
// Make newform 1058.2.a.n in Magma, downloaded from the LMFDB on 08 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1058_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_1058_2_a_n();".
// 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_1058_2_a_n();".
// 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 := [-146, 532, -298, -440, 151, 80, -22, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [19, 14, -13, -2, 1, 0, 0, 0], [1223, -1845, -1724, 889, 380, -131, -21, 6], [-2196, 1572, 2855, -553, -562, 90, 29, -5], [-3371, 3541, 2912, -1469, -545, 190, 28, -8], [4649, -7082, -5525, 2938, 1090, -380, -56, 16], [4649, -7381, -5525, 2938, 1090, -380, -56, 16], [12287, -19884, -14230, 7672, 2821, -986, -147, 42]];
    Rf_basisdens := [1, 13, 299, 299, 299, 299, 299, 299];
    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_1058_a();"
function MakeCharacter_1058_a()
    N := 1058;
    order := 1;
    char_gens := [5];
    v := [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_1058_a_Hecke(Kf)
    return MakeCharacter_1058_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, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0], [0, 0, 1, -1, 0, 0, -1, 0], [0, 0, 1, 0, 0, 0, 1, 0], [2, 1, 0, 0, 0, -1, 0, 0], [0, 0, 0, 1, -1, 0, 2, 0], [0, 0, -2, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, -1, 0, -1], [0, 2, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 0, 0, 1, 0], [2, -3, 0, 0, 0, -1, 0, 0], [0, 0, -1, -1, -1, 0, -3, 0], [-1, 2, 0, 0, 0, 0, 0, -1], [0, 0, 2, 1, 1, 0, -1, 0], [2, 3, 0, 0, 0, 1, 0, 1], [0, 0, 4, -3, 1, 0, -7, 0], [0, 0, -3, 0, -1, 0, 3, 0], [-1, -2, 0, 0, 0, 2, 0, -3], [2, -4, 0, 0, 0, -2, 0, 1], [0, 0, -4, -2, 2, 0, 6, 0], [0, 0, -3, 0, -1, 0, 1, 0], [0, 0, 2, 2, 1, 0, -3, 0], [0, 0, -4, 0, 1, 0, -1, 0], [8, 3, 0, 0, 0, -3, 0, 2], [0, 0, -6, 2, 0, 0, 0, 0], [0, 0, -5, 0, 2, 0, 5, 0], [0, 0, 5, 2, -1, 0, -6, 0], [0, 0, 0, -1, 2, 0, -3, 0], [0, 2, 0, 0, 0, -2, 0, 2], [-6, -3, 0, 0, 0, -1, 0, 1], [0, 0, -1, 1, 0, 0, 0, 0], [1, -5, 0, 0, 0, 3, 0, 0], [0, 0, 2, 3, -3, 0, -1, 0], [1, 4, 0, 0, 0, 0, 0, 3], [0, 0, 5, 0, -3, 0, -2, 0], [-1, -4, 0, 0, 0, 4, 0, 