
// Make newform 4158.2.a.bw in Magma, downloaded from the LMFDB on 04 July 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_4158_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_4158_2_a_bw();".
// 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_4158_2_a_bw();".
// 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
        poly := [-6, -10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-7, -1, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_4158_a();"
function MakeCharacter_4158_a()
    N := 4158;
    order := 1;
    char_gens := [1541, 2971, 1135];
    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_4158_a_Hecke( : Kf := Kf)
    return MakeCharacter_4158_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<X, Y> := 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<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    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, 0, 0], [0, 0, 0], [0, 1, 0], [-1, 0, 0], [1, 0, 0], [0, 1, -1], [1, 0, 0], [0, 0, 0], [1, 1, 0], [0, -2, -1], [4, 1, 1], [1, -2, 1], [-1, -1, -1], [1, 0, 0], [4, 0, 0], [5, 0, 2], [0, 1, 1], [1, 1, 1], [3, 1, -2], [2, -4, -1], [-7, -2, -1], [5, 0, -1], [2, -4, -2], [5, 1, 0], [-2, 3, 0], [7, 0, 1], [2, -1, 1], [11, 0, 1], [-6, 1, 0], [2, 2, 0], [-1, 0, -3], [-3, 1, -2], [5, -4, 1], [-2, 4, 0], [3, -1, -2], [-8, -2, -4], [-1, -5, 2], [-2, -2, 1], [10, 2, 4], [4, -4, -2], [10, 4, 0], [-4, -2, 1], [1, 1, 1], [-4, 0, 3], [6, -6, 0], [0, 3, -1], [8, -3, 1], [-3, -3, -1], [-8, 6, -1], [-6, 2, 0], [12, 1, -2], [3, 3, 1], [-6, -2, -2], [-1, -1, 1], [-6, -6, 0], [-5, -2, -4], [-4, -3, 2], [-2, -4, 5], [10, 1, 2], [4, -2, 4], [-4, -5, 0], [-5, 6, 1], [-4, 4, 4], [-12, 0, 2], [-10, 1, -4], [11, -3, -1], [12, 0, 5], [-17, -1, 0], [5, -4, 3], [-3, 6, 4], [12, 4, 1], [10, 9, -3], [21, 2, 4], [-18, 4, 2], [4, -4, -4], [-7, -6, 1], [-19, 5, 1], [-12, -2, -2], [3, 2, -1], [0, 4, 0], [1, 0, -4], [5, -8, -1], [5, 7, -5], [-8, -3, -4], [7, -3, 0], [18, 6, 0], [9, 4, 1], [0, -4, -3], [3, 0, 3], [-2, 0, 6], [-7, -5, 3], [-2, -1, -4], [-12, 6, -2], [-24, 6, 0], [-15, 1, -7], [20, -1, -2], [-8, -4, 2], [-26, -4, 1], [-16, -6, 0], [-14, -6, 2], [20, 2, 4], [9, 7, 0], [-16, -8, -1], [16, -3, -2], [0, -3, -1], [6, 3, 4], [-17, 10, 2], [-17, -7, 3], [-6, -10, -1], [-14, 4, -6], [2, -4, 5], [-18, -3, 2], [4, 8, 0], [24, 2, 0], [18, -1, 2], [-3, 10, -5], [12, -4, 4], [-8, -10, 0], [-2, -7, 1], [-9, -2, -1], [7, -5, -3], [33, -1, -2], [0, -8, 0], [16, 3, 4], [-10, 10, 0], [-3, -7, -1], [29, -6, -1], [-15, 4, -1], [-3, 0, -8], [-33, 0, -2], [-17, -7, -3], [16, -3, 1], [-26, -4, 2], [16, 6, 2], [-22, -3, 5], [1, -16, -1], [-16, -4, -2], [14, -11, -2], [16, 1, -2], [-28, 6, 2], [-8, -9, 8], [14, -8, 3], [10, -12, 2], [8, 10, 2], [8, -4, -6], [-9, -10, 5], [-14, -13, 7], [12, 1, 3], [-5, -10, 5], [7, 5, 4], [-2, 11, 8], [20, 6, 5], [3, 11, 4], [-32, 1, -6], [16, -2, 0], [4, -6, 4], [-2, 12, -2], [0, 6, -6], [-4, -8, 6], [-5, 8, 1], [22, -5, -2], [2, -6, 4], [-7, -8, -5], [-36, -5, 3], [-30, -2, 0], [-23, -2, -1], [4, 10, 4], [-5, -4, 0], [-36, 4, -3], [10, -5, -2], [-12, -12, 6], [30, -6, 1], [7, -2, 0], [-4, -2, 5], [32, 0, 0], [-8, 14, 1], [15, -8, -4], [-34, 1, -7], [1, 1, -7], [8, -3, -5], [8, -5, -6], [-4, -10, -9], [-1, 12, 1], [-21, 10, -4], [-4, 2, 5], [6, -12, 0], [-26, -5, -2], [-17, 10, -1], [-1, -5, 10], [-20, 11, 4], [-2, 12, 0], [35, 6, 9], [15, 15, -8], [-20, -4, 4], [3, 2, 9], [5, 15, 2], [18, 12, 1], [-22, -6, -6], [36, -3, -3], [-4, 0, 2], [-7, 3, 0], [-7, -1, 6], [38, 8, 0], [-6, 4, 2], [19, -14, 2], [-8, -9, 4], [-10, 9, 3], [-4, 2, 4], [30, 8, 2], [26, 2, -4], [11, 