// Make newform 4026.2.a.f in Magma, downloaded from the LMFDB on 28 March 2024. // To make the character of type GrpDrchElt, type "MakeCharacter_4026_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_4026_2_a_f();". // 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_4026_2_a_f();". // 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_4026_a();" function MakeCharacter_4026_a() N := 4026; order := 1; char_gens := [1343, 1465, 3235]; 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_4026_a_Hecke(Kf) return MakeCharacter_4026_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 := [[1], [-1], [-1], [2], [-1], [-3], [8], [4], [4], [-9], [-3], [6], [3], [4], [0], [8], [-3], [-1], [4], [6], [12], [-4], [4], [-17], [13], [-1], [12], [13], [3], [2], [0], [1], [14], [-19], [20], [4], [-7], [-5], [24], [11], [14], [1], [18], [-17], [-14], [14], [-25], [-12], [24], [-4], [-20], [-16], [-8], [-19], [26], [-6], [23], [20], [26], [20], [-32], [-8], [-11], [16], [-4], [3], [-4], [15], [-25], [36], [6], [-33], [-22], [-20], [-13], [14], [36], [-29], [25], [31], [-36], [-31], [18], [-4], [13], [-26], [-40], [-14], [-22], [6], [33], [0], [20], [-28], [-22], [38], [-4], [-18], [-13], [16], [11], [27], [-32], [-35], [-44], [8], [39], [6], [0], [-14], [1], [-33], [-19], [-31], [-8], [-6], [-34], [-10], [-12], [-39], [-21], [11], [-50], [-30], [33], [15], [21], [0], [-12], [14], [-31], [-5], [2], [-52], [-12], [13], [21], [-35], [42], [25], [44], [-38], [11], [-4], [-20], [9], [3], [17], [31], [17], [-36], [52], [4], [-24], [16], [15], [-49], [0], [-26], [27], [-48], [-42], [-23], [-36], [44], [6], [-10], [40], [55], [20], [-22], [-22], [-45], [-10], [-24], [4], [20], [0], [36], [14], [-53], [-12], [-24], [30], [-15], [-4], [26], [-45], [-54], [-11], [56], [39], [-51], [27], [38], [38], [36], [-31], [-42], [-42], [34], [-2], [43], [-39], [-12], [41], [-1], [46], [36], [50], [30], [-42], [10], [15], [-21], [-40], [-35], [10], [67], [-14], [55], [-23], [43], [58], [-58], [-24], [-41], [-24], [-14], [-60], [11], [28], [26], [58], [17], [-69], [50], [51], [-15], [-53], [-63], [68], [1], [2], [44], [-28], [12], [10], [20], [-4], [64], [63], [4], [18], [56], [12], [5], [16], [6], [-3], [-17], [45], [42], [51], [-30], [71], [31], [18], [-6], [-16], [14], [-39], [13], [-80], [-17], [-16], [-5], [-58], [-26], [29], [51], [-68], [24], [20], [9], [-69], [-8], [-42], [0], [-26], [-16], [-24], [39], [42], [-35], [20], [4], [2], [-84], [20], [14], [10], [-8], [84], [-38], [18], [-60], [20], [-24], [-16], [48], [-6], [-54], [-27], [22], [52], [-16], [-28], [-51], [6], [84], [-32], [-50], [19], [-20], [-76], [-24], [44], [-48], [34], [3], [18], [62], [-61], [9], [15], [-84], [-61], [-54], [46], [64], [-69], [40], [61], [-66], [-80], [-74], [26], [32], [49], [32], [8], [-78], [0], [90], [-86], [-55], [32], [72], [70], [86], [-38], [-44], [83], [-1], [62], [-37], [-24], [20], [-75], [30], [-12], [6], [0], [-16], [-7], [76], [11], [10], [-22], [2], [-81], [-76], [-43], [8], [-102], [-26], [-59], [-18], [75], [37], [54], [10], [-66], [-40], [-67], [-18], [-2], [-45], [-54], [-53], [87], [79], [10], [-23], [-25], [92], [-30], [7], [64], [-52], [46], [15], [18], [58], [-58], [38], [4], [47], [-48], [54], [-24], [-73], [60], [47], [-54], [39], [95], [-74], [-87]]; aps := ConvertToHeckeField(raw_aps); chi := MakeCharacter_4026_a_Hecke(Universe(aps)); return ExtendMultiplicatively(weight, aps, chi); end function; // To make the newform (type ModFrm), type "MakeNewformModFrm_4026_2_a_f();". // 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_4026_2_a_f(:prec:=1) chi := MakeCharacter_4026_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_4026_2_a_f();". // 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_4026_2_a_f( : sign := -1) R := PolynomialRing(Rationals()); chi := MakeCharacter_4026_a(); // SetVerbose("ModularSymbols", true); Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign))); Vf := Kernel([<5,R![1, 1]>,<7,R![-2, 1]>,<13,R![3, 1]>],Snew); return Vf; end function;