1], [4, -6, 0, 0, 0, 2, 0, -4], [7, 1, 0, 0, 0, -1, 0, -1], [-3, 0, 0, 0, 0, 7, 0, -1], [0, 0, -11, -1, 0, 0, -1, 0], [0, 0, 8, 0, 0, 0, -4, 0], [-2, -1, 0, 0, 0, 4, 0, 2], [-5, -3, 0, 0, 0, -5, 0, 3], [0, 0, -1, 1, -4, 0, 5, 0], [4, -1, 0, 0, 0, -3, 0, 1], [7, -6, 0, 0, 0, 2, 0, 1], [0, 0, 5, -3, -3, 0, -1, 0], [0, 0, -3, 3, -6, 0, -1, 0], [-9, 8, 0, 0, 0, 0, 0, 0], [-14, 6, 0, 0, 0, 2, 0, 2], [0, 0, 6, -3, 2, 0, -9, 0], [0, 0, -10, 2, -1, 0, 2, 0], [-1, 1, 0, 0, 0, 2, 0, -3], [0, 0, 9, -1, 0, 0, -13, 0], [10, 0, 0, 0, 0, 2, 0, -4], [-5, 6, 0, 0, 0, -2, 0, -3], [6, -10, 0, 0, 0, -4, 0, -2], [0, 0, -6, 2, -5, 0, -2, 0], [0, 0, -1, 2, -1, 0, 9, 0], [0, 0, 2, 1, -1, 0, 7, 0], [18, 10, 0, 0, 0, -5, 0, 2], [-13, -4, 0, 0, 0, -2, 0, 1], [0, 0, 13, 1, 2, 0, -2, 0], [13, -3, 0, 0, 0, -5, 0, -3], [-5, -5, 0, 0, 0, -5, 0, 0], [0, 0, -10, 4, 6, 0, 7, 0], [8, -2, 0, 0, 0, -5, 0, 4], [10, -2, 0, 0, 0, -2, 0, 2], [0, -9, 0, 0, 0, 4, 0, -4], [0, 0, -3, 3, -2, 0, -7, 0], [0, 0, 5, -1, 6, 0, 11, 0], [0, 0, -9, 3, -4, 0, -3, 0], [0, 0, -10, -1, 3, 0, 6, 0], [0, 0, -4, -2, 0, 0, 6, 0], [0, 0, -1, -3, 0, 0, -7, 0], [8, 5, 0, 0, 0, -3, 0, 0], [0, 0, 3, 2, -6, 0, -10, 0], [17, -2, 0, 0, 0, 1, 0, 5], [0, 0, -3, -7, 1, 0, 7, 0], [0, 0, 7, -1, 4, 0, 7, 0], [0, 0, -1, 1, 0, 0, 15, 0], [0, 0, -3, 1, 1, 0, 8, 0], [-10, 6, 0, 0, 0, 2, 0, 4], [-11, 6, 0, 0, 0, 5, 0, -1], [9, 1, 0, 0, 0, 10, 0, -1], [0, 0, -7, 1, -5, 0, -6, 0], [-1, 1, 0, 0, 0, 1, 0, 1], [4, -4, 0, 0, 0, -4, 0, -2], [0, 0, -2, -5, 4, 0, -14, 0], [0, 0, -12, 2, -2, 0, 22, 0], [17, 2, 0, 0, 0, 8, 0, -5], [0, 4, 0, 0, 0, -2, 0, -4], [-15, -8, 0, 0, 0, 7, 0, -1], [0, 0, 7, 5, -2, 0, 5, 0], [-1, -3, 0, 0, 0, 9, 0, -1], [0, 0, 1, 8, -6, 0, -15, 0], [0, 0, -12, 3, -4, 0, -12, 0], [-11, -13, 0, 0, 0, -1, 0, 1], [10, 8, 0, 0, 0, -4, 0, 2], [0, 0, 0, 9, -1, 0, -11, 0], [0, 0, 7, 0, 1, 0, -3, 0], [0, 0, -2, -3, 2, 0, -1, 0], [0, 0, 8, -3, -5, 0, -10, 0], [13, -1, 0, 0, 0, -7, 0, -2], [9, 5, 0, 0, 0, 1, 0, 2], [1, 12, 0, 0, 0, 7, 0, -3], [-25, 2, 0, 0, 0, 6, 0, -1], [13, -1, 0, 0, 0, 3, 0, 3], [11, -4, 0, 0, 0, -6, 0, 3], [0, 0, 2, -3, 11, 0, 5, 0], [0, 0, -16, -2, 1, 0, 14, 0], [0, 0, 8, 