// Make newform 2352.4.a.x in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_2352_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_2352_4_a_x();". // 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_2352_4_a_x();". // 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_2352_a();" function MakeCharacter_2352_a() N := 2352; order := 1; char_gens := [1471, 1765, 785, 2257]; 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_2352_a_Hecke(Kf) return MakeCharacter_2352_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], [3], [-12], [0], [60], [-44], [128], [52], [160], [-230], [136], [-318], [192], [-220], [184], [-498], [-492], [-20], [-380], [264], [560], [-104], [1508], [-1144], [904], [500], [248], [1028], [1066], [466], [-1616], [1700], [1958], [-1284], [-306], [2264], [2068], [-1908], [-3128], [-532], [1068], [4164], [288], [1282], [2622], [776], [5476], [5680], [-5180], [2996], [-1818], [-2264], [2952], [2460], [2952], [5448], [-8052], [3872], [6062], [-7018], [-3596], [8100], [-7164], [-544], [-2792], [-3642], [-2380], [-12306], [-8244], [-9276], [-10648], [-2000], [-848], [-1810], [1668], [11736], [-10046], [7636], [-1842], [-664], [6572], [-2370], [-9048], [-3208], [8832], [16756], [16994], [11194], [-4292], [4328], [3844], [12712], [13912], [-11340], [3444], [7856], [12460], [3552], [-14380], [14118], [3052], [-17162], [15948], [15958], [4548], [-12304], [28236], [-4888], [5208], [27040], [-4032], [21602], [-2554], [12580], [-14784], [1278], [24428], [424], [15754], [5860], [-10580], [-8290], [-14500], [7572], [-17020], [-12422], [22370], [-14936], [-24456], [34796], [-14228], [-14368], [7360], [-4558], [13008], [-14168], [-14700], [8068], [-4228], [-36934], [-16540], [40718], [34832], [-38300], [20476], [-42024], [-412], [42816], [-43572], [16208], [34378], [7416], [16772], [-23336], [22732], [-16584], [41616], [-41680], [-13248], [-6204], [44996], [-24918], [8984], [-15020], [-53138], [-2120], [-18520], [-46996], [44882], [8092], [-540], [-41020], [-42664], [-29082], [-66680], [47112], [-21748], [-16254], [784], [-8420], [-38784], [-40932], [-26238], [-16920], [26456], [25052], [12166], [12948], [15494], [-60776], [7520], [-5164], [61076], [-6388], [-58916], [75432], [20302], [-73286], [8232], [65088], [33750], [54776], [75692], [75920], [-60684], [-30588], [12376], [-37692], [-43686], [-6676], [-79794], [-68892], [-52648], [22708], [-31104], [-16600], [19528], [60416], [58152], [64538], [-13694], [7680], [-49182], [35296], [101844], [15058], [39368], [-69640], [7216], [-66340], [-48566], [4796], [29416], [69306], [-13692], [40032], [17464], [-55698], [-86652], [21840], [-36748], [-47500], [43408], [-38762], [48456], [16936], [-26656], [-2124], [-1180], [-89280], [96006], [34984], [-16184], [-90320], [-19788], [-17396], [-93742], [-28820], [-100428], [-45064], [-49360], [-26188], [-117380], [-26948], [26416], [82988], [-14986], [37816], [-90404], [-2526], [12420], [32236], [-39424], [33952], [53776], [-73712], [6860], [-56090], [14426], [146260], [4080], [-24592], [-154728], [22012], [81572], [102632], [-91022], [-86514], [-62920], [35000], [-126250], [51380], [-23338], [-46508], [70518], [71196], [-130592], [-96916], [-106164], [119068], [1496], [-35126], [-23896], [32564], [110868], [140062], [83404], [-27668], [-33968], [-33602], [-65712], [165234], [-41090], [165772], [-92808], [-39824], [130412], [23584], [65872], [71054], [-63236], [-47690], [29188], [-78520], [-103686], [-88632], [-9380], [-116044], [-28568], [-31454], [-113290], [87450], [-18664], [-56180], [-32916], [-205996], [-63494], [116056], [31232], [90808], [140590], [-68394], [171148], [8832], [-111078], [153364], [207348], [-119852], [-39888], [-98476], [-60084], [-56422], [165462], [-29064], [23182], [42696], [76040], [-206652], [-223182], [206752], [-136034], [-88344], [93816], [-78828], [113492], [68166], [170460], [-202408], [76714], [-40724], [-111780], [28224], [120558], [193568], [142694], [-16460], [-48664], [133472], [126128], [-45176], [107148], [-182630], [-142112], [-127454], [-79076], [-159552], [-167160], [218316], [211804], [39672], [-171170], [-145212], [150596], [47292], [180256], [54518], [-209592], [-27456], [221852], [-177742], [83324], [-33374], [-183416], [-220416], [-175108], [4216], [97814], [11154], [130700], [-106164], [51008], [212050], [-147732], [132884], [171462], [-29764], [192096], [270136], [-128144], [-248848], [-258534], [-147284], [59536], [21436], [-58184], [3300], [-239804], [110742], [-171780], [-122136]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_2352_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_2352_4_a_x();". // 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_2352_4_a_x(:prec:=1) chi := MakeCharacter_2352_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_2352_4_a_x();". // 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_2352_4_a_x( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_2352_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![12, 1]>,<11,R![-60, 1]>],Snew); return Vf; end function;