// Make newform 1225.4.a.l in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_1225_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_1225_4_a_l();". // 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_1225_4_a_l();". // 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_1225_a();" function MakeCharacter_1225_a() N := 1225; order := 1; char_gens := [1177, 101]; v := [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_1225_a_Hecke(Kf) return MakeCharacter_1225_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 := [[5], [0], [0], [0], [-68], [0], [0], [0], [40], [-166], [0], [-450], [0], [180], [0], [-590], [0], [0], [740], [688], [0], [-1384], [0], [0], [0], [0], [0], [1580], [-54], [670], [2000], [0], [-3110], [0], [814], [-2952], [0], [-1780], [0], [0], [-2084], [0], [-4072], [4590], [2210], [0], [5868], [0], [0], [0], [4730], [-7376], [0], [0], [0], [-7520], [0], [0], [-7310], [4342], [0], [0], [0], [0], [0], [6970], [10908], [3330], [4100], [0], [0], [-8104], [0], [13970], [11916], [0], [-10526], [0], [1598], [0], [0], [15262], [-8608], [0], [0], [-18580], [-2686], [-8010], [0], [8440], [0], [0], [-21240], [20372], [-7236], [0], [0], [0], [0], [15878], [-12980], [-20470], [0], [-26906], [-6788], [0], [0], [0], [-24736], [0], [0], [-15010], [-30550], [0], [-26192], [8878], [0], [0], [-27050], [-1804], [0], [33570], [0], [34060], [0], [-4198], [12546], [0], [0], [0], [-25324], [-25160], [-2448], [34830], [0], [0], [0], [0], [0], [37354], [0], [-43538], [46240], [23980], [0], [0], [0], [0], [0], [20200], [6550], [0], [-30060], [0], [-52740], [-39632], [21744], [0], [0], [0], [-48820], [-29290], [-52040], [0], [37490], [0], [57528], [0], [-6046], [0], [18356], [0], [64192], [-29990], [0], [0], [-53548], [66082], [0], [0], [-53440], [0], [-4930], [0], [73120], [0], [66330], [0], [-63866], [0], [0], [77180], [-79812], [0], [14180], [0], [-49698], [12710], [0], [0], [-72266], [0], [0], [0], [0], [0], [0], [-68980], [-29174], [0], [-92990], [0], [-91800], [0], [0], [0], [-81720], [0], [320], [8230], [24642], [0], [23746], [-94320], [0], [74034], [0], [-105376], [0], [-108148], [72630], [0], [55512], [76442], [0], [0], [0], [87890], [-8636], [0], [-96980], [0], [0], [112086], [0], [0], [0], [0], [-18884], [-125960], [116730], [0], [126320], [0], [0], [-69844], [45362], [0], [0], [0], [112880], [134860], [0], [0], [0], [0], [-130954], [0], [-37060], [-142690], [0], [98460], [0], [142704], [0], [0], [31540], [66726], [32202], [0], [0], [-143008], [0], [0], [0], [-97088], [95310], [-151790], [0], [0], [-165002], [0], [115610], [0], [75850], [0], [0], [0], [0], [0], [0], [174550], [153736], [-94180], [19532], [154530], [24260], [0], [-48424], [174690], [0], [-54094], [172958], [158940], [96080], [0], [0], [124168], [0], [-27634], [0], [152010], [0], [89640], [-190570], [0], [177284], [0], [161600], [189950], [-45162], [199750], [0], [0], [210548], [0], [-13606], [0], [0], [0], [-67790], [-43430], [0], [-219168], [-195610], [162724], [0], [206820], [0], [0], [0], [122470], [34358], [0], [-147026], [0], [0], [0], [237550], [-226760], [-172510], [0], [-75280], [100004], [0], [61290], [0], [-20680], [250938], [-30308], [0], [60520], [-21586], [0], [-221510], [0], [-120272], [0], [0], [0], [0], [263030], [122400], [81310], [0], [0], [178488], [0], [-38180], [0], [110126], [0], [280436], [0], [-279032], [-231030], [0], [0], [242692], [-203278], [0], [-273730], [261560], [0], [0], [0], [288650], [-16798], [0], [0], [0], [-288850], [-147100], [-261252], [101050], [0], [-226616], [0], [0], [0], [-10246], [0], [-287240], [0], [0], [0], [39100], [214006], [0], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_1225_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_1225_4_a_l();". // 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_1225_4_a_l(:prec:=1) chi := MakeCharacter_1225_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_1225_4_a_l();". // 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_1225_4_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_1225_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign))); Vf := Kernel([<2,R![-5, 1]>,<3,R![0, 1]>,<19,R![0, 1]>],Snew); return Vf; end function;