// Make newform 4200.2.a.w in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4200_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_4200_2_a_w();". // 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_4200_2_a_w();". // 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_4200_a();" function MakeCharacter_4200_a() N := 4200; order := 1; char_gens := [3151, 2101, 2801, 1177, 3601]; v := [1, 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_4200_a_Hecke(Kf) return MakeCharacter_4200_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 := 2; raw_aps := [[0], [1], [0], [-1], [2], [2], [0], [-6], [0], [-6], [10], [0], [6], [8], [12], [6], [0], [-6], [-4], [6], [14], [4], [0], [-6], [2], [-10], [-16], [-4], [10], [-10], [4], [12], [2], [-6], [14], [0], [22], [16], [24], [0], [18], [-26], [10], [16], [18], [-22], [-8], [16], [-8], [-6], [6], [30], [-26], [-8], [20], [8], [14], [30], [0], [-26], [12], [16], [4], [-8], [-14], [-26], [0], [16], [-4], [-14], [-24], [-22], [-16], [-4], [-20], [16], [-10], [2], [2], [-6], [-4], [10], [14], [-30], [10], [28], [18], [-12], [18], [-36], [-32], [24], [12], [-22], [-20], [24], [-30], [-38], [-20], [-34], [-36], [-2], [20], [34], [28], [-22], [28], [-40], [42], [-10], [32], [24], [-18], [-22], [-4], [46], [-44], [0], [-38], [22], [-38], [12], [-48], [-12], [-10], [-46], [26], [20], [16], [-14], [-4], [-16], [4], [-12], [-10], [-42], [-32], [12], [-48], [54], [-38], [26], [-44], [36], [30], [-12], [10], [28], [-2], [-32], [56], [-54], [-36], [12], [28], [-30], [44], [30], [-18], [-6], [4], [-18], [-40], [32], [30], [-28], [52], [-22], [-14], [40], [-6], [-18], [22], [32], [-26], [-26], [-32], [-10], [8], [6], [-32], [56], [-4], [-24], [-16], [26], [0], [-28], [-50], [-60], [2], [-12], [-52], [62], [12], [-16], [66], [-4], [12], [64], [-6], [-62], [-50], [-42], [-8], [56], [22], [-36], [30], [-10], [60], [10], [0], [8], [12], [-50], [4], [70], [32], [2], [58], [50], [-50], [56], [12], [26], [12], [-46], [-40], [-30], [-52], [-38], [24], [62], [36], [-20], [42], [-30], [-6], [-28], [12], [2], [-64], [38], [-60], [36], [-56], [-60], [-16], [40], [40], [-42], [-40], [74], [-8], [-14], [2], [28], [-32], [-22], [36], [-12], [10], [-42], [8], [14], [-14], [-70], [68], [-54], [62], [-32], [34], [-4], [-14], [0], [52], [34], [6], [-56], [44], [-52], [36], [14], [36], [10], [76], [-26], [58], [14], [18], [-56], [46], [-68], [28], [-14], [62], [52], [-72], [20], [-26], [-62], [-8], [-44], [68], [80], [12], [10], [62], [-60], [12], [74], [66], [68], [-48], [-66], [-60], [-6], [-54], [-38], [-26], [-36], [-50], [-48], [2], [30], [68], [-76], [64], [30], [-2], [-54], [-58], [-28], [88], [-28], [30], [76], [-18], [-16], [76], [54], [38], [40], [90], [-90], [50], [4], [84], [-48], [2], [4], [-78], [8], [-2], [48], [20], [-84], [-50], [56], [92], [10], [-48], [6], [-52], [68], [-12], [16], [-74], [66], [-90], [-88], [-50], [14], [8], [-4], [50], [10], [66], [-58], [30], [22], [68], [18], [-70], [48], [20], [34], [20], [12], [54], [-92], [30], [100], [-42], [96], [-10], [2], [-20], [-70], [-50], [6], [-56], [36], [-30], [38], [8], [-6], [28], [8], [-74], [-18], [12], [-72], [24], [-18], [-18], [96], [-8], [-20], [-10], [-54], [-40], [88], [-70], [32], [36], [-46], [-18], [-60]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4200_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4200_2_a_w();". // 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_4200_2_a_w(:prec:=1) chi := MakeCharacter_4200_a(); f_vec := qexpCoeffs(); Kf := Universe(f_vec); // SetVerbose("ModularForms", true); // SetVerbose("ModularSymbols", true); S := CuspidalSubspace(ModularForms(chi, 2)); 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_4200_2_a_w();". // 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_4200_2_a_w( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4200_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<11,R![-2, 1]>,<13,R![-2, 1]>,<17,R![0, 1]>,<19,R![6, 1]>,<23,R![0, 1]>],Snew); return Vf; end function;