// Make newform 6400.2.a.u in Magma, downloaded from the LMFDB on 29 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_6400_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_6400_2_a_u();". // 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_6400_2_a_u();". // 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_6400_a();" function MakeCharacter_6400_a() N := 6400; order := 1; char_gens := [4351, 4101, 5377]; 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_6400_a_Hecke(Kf) return MakeCharacter_6400_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], [2], [5], [-6], [-3], [1], [-4], [-6], [8], [-2], [-7], [4], [-2], [-4], [4], [-10], [3], [2], [-1], [-10], [-9], [-5], [2], [-10], [-14], [17], [14], [-9], [-12], [0], [-7], [-11], [6], [22], [-8], [11], [-18], [4], [21], [-20], [8], [1], [8], [0], [15], [-16], [-12], [-4], [-6], [-10], [-3], [15], [22], [-24], [-16], [8], [-22], [-2], [19], [-14], [13], [2], [-6], [12], [5], [-33], [17], [34], [6], [-30], [-12], [-4], [-21], [-6], [16], [-8], [-3], [-15], [-19], [20], [18], [41], [-10], [-1], [-15], [-27], [-20], [4], [28], [20], [12], [-20], [16], [-24], [24], [-27], [-31], [20], [-7], [-18], [36], [-35], [20], [7], [-33], [-29], [30], [23], [28], [16], [-42], [4], [12], [2], [36], [-48], [-6], [-9], [-10], [-14], [28], [9], [45], [10], [-4], [30], [22], [-6], [36], [36], [-2], [28], [53], [-15], [-54], [-32], [-48], [50], [-20], [20], [36], [-53], [-46], [40], [-44], [33], [29], [-6], [52], [-18], [-19], [-8], [52], [-12], [-20], [10], [-37], [-10], [-12], [-31], [52], [-15], [-43], [-14], [-2], [8], [25], [6], [-1], [-10], [-18], [-41], [50], [-5], [-25], [-30], [36], [-16], [28], [-35], [26], [-37], [34], [56], [42], [21], [5], [-2], [11], [39], [-65], [-60], [33], [54], [-13], [44], [-33], [26], [24], [28], [68], [35], [-56], [12], [40], [-24], [30], [60], [-18], [-30], [46], [-23], [0], [-47], [8], [57], [-18], [-6], [20], [10], [-15], [-6], [-27], [-54], [39], [30], [-18], [15], [14], [-69], [-22], [43], [44], [-42], [5], [-54], [-21], [-68], [1], [15], [76], [-36], [-69], [60], [38], [60], [19], [34], [-48], [-13], [-58], [15], [54], [11], [-10], [27], [-62], [-7], [14], [23], [26], [-26], [77], [1], [-56], [78], [9], [-34], [50], [23], [39], [30], [22], [56], [77], [44], [-22], [-75], [-36], [72], [12], [0], [-33], [-72], [-54], [-2], [-40], [30], [-10], [-52], [-21], [-65], [44], [-36], [58], [66], [44], [23], [-31], [-18], [-20], [41], [55], [-23], [12], [34], [60], [26], [-6], [-24], [57], [16], [12], [-50], [21], [-12], [-49], [-45], [-20], [13], [-10], [64], [-6], [-38], [-49], [44], [-82], [66], [-90], [22], [10], [-79], [-55], [12], [34], [46], [-37], [18], [16], [13], [36], [22], [84], [36], [-10], [-73], [78], [-22], [35], [-22], [30], [-46], [36], [19], [10], [65], [-18], [-84], [-62], [-77], [-42], [24], [-37], [19], [62], [-44], [-27], [20], [-11], [-96], [-44], [-28], [-38], [1], [-82], [11], [45], [-22], [-80], [-11], [62], [-33], [56], [-74], [28], [-32], [4], [-12], [-65], [36], [39], [-57], [12], [89], [-60], [5], [-35], [-80], [-86], [-39], [-22], [-87], [-84], [-18], [-8], [2], [76], [-49], [-29], [88], [-81], [0], [43], [90], [40], [-88], [-63], [6], [-86], [-2], [-12], [-21], [9], [-58], [-4], [15], [-55], [0]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_6400_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_6400_2_a_u();". // 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_6400_2_a_u(:prec:=1) chi := MakeCharacter_6400_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_6400_2_a_u();". // 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_6400_2_a_u( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_6400_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<3,R![-1, 1]>,<7,R![-2, 1]>,<11,R![-5, 1]>,<13,R![6, 1]>,<17,R![3, 1]>,<29,R![6, 1]>,<31,R![-8, 1]>],Snew); return Vf; end function;