// Make newform 6825.2.a.l in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6825_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_6825_2_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_6825_2_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_6825_a();" function MakeCharacter_6825_a() N := 6825; order := 1; char_gens := [2276, 3277, 976, 4201]; 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_6825_a_Hecke(Kf) return MakeCharacter_6825_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 := [[2], [1], [0], [-1], [-2], [-1], [4], [3], [9], [-1], [-5], [8], [6], [9], [3], [-3], [0], [10], [2], [12], [-5], [-13], [11], [1], [-1], [-6], [12], [-12], [-2], [15], [4], [4], [0], [-18], [18], [4], [-24], [-8], [-7], [-12], [-5], [14], [-16], [6], [6], [-4], [19], [23], [-4], [26], [-19], [-24], [25], [30], [-14], [27], [-20], [0], [7], [-22], [-24], [33], [7], [-18], [-30], [-20], [-10], [-9], [-24], [-35], [14], [12], [-10], [2], [-18], [4], [-26], [19], [-20], [-11], [-2], [20], [30], [8], [-34], [-23], [16], [-8], [-2], [-34], [18], [39], [18], [-36], [24], [26], [-17], [0], [34], [12], [-25], [12], [0], [-29], [-1], [34], [-13], [17], [-25], [22], [46], [-28], [-10], [36], [-34], [-19], [-8], [-22], [6], [37], [9], [23], [14], [12], [45], [-7], [-38], [6], [-14], [27], [-22], [24], [-25], [-1], [35], [-45], [-10], [5], [38], [-27], [-28], [-34], [0], [8], [2], [0], [45], [-38], [-26], [-40], [40], [10], [20], [48], [-39], [5], [48], [-9], [-44], [-35], [-18], [3], [46], [-30], [-54], [-43], [4], [-52], [18], [-12], [30], [-14], [5], [0], [60], [36], [48], [-30], [14], [6], [16], [-30], [-14], [-51], [-50], [-54], [-7], [-11], [26], [-21], [8], [-34], [-65], [-25], [15], [48], [-2], [-13], [7], [54], [-60], [32], [-49], [-40], [69], [52], [31], [-19], [44], [-2], [13], [-12], [64], [-28], [-11], [71], [13], [-64], [-12], [24], [-19], [16], [28], [38], [18], [55], [12], [-55], [-22], [-46], [-66], [-40], [-16], [-18], [-22], [3], [-70], [32], [-19], [-2], [32], [46], [16], [-66], [-21], [-72], [47], [3], [-28], [-33], [-8], [50], [-42], [10], [-6], [-68], [-73], [4], [-12], [-13], [-3], [-49], [-19], [2], [-18], [-38], [-34], [45], [-14], [38], [26], [24], [26], [53], [0], [8], [-28], [16], [-8], [0], [66], [74], [20], [53], [-44], [-5], [53], [-48], [32], [20], [15], [0], [-66], [-33], [11], [-8], [42], [-16], [20], [1], [22], [32], [55], [61], [-7], [42], [-20], [76], [30], [-39], [28], [-54], [-50], [71], [-32], [56], [-69], [-50], [-55], [33], [-64], [68], [-18], [-12], [0], [36], [56], [-37], [20], [-6], [50], [69], [-38], [-77], [-74], [5], [40], [-86], [14], [62], [-24], [-70], [-34], [-64], [40], [48], [-14], [56], [-85], [-46], [63], [-18], [60], [-50], [-10], [-62], [29], [21], [-66], [14], [-66], [78], [-27], [-72], [38], [17], [-75], [-58], [-57], [60], [-18], [32], [-33], [-60], [-27], [-36], [96], [38], [4], [-6], [-21], [-6], [64], [12], [25], [59], [-60], [96], [4], [-68], [-70], [-65], [-40], [28], [-100], [23], [-40], [78], [37], [72], [83], [33], [-10], [63], [35], [8], [68], [-64], [-33], [9], [40], [-53], [51], [52], [33], [-60], [-4], [-56], [42], [28], [-5], [102], [-68], [-64], [-86], [-67], [-15], [-48], [65], [42]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6825_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6825_2_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_6825_2_a_l(:prec:=1) chi := MakeCharacter_6825_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_6825_2_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_6825_2_a_l( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6825_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<2,R![-2, 1]>,<11,R![2, 1]>,<17,R![-4, 1]>,<19,R![-3, 1]>],Snew); return Vf; end function;