// Make newform 1344.4.a.s in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1344_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_1344_4_a_s();". // 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_1344_4_a_s();". // 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_1344_a();" function MakeCharacter_1344_a() N := 1344; order := 1; char_gens := [127, 1093, 449, 577]; 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_1344_a_Hecke(Kf) return MakeCharacter_1344_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], [-4], [-7], [26], [-2], [-36], [76], [-114], [-6], [-256], [86], [160], [220], [308], [-258], [-264], [-606], [520], [-286], [-530], [-44], [-1012], [768], [222], [320], [-592], [-1782], [-230], [-1718], [-2444], [-1996], [-1746], [-1804], [2814], [-792], [2778], [-2880], [-1060], [1440], [-1210], [-1618], [4018], [3382], [4302], [-2640], [-3396], [3480], [1504], [-5122], [1630], [-2522], [3022], [-3480], [3496], [-5058], [-3596], [-2424], [-3634], [-2970], [7028], [528], [-4300], [-4580], [2266], [7926], [4132], [-3622], [10254], [8178], [9932], [6546], [-12176], [8210], [-6908], [-10248], [274], [10010], [1998], [-12842], [10400], [-15586], [8066], [-5222], [-10920], [-1650], [11858], [-17894], [2088], [13532], [-6344], [9948], [20144], [-10530], [5548], [-2152], [19780], [23052], [13364], [9766], [-3768], [-1618], [5768], [-8114], [-7816], [-18278], [24960], [10892], [-20022], [10370], [-8992], [-12318], [-26026], [18332], [10572], [24822], [6620], [15956], [30802], [12714], [-5822], [28510], [21864], [17862], [6068], [-7734], [-19902], [12160], [3088], [-10178], [5840], [-14414], [-12608], [-22606], [5952], [14138], [-12616], [12292], [20784], [-13574], [-32308], [2390], [28020], [-34006], [6310], [-16524], [-36662], [-42000], [-28388], [23918], [2346], [51876], [11372], [38764], [11212], [1458], [-38608], [7376], [-24878], [26148], [-18350], [-57510], [-33364], [-26892], [-31970], [-19728], [-21400], [4754], [-52882], [27312], [-3630], [50906], [-486], [-28618], [-24536], [30564], [43904], [-29534], [30504], [2506], [-50468], [-27324], [478], [-8628], [-18722], [-41736], [47730], [-46372], [73030], [-6440], [56558], [20746], [-10348], [35016], [-61302], [6276], [-13762], [-46362], [71104], [-12732], [-16154], [-25768], [39654], [46638], [69548], [-9776], [26040], [17042], [11850], [-32772], [40034], [-62220], [-30832], [-2604], [-60036], [47714], [26664], [-33996], [-34194], [-82262], [57490], [-13328], [28106], [20400], [20500], [-86202], [-26784], [-83066], [13456], [96454], [-21254], [-80220], [-9112], [16490], [-17140], [80896], [21950], [90806], [-30666], [-112040], [-14962], [-2660], [88784], [21958], [-40128], [-7620], [48272], [40000], [83288], [81198], [-3214], [-19428], [68954], [46526], [-55392], [-67014], [66110], [-70964], [56456], [13506], [25588], [-92754], [-92578], [-878], [21476], [78980], [-138946], [60412], [-31536], [-75694], [90578], [-66012], [128126], [3832], [-110586], [37424], [-78334], [-18490], [-74978], [67632], [132648], [-51428], [72752], [122050], [22260], [-4574], [-64270], [-121418], [-58960], [-97960], [-86346], [84804], [-79162], [-16896], [-7654], [-121424], [-104360], [-91164], [138968], [113836], [78714], [-23894], [-83572], [-70806], [55384], [-57734], [-80042], [74726], [95634], [51118], [-120216], [119482], [26158], [117596], [-67698], [55822], [-162092], [-33450], [167158], [-61034], [157420], [82342], [99160], [29848], [1394], [-92954], [-83576], [10612], [175482], [73778], [53554], [-142142], [73336], [69632], [34544], [-110600], [-79014], [177192], [199942], [-41416], [6898], [-120762], [59508], [56848], [-42326], [-145622], [-16618], [-86484], [152824], [-44672], [-217980], [188310], [14902], [-13848], [149554], [-145276], [138416], [41220], [-83526], [-139702], [152562], [17752], [-93454], [185810], [8324], [20454], [7980], [145948], [-206034], [-232290], [-16196], [64154], [133814], [-194608], [-91802], [18804], [-179562], [-132198], [-197364], [-67394], [-182952], [86730], [-115872], [57210], [-63636], [137664], [-134192], [168342], [944], [-79780], [96062], [20636], [77370], [-73772], [-251758], [112050], [-157624], [162644], [-228928], [9770], [150050], [219738], [-246012], [57584], [45068], [-122600], [-173130], [93138], [8924], [-49200], [-185758], [-5118], [33270], [40308], [-50890], [-256660], [-24178], [-219032], [-271836], [-213252], [-870], [-15166], [-229182], [-292004], [54282], [99584], [178094], [319782], [-145788], [316524]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1344_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1344_4_a_s();". // 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_1344_4_a_s(:prec:=1) chi := MakeCharacter_1344_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_1344_4_a_s();". // 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_1344_4_a_s( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1344_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![4, 1]>,<11,R![-26, 1]>],Snew); return Vf; end function;