5, 6, 0, 2, 0], [0, 0, -8, -2, 6, 0, 12, 0], [0, 0, 6, 4, 2, 0, -7, 0], [0, 0, 9, -10, -4, 0, -7, 0], [-1, 6, 0, 0, 0, -10, 0, 3], [2, -5, 0, 0, 0, -1, 0, 0], [0, 0, -9, 6, -5, 0, 5, 0], [0, 0, -2, -1, 7, 0, 1, 0], [-11, 0, 0, 0, 0, 3, 0, 1], [0, 0, 20, -3, -5, 0, 1, 0], [-1, -17, 0, 0, 0, -5, 0, -2], [30, -2, 0, 0, 0, -2, 0, 2], [0, 0, 5, 3, 2, 0, 15, 0], [0, 0, -6, -3, 7, 0, 3, 0], [-2, -4, 0, 0, 0, -8, 0, 6], [0, 0, -3, 9, 0, 0, -7, 0], [0, 0, 7, 0, -3, 0, 0, 0], [11, 16, 0, 0, 0, -4, 0, 1], [0, 0, 3, 3, 0, 0, -9, 0], [0, 0, 7, -3, -2, 0, -23, 0], [0, 0, 12, -1, -5, 0, -1, 0], [-10, 13, 0, 0, 0, 0, 0, 2], [0, 0, 3, -10, -2, 0, -1, 0], [0, 0, 19, -3, 0, 0, -5, 0], [0, 0, -8, -5, -7, 0, 2, 0], [0, 0, -3, -7, -4, 0, 3, 0], [-9, -4, 0, 0, 0, 15, 0, 3], [-6, 5, 0, 0, 0, -1, 0, 1], [-22, -1, 0, 0, 0, -1, 0, 6], [11, -8, 0, 0, 0, -2, 0, 1], [0, 0, -20, 0, 7, 0, 24, 0], [-11, 3, 0, 0, 0, 1, 0, 5], [0, 0, -23, 1, 2, 0, 9, 0], [10, -16, 0, 0, 0, 2, 0, -2], [9, 4, 0, 0, 0, -8, 0, 2], [-4, -2, 0, 0, 0, 11, 0, 4], [-23, 14, 0, 0, 0, 0, 0, 7], [8, -6, 0, 0, 0, 6, 0, -4], [0, 0, 6, -4, 9, 0, 17, 0], [-4, -9, 0, 0, 0, 1, 0, -5], [-30, -10, 0, 0, 0, 8, 0, -2], [0, 0, -12, 7, -3, 0, 0, 0], [0, 0, -16, -4, -2, 0, 12, 0], [0, 0, -9, -1, -4, 0, 5, 0], [-4, -12, 0, 0, 0, 0, 0, -4], [0, 0, -26, 2, 9, 0, 14, 0], [0, 0, -9, -3, -2, 0, 11, 0], [-23, -4, 0, 0, 0, 7, 0, 3], [0, 0, 13, 1, 3, 0, 14, 0], [29, 8, 0, 0, 0, -6, 0, 5], [0, 0, -12, 0, 15, 0, 12, 0], [0, 0, -13, 1, -7, 0, -2, 0], [0, 0, 12, 0, -8, 0, -10, 0], [-14, 2, 0, 0, 0, -10, 0, 8], [11, 3, 0, 0, 0, -11, 0, -1], [0, 0, 19, -3, 8, 0, -14, 0], [-16, 0, 0, 0, 0, 2, 0, -6], [0, 0, 0, -4, 1, 0, 16, 0], [15, 5, 0, 0, 0, 3, 0, 3], [0, 0, 13, -3, -2, 0, 3, 0], [0, 0, 13, -11, 6, 0, -14, 0], [-7, -4, 0, 0, 0, -8, 0, -5], [0, 0, 24, 2, -2, 0, -11, 0], [-8, 2, 0, 0, 0, -8, 0, 8], [-28, 5, 0, 0, 0, 11, 0, 0], [0, 0, 4, 0, 2, 0, 0, 0], [0, 0, 12, -7, -1, 0, 7, 0], [13, -4, 0, 0, 0, 10, 0, 3], [0, 0, -10, 1, 5, 0, 22, 0], [-9, 9, 0, 0, 0, -3, 0, -3], [29, 5, 0, 0, 0, -7, 0, -3], [0, 0, 5, 3, 4, 0, 13, 0], [0, 0, -9, -4, 9, 0, 20, 0], [4, 3, 0, 0, 0, 5, 0, -2], [0, 0, 19, -1, 7, 0, 13, 0], [2, 