11, -4], [-8, 4, 6], [26, -1, 2], [-6, 1, -9], [0, 13, 10], [-12, 7, 6], [26, 2, -4], [15, 5, -1], [37, 3, 4], [2, 11, -3], [-2, 0, -8], [11, -13, -9], [34, -2, 8], [20, -7, 6], [49, -3, 1], [4, 10, 4], [23, 9, 6], [6, 0, 6], [-1, 13, 8], [8, 2, -6], [-16, 14, -4], [6, -4, 0], [-5, 0, -9], [-30, -10, 2], [-8, -4, 6], [28, 2, -6], [-26, -6, 4], [-8, 0, -5], [-6, -9, -6], [-4, -18, -2], [-12, -4, 2], [6, -10, -2], [-2, 7, -7], [45, -2, 5], [1, 18, 4], [-8, 6, 10], [43, 8, -4], [-29, -13, 1], [-14, 5, 3], [18, -7, -7], [22, -6, -8], [-42, 11, 1], [33, -7, 0], [-42, -7, -2], [14, 10, -10], [18, -6, -6], [0, -10, 0], [0, -10, -8], [8, 10, -10], [-9, -14, -1], [16, 6, 0], [-10, 0, -6], [-9, 7, -5], [35, -13, -1], [12, 10, -5], [21, 10, 1], [9, 13, 13], [46, 12, -4], [-20, -3, 3], [18, 12, -4], [-1, -5, 14], [15, 15, -1], [-12, -17, 4], [11, -4, 5], [35, -2, 11], [43, -6, -4], [36, 7, 4], [-46, 3, -6], [-24, -10, -2], [16, 0, 13], [-8, 3, -2], [34, -14, 0], [18, -9, 6], [27, 16, 0], [2, 5, 0], [18, -16, 2], [-25, -13, 3], [15, -11, 2], [-33, -10, -4], [21, 10, 16], [-9, 21, 3], [-15, -8, 0], [-10, -3, 2], [57, 7, 0], [-28, 3, -8], [-16, 4, 2], [5, -15, -1], [-20, 2, -8], [-45, 1, -6], [-38, 18, 4], [-56, 0, -4], [13, 5, 10], [-23, -4, 7], [24, 9, 12], [22, 4, 3], [-6, -23, -1], [-2, 5, -14], [60, -6, -1], [18, 12, -6], [24, -1, -4], [10, 7, 14], [48, 1, -3], [70, -5, -2], [16, 4, 5], [0, -21, -3], [-13, 10, 9], [5, -6, 12], [-63, 5, -1], [-32, 11, 0], [66, 4, -2], [-14, -13, 4], [-25, -9, 3], [-25, 10, -5], [-16, 17, 0], [6, 9, 6], [34, -19, 0], [-14, -12, -1], [-13, 18, -1], [25, 18, -8], [-24, 12, 6], [12, 16, -2], [11, 21, -5], [-6, 6, -2], [23, 3, 7], [16, -1, 5], [18, 13, -5], [11, 18, -5], [18, -1, -1], [8, 5, -2], [34, 6, -2], [-34, 12, 0], [10, 8, -6], [6, -10, 10], [-54, -7, 4], [12, -9, -3], [10, -28, 0], [24, -8, -7], [-17, 23, 2], [-8, 11, -12], [20, -4, 10], [-22, 7, -6], [-14, 1, -15], [19, -20, 0], [-11, -7, 4], [14, 20, -8], [-5, 9, -5], [34, 8, 4], [14, 0, 6], [22, 3, 8], [-2, -10, 6], [40, 0, -4], [-33, 4, -10], [25, -9, -4], [-21, -6, -9], [16, -20, 6], [27, 1, 7], [48, 10, -4], [-21, 20, 1], [34, -2, -3], [-18, 3, 0], [63, 6, 7], [-50, -7, -11], [-11, 15, -6], [47, -11, 4], [64, -4, 0], [36, -3, -3], [18, 21, -1], [-24, 22, 2], [-7, -29, 5], [-28, -8, -12], [26, -6, -2], [-7, -8, 1], [-6, -12, -8], [-54, -7, -14], [50, 11, 6], [34, -16, -2], [-26, 9, 8], [21, -11, -8], [-12, -21, -1], [3, -2, 3], [-36, -19, 7], [34, -14, 6], [52, -9, -4], [38, -12, -6], [-39, 11, 2], [-49, 11, -2], [17, 1, -8], [-41, 1, 10], [41, 17, -4], [-34, 18, 7], [8, -5, -9], [-27, -18, 3], [-14, 4, 2], [69, -3, 6], [45, -4, 5], [-2, -8, 10], [17, 6, 13], [-16, -18, -16], [-42, 14, 0], [2, 1, -11], [-41, 10, -1], [-1, 11, 4], [22, 15, -2], [26, -19, -3], [11, 15, 16], [-28, 9, -10], [32, 11, 9], [-3, 13, -1], [-16, -4, -1], [-60, 15, 2], [17, 14, 14], [14, -1, 13], [-50, 0, 2], [4, -4, 1], [16, 8, -2], [-46, -8, 6], [-28, -12, 2], [-12, 14, 2], [26, -12, 6], [-24, 22, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4158_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4158_2_a_bw();".
// 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_4158_2_a_bw(:prec:=3)
    chi := MakeCharacter_4158_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_4158_2_a_bw();".
// 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_4158_2_a_bw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4158_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-6, -10, 0, 1]>,<13,R![73, -37, -1, 1]>,<17,R![-1, 1]>,<19,R![0, 1]>,<41,R![-60, -32, 2, 1]>],Snew);
    return Vf;
end function;