
// Make newform 4158.2.a.q 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_q();".
// 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_q();".
// 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_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], [-3], [1], [-1], [5], [3], [-4], [-3], [0], [8], [-4], [-3], [2], [0], [3], [0], [-1], [11], [0], [14], [11], [9], [0], [-1], [0], [2], [9], [14], [-12], [-13], [21], [-6], [2], [12], [-19], [14], [11], [12], [-18], [24], [20], [-15], [-4], [0], [20], [-22], [-10], [24], [8], [-3], [-18], [-10], [30], [6], [-18], [-15], [11], [5], [-15], [-16], [-12], [-28], [-3], [-25], [6], [-4], [2], [-12], [-10], [0], [-12], [14], [11], [8], [24], [-30], [38], [6], [26], [36], [20], [-24], [29], [-1], [-24], [-24], [-34], [18], [-4], [-18], [-18], [2], [-24], [-7], [-6], [-27], [0], [-16], [-13], [26], [-18], [-45], [33], [-4], [-34], [-18], [18], [39], [8], [-19], [29], [-6], [-16], [20], [12], [-16], [-24], [-27], [39], [8], [-22], [12], [24], [5], [48], [-46], [39], [-28], [-43], [-16], [54], [14], [-16], [30], [20], [-6], [14], [-39], [-6], [-16], [30], [32], [27], [32], [48], [2], [-33], [-31], [21], [-13], [54], [-28], [24], [23], [-3], [5], [-36], [2], [-18], [18], [-27], [41], [36], [6], [-9], [-4], [-7], [-58], [18], [12], [-22], [36], [-34], [-22], [-48], [-10], [-6], [-7], [-46], [50], [-45], [-22], [18], [27], [-30], [38], [-4], [-58], [-18], [-19], [-3], [-25], [-27], [63], [-42], [-22], [-28], [9], [-36], [54], [-40], [-40], [44], [-6], [3], [-34], [27], [3], [23], [-22], [-6], [62], [-60], [6], [38], [5], [-18], [51], [-27], [-55], [-40], [-12], [2], [-51], [-10], [-30], [9], [56], [-3], [8], [-70], [-40], [51], [-55], [30], [-1], [24], [54], [-21], [-42], [-46], [68], [-34], [-69], [6], [56], [36], [50], [-30], [53], [75], [-24], [-49], [-24], [-63], [-22], [32], [-33], [2], [-37], [21], [65], [41], [24], [71], [33], [-48], [68], [54], [-10], [-7], [26], [44], [50], [38], [-30], [77], [-52], [81], [-6], [-40], [0], [-43], [32], [-60], [-46], [-6], [-22], [-18], [75], [48], [6], [75], [71], [-36], [56], [-39], [-12], [2], [14], [30], [-16], [-54], [-25], [-49], [18], [68], [75], [-58], [-48], [81], [-81], [-4], [48], [80], [78], [-84], [-67], [75], [56], [-61], [18], [2], [63], [-43], [11], [-7], [54], [54], [14], [3], [-55], [45], [2], [24], [-70], [-21], [47], [59], [-10], [-30], [48], [-64], [-45], [-15], [80], [17], [-54], [69], [14], [38], [-9], [-73], [38], [21], [27], [-63], [-63], [-57], [62], [-45], [27], [-15], [-55], [-55], [72], [-7], [-52], [72], [-76], [66], [-30], [-67], [32], [-78], [66], [20], [-78], [26], [66], [-60], [56], [54], [86], [-51], [-28], [86], [-88], [57], [62], [75], [30], [-43], [-24], [-4], [-7], [0], [-28], [-24], [-13], [-102], [-49], [54], [-72], [32], [-28], [-90], [-28], [-30], [-28], [-30], [-90], [8], [20], [99], [-24], [74], [36], [-30], [9], [-73], [-3], [57], [47], [78], [-6], [-63], [-79], [-36]];
    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_q();".
// 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_q(:prec:=1)
    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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4158_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 1]>,<13,R![-5, 1]>,<17,R![-3, 1]>,<19,R![4, 1]>,<41,R![3, 1]>],Snew);
    return Vf;
end function;