// Make newform 448.4.a.e in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_448_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_448_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_448_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_448_a();" function MakeCharacter_448_a() N := 448; order := 1; char_gens := [127, 197, 129]; 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_448_a_Hecke(Kf) return MakeCharacter_448_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], [-2], [-16], [7], [-8], [-28], [54], [-110], [-48], [110], [-12], [246], [182], [128], [-324], [162], [810], [488], [244], [768], [-702], [-440], [-1302], [730], [294], [688], [-1388], [244], [-90], [1318], [1776], [-1118], [2274], [-210], [2010], [-1112], [-124], [2008], [-2884], [-2228], [-820], [-3892], [5048], [-2962], [-3334], [-1860], [-4268], [5432], [-2046], [2980], [4458], [-4440], [3302], [1582], [2354], [3872], [-180], [-2032], [5426], [842], [-3782], [4312], [2674], [3768], [2438], [3186], [8672], [814], [9344], [5180], [12178], [-440], [9816], [442], [-3960], [-6708], [13350], [1356], [6222], [5150], [2310], [-1262], [4488], [17038], [-16200], [-8772], [2130], [10534], [9268], [9392], [-10806], [-4940], [5216], [4412], [19060], [-12768], [5500], [-7338], [-17582], [1618], [16144], [-4654], [10078], [-5930], [-19048], [-14366], [-3626], [-1062], [10200], [-25158], [-25664], [-19018], [17334], [18730], [6928], [16302], [4718], [21436], [-4458], [-26640], [-7432], [58], [21516], [18108], [-10078], [-18762], [-6810], [-4860], [13636], [-2088], [-5160], [28152], [16808], [-21674], [7422], [13790], [6232], [-1766], [-1204], [-7050], [23282], [-10142], [9192], [-46716], [-11240], [-700], [37492], [28894], [-2770], [-17688], [33566], [-16758], [11468], [50356], [-8716], [-7632], [23080], [45110], [16674], [-43832], [-736], [38138], [-26224], [18762], [38394], [-5388], [-25472], [17096], [-8930], [-53468], [-33300], [-50652], [7008], [-63862], [8500], [-39550], [15892], [-30442], [29232], [44840], [6616], [65982], [-338], [-40486], [-53808], [16060], [39866], [-6302], [55810], [13148], [7618], [73068], [-47188], [-2172], [22924], [-74622], [5102], [48462], [19194], [-2088], [-230], [-5012], [5976], [74170], [57890], [-19304], [64860], [-41112], [48750], [29062], [8574], [49728], [42152], [-28166], [12280], [-60338], [-21104], [-72578], [-56104], [-32538], [-66002], [73640], [-65710], [-56888], [20594], [-6050], [52138], [46600], [-12224], [-12888], [21302], [-45530], [-13232], [-107318], [58338], [-86784], [-26570], [12202], [55260], [7168], [21888], [90162], [-94848], [17690], [92778], [-40420], [69776], [68402], [-31760], [84432], [78706], [-55998], [-23344], [92890], [34232], [-49540], [-52562], [67254], [84336], [92654], [-72008], [-25576], [85360], [-120008], [34214], [24950], [-47070], [-77798], [-87932], [99422], [-22612], [-128416], [-21422], [5440], [73234], [122512], [-8576], [-128190], [108142], [56282], [-74408], [48168], [74116], [89768], [51074], [-75312], [82018], [34746], [-53660], [17010], [-122422], [-75686], [-63162], [-5838], [143162], [79340], [-33952], [-49028], [22430], [63854], [130398], [-46794], [40160], [92248], [-42288], [24474], [-4656], [42000], [119120], [2402], [-23928], [113590], [-84918], [12648], [115456], [-173350], [-11970], [-167712], [82138], [-86910], [122582], [154594], [-145992], [-91688], [-59802], [-120818], [-142640], [-189502], [83056], [-176178], [-72862], [-177834], [-162260], [-79242], [-22268], [14434], [85510], [72018], [-41398], [-82084], [82202], [-150106], [-116340], [-13792], [109222], [197920], [-75892], [-202616], [-172872], [129736], [40462], [53514], [188858], [198732], [41210], [-119182], [-154380], [-184998], [-13966], [123072], [33066], [-155678], [95376], [85700], [42194], [132838], [11116], [-147328], [53622], [126972], [-174690], [-164688], [115050], [241768], [-4654], [118690], [55928], [-63142], [-103290], [22974], [-41732], [159798], [-60664], [49874], [-101010], [-18968], [190408], [-19024], [6908], [129276], [148990], [-196128], [-155100], [164054], [114808], [-134702], [209540], [-54810], [-245628], [-12522], [146200], [155378], [67376], [8094], [230820], [-199852], [57786], [-94498], [275798], [-53730], [17978], [-33074], [-209032], [-119708], [-2406], [295188], [-125880], [-1244], [225414], [89472], [-159690], [189656], [-128144], [269990], [129878], [-145584], [254848], [197850], [122022], [-205140]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_448_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_448_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_448_4_a_e(:prec:=1) chi := MakeCharacter_448_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_448_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_448_4_a_e( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_448_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<3,R![2, 1]>,<5,R![16, 1]>],Snew); return Vf; end function;