334894040], [316677474], [195900888], [626571374], [340494198], [-198757152], [334322292], [694299706], [786229694], [-100556516], [-496612102], [797751412], [-743645688]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_560_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_560_6_a_g();". // 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_560_6_a_g(:prec:=1) chi := MakeCharacter_560_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 6)); 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_560_6_a_g();". // 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_560_6_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_560_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign))); Vf := Kernel([<3,R![-12, 1]>],Snew); return Vf; end function;