// Make newform 1008.4.a.i in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1008_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_1008_4_a_i();". // 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_1008_4_a_i();". // 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_1008_a();" function MakeCharacter_1008_a() N := 1008; order := 1; char_gens := [127, 757, 785, 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_1008_a_Hecke(Kf) return MakeCharacter_1008_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], [0], [-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_1008_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1008_4_a_i();". // 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_1008_4_a_i(:prec:=1) chi := MakeCharacter_1008_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_1008_4_a_i();". // 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_1008_4_a_i( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1008_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;