// Make newform 1008.4.a.j 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_j();". // 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_j();". // 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], [-2], [7], [-8], [-42], [2], [124], [76], [-254], [72], [398], [-462], [-212], [-264], [162], [-772], [30], [764], [-236], [418], [-552], [1036], [-30], [-1190], [-1370], [-464], [-2136], [-1226], [-338], [-2088], [-292], [-818], [2156], [2850], [-1672], [446], [-2708], [896], [-4034], [-3480], [-2898], [2652], [146], [2546], [2536], [1300], [-2576], [-1836], [-1874], [-3730], [2004], [-646], [1260], [-5910], [2988], [1318], [5640], [6446], [-4930], [6260], [2310], [-196], [-6736], [394], [6714], [-692], [-1566], [-5328], [11326], [2130], [3044], [-12416], [-7442], [-100], [8080], [5482], [10446], [11334], [8594], [10500], [-12066], [4332], [-1918], [7992], [3184], [-11426], [-16934], [17038], [13592], [8612], [7432], [6616], [17040], [2948], [17304], [-4650], [-16854], [124], [5382], [-17460], [9514], [3988], [-11346], [8436], [2098], [9436], [1314], [-8940], [16058], [-3936], [174], [-16018], [3068], [-24656], [-7594], [3724], [3792], [-24702], [-20144], [-2522], [-10414], [22230], [18192], [-8108], [5794], [-1954], [-32016], [23072], [31782], [24396], [-32604], [7680], [366], [-29374], [-38990], [20470], [-29916], [-4914], [-34250], [41804], [-30862], [-10576], [-10680], [-1178], [-5600], [-826], [-45918], [-42380], [-26524], [20614], [23730], [9028], [37200], [-23988], [15276], [10760], [52890], [-6118], [32230], [-18544], [-25930], [-8192], [-54444], [25446], [33192], [-11024], [-40714], [28226], [-48330], [44456], [61054], [-3900], [-32166], [-27984], [11242], [-7548], [22730], [-16688], [-24946], [2136], [39676], [-62690], [17386], [-62140], [27078], [2022], [36052], [32346], [65280], [-13718], [-70824], [28484], [5150], [-14736], [27386], [16610], [24646], [77658], [33840], [58122], [-17472], [-36706], [76410], [-73164], [-35922], [-9872], [-52704], [-25482], [-30140], [-75118], [18046], [-8472], [-23980], [-27792], [-74086], [-29056], [-24526], [-62084], [42530], [-7778], [35840], [42414], [95264], [68796], [-56866], [49250], [106196], [-54336], [20208], [-11754], [-25828], [14096], [36870], [-52556], [27312], [79018], [-56622], [38880], [46760], [48712], [75388], [2872], [39990], [28490], [-92160], [24360], [-12140], [-1532], [-9084], [-125018], [44658], [-3564], [-120638], [56702], [19984], [3758], [-9420], [-14378], [-94582], [22184], [94376], [68878], [34846], [107298], [-109132], [-27414], [42490], [-75196], [-44022], [43710], [-69404], [51378], [-120224], [-47894], [18912], [-69560], [-45690], [90250], [-139452], [-22752], [83616], [-40824], [-69034], [-9732], [95500], [94674], [-17950], [-26864], [93450], [-13110], [-47028], [56190], [101948], [-21418], [-131346], [114784], [-161154], [-13956], [67620], [32650], [55266], [41336], [51648], [-55628], [73586], [6656], [-137242], [-28652], [-98050], [113808], [-125894], [78310], [73396], [-110164], [-11230], [20412], [-30556], [21770], [-60090], [-107924], [-149318], [-8610], [-118288], [-14762], [118442], [121276], [-135004], [92476], [41690], [-56554], [45074], [31416], [201892], [-52828], [-44884], [-17610], [81226], [-36526], [488], [141518], [-79746], [-176274], [-152952], [-80814], [63408], [29678], [-71924], [10080], [-96714], [61276], [23498], [117738], [93312], [-70610], [136010], [-5920], [-172524], [-71602], [173812], [-51266], [83522], [-2836], [51400], [38180], [-133750], [60438], [-145872], [-122246], [11416], [-62764], [208284], [-65774], [118736], [40102], [-114900], [40084], [-243854], [96290], [249074], [116958], [202774], [-17320], [-126930], [-158284], [27032], [-243536], [104494], [178340], [-66584], [-178078], [-64962], [80296], [86140], [92420], [47802], [-149056], [-129766], [53940], [-121078], [143054], [-41922], [-67864], [-11678], [71934], [5752], [157366], [208542], [-39212], [201028], [-48830], [252218], [120304], [251364], [77786], [56726], [119860], [-22048], [-271950], [-293536], [-167950], [150278], [-183524], [-178332], [-203350], [-156098], [-233752], [-190290], [315660], [191056]]; 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_j();". // 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_j(: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_j();". // 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_j( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1008_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<5,R![2, 1]>,<11,R![8, 1]>],Snew); return Vf; end function;