// Make newform 1050.4.a.g in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1050_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_1050_4_a_g();". // 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_1050_4_a_g();". // 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_1050_a();" function MakeCharacter_1050_a() N := 1050; order := 1; char_gens := [701, 127, 451]; 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_1050_a_Hecke(Kf) return MakeCharacter_1050_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 := [[-2], [3], [0], [-7], [-72], [34], [-6], [92], [180], [-114], [56], [34], [6], [-164], [-168], [-654], [-492], [-250], [124], [36], [-1010], [56], [-228], [390], [70], [-1350], [-2000], [-696], [-1114], [462], [-1064], [180], [2718], [-1348], [558], [1928], [2410], [-740], [-3984], [1038], [-2568], [-2698], [-4116], [3310], [-1278], [2936], [-3508], [1888], [-3564], [1334], [-2658], [-588], [5690], [180], [-5310], [-828], [-4134], [-2968], [4786], [-4398], [-4772], [-6522], [6244], [-528], [5830], [-5046], [-5020], [7486], [10032], [5942], [90], [10596], [-4016], [-3278], [4628], [2880], [7974], [12346], [9738], [-430], [-1812], [-10690], [-4116], [-9938], [1784], [11712], [7650], [-3674], [-3102], [-8984], [-3612], [-9288], [5848], [-5952], [10748], [16488], [14058], [-14466], [-18524], [4358], [2140], [-2022], [-7356], [11202], [-10564], [18574], [-13188], [22506], [10596], [14618], [-5168], [-5726], [7806], [-18052], [-6208], [-21510], [11140], [-9312], [-4878], [-9744], [2990], [-33266], [-5370], [-384], [-14524], [24750], [-1042], [-36960], [16288], [7810], [-36700], [-29508], [-15136], [-3422], [31446], [-18718], [1686], [-5492], [17310], [35754], [33644], [28734], [28672], [15912], [17534], [40656], [-23870], [29610], [-45484], [-46164], [2986], [6534], [-29756], [-29952], [36268], [-23604], [34184], [-53922], [-40538], [-3606], [-14064], [-33066], [26368], [55884], [51126], [14184], [51680], [-52094], [-14398], [18918], [28728], [56342], [-17100], [-38522], [7472], [49038], [-38788], [17862], [3040], [54326], [-21224], [-60252], [-17102], [-8814], [50172], [53514], [-35750], [-45380], [-934], [44880], [57958], [7800], [20828], [36114], [-26304], [-22782], [-25630], [12490], [37266], [-30288], [25734], [5888], [81034], [-16918], [34956], [50862], [-76336], [61344], [-28950], [-34972], [-5042], [17970], [-27128], [44724], [-22416], [-5974], [-28208], [-54090], [-16716], [-70062], [-51154], [77744], [60210], [79936], [-36564], [104366], [-18438], [-52716], [-40016], [10032], [-19622], [48044], [-11968], [-50406], [49372], [-52272], [48746], [83778], [110256], [20232], [-33480], [56252], [28168], [11990], [-69054], [19536], [-41096], [-50388], [-60244], [78156], [51802], [-20922], [-1668], [7634], [-4482], [-59712], [25598], [38284], [2070], [-13754], [53224], [-74856], [51014], [9466], [44250], [12404], [82854], [60798], [-30044], [-96582], [107606], [-4844], [104542], [5024], [-10010], [-272], [48600], [-141370], [-116278], [11580], [-65856], [-149200], [-25464], [-127906], [86020], [-118932], [-103826], [41682], [129584], [97470], [110790], [90876], [-78066], [26196], [-139286], [-122622], [-124144], [-8706], [12036], [105244], [36646], [113874], [-45160], [-87648], [-68452], [-85682], [-2016], [-94210], [23508], [-167870], [63408], [-70122], [31722], [-160220], [135396], [-147598], [-87036], [13332], [67174], [43530], [-34756], [37222], [-31806], [-4160], [9270], [-119158], [-181708], [-78644], [9444], [36234], [121142], [102210], [166376], [76788], [28892], [-114948], [-101338], [-93438], [60770], [-25112], [41074], [-151698], [-61806], [85160], [72642], [-4464], [-188458], [119932], [-174336], [-75594], [171596], [8566], [-122106], [172672], [162542], [167634], [-102960], [-26580], [-135666], [140748], [-115166], [192390], [32916], [-142248], [-164468], [179062], [115638], [-126128], [-31174], [-50904], [181460], [-97452], [-173922], [92576], [84634], [-72060], [250164], [40750], [-165498], [-200882], [201330], [139638], [32776], [-151794], [-245644], [-21336], [165440], [-132806], [138148], [-40056], [96770], [-196818], [13560], [-181964], [33540], [55366], [62624], [273198], [-56212], [122502], [140438], [-8130], [-178184], [32490], [124098], [-182104], [103786], [254370], [173356], [62700], [142558], [109290], [-142704], [-15748], [195910], [21738], [96132], [209824], [8874], [-161856], [-738], [59794], [42324], [74508], [-113402], [-116130], [-210984], [-110142], [-26692], [222336]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1050_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1050_4_a_g();". // 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_1050_4_a_g(:prec:=1) chi := MakeCharacter_1050_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_1050_4_a_g();". // 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_1050_4_a_g( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1050_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<11,R![72, 1]>,<13,R![-34, 1]>],Snew); return Vf; end function;