4, 0, 0, 0, -6, 0, 5], [-13, 10, 0, 0, 0, -6, 0, -1], [-7, -9, 0, 0, 0, -6, 0, 3], [-13, 5, 0, 0, 0, -11, 0, 10], [0, 0, -6, 11, -8, 0, 18, 0], [2, -13, 0, 0, 0, -7, 0, 6], [0, 0, 27, -1, -5, 0, -23, 0], [0, 0, -18, 4, -9, 0, 4, 0], [0, 0, 17, 3, -11, 0, -12, 0], [0, 0, 15, 3, -4, 0, -25, 0], [0, 0, 21, -1, -3, 0, -28, 0], [0, 0, 0, 0, 0, 4, 0, 2], [0, 0, -7, 3, -10, 0, 9, 0], [7, -8, 0, 0, 0, -8, 0, -3], [7, -5, 0, 0, 0, 7, 0, 3], [0, 0, 13, -2, 2, 0, -23, 0], [0, 0, 13, -4, 11, 0, 1, 0], [6, -12, 0, 0, 0, 4, 0, -4], [0, 0, -26, 4, 6, 0, 16, 0], [20, -10, 0, 0, 0, -6, 0, -4], [9, -13, 0, 0, 0, -6, 0, 3], [14, -16, 0, 0, 0, -10, 0, -6], [-17, 7, 0, 0, 0, -6, 0, 9], [12, 1, 0, 0, 0, 11, 0, -2], [0, 0, 32, -6, -4, 0, -22, 0], [0, 0, -10, 7, -6, 0, 12, 0], [-4, 10, 0, 0, 0, -14, 0, 4], [0, 0, -10, 10, 0, 0, 13, 0], [29, 8, 0, 0, 0, -8, 0, 5], [0, -13, 0, 0, 0, 2, 0, -8], [0, 0, 23, -5, 6, 0, -33, 0], [6, -17, 0, 0, 0, -1, 0, -2], [2, 11, 0, 0, 0, -15, 0, -4], [0, 0, -31, 5, 2, 0, 21, 0], [-28, 7, 0, 0, 0, -8, 0, 4], [0, 0, 1, -9, 0, 0, 9, 0], [-23, 4, 0, 0, 0, -3, 0, 7], [-14, 12, 0, 0, 0, -6, 0, -2], [0, 0, 13, 4, 10, 0, 1, 0], [13, 8, 0, 0, 0, -4, 0, -7], [0, 0, 9, 1, -4, 0, -11, 0], [8, 12, 0, 0, 0, -3, 0, -4], [11, 5, 0, 0, 0, -3, 0, 3], [0, 0, 24, -6, 3, 0, -22, 0], [0, 0, -18, 0, 0, 0, -22, 0], [14, 10, 0, 0, 0, -8, 0, 1], [0, 0, -8, 1, 14, 0, 14, 0], [0, 0, -14, 2, -12, 0, 12, 0], [0, 0, 6, 3, -10, 0, 9, 0], [0, 0, -13, -8, 5, 0, 28, 0], [31, 0, 0, 0, 0, -2, 0, -3], [16, -6, 0, 0, 0, -10, 0, -4], [0, 0, -17, 6, 6, 0, 13, 0], [-16, -16, 0, 0, 0, -3, 0, -4], [11, -2, 0, 0, 0, 16, 0, 1], [-26, -1, 0, 0, 0, 7, 0, 0], [15, -3, 0, 0, 0, -9, 0, 6], [-5, 6, 0, 0, 0, 6, 0, 7], [10, 18, 0, 0, 0, -2, 0, 6], [0, 0, -15, -1, 3, 0, 17, 0], [0, 0, -14, 3, 0, 0, -18, 0], [0, 0, -18, 10, -12, 0, 10, 0], [0, 0, 3, -1, 14, 0, 5, 0], [0, 0, 12, 0, -7, 0, -20, 0], [0, 0, -1, -11, 2, 0, 7, 0], [0, 0, -19, 9, -7, 0, 16, 0], [-16, -7, 0, 0, 0, 13, 0, -2], [2, -34, 0, 0, 0, -3, 0, 2], [0, 0, 18, 1, -1, 0, -27, 0], [0, 0, 5, -8, 7, 0, 1, 0], [59, 3, 0, 0, 0, -7, 0, 3], [-7, 11, 0, 0, 0, 3, 0, -3], [0, 0, -3, -7, 4, 0, -33, 0], [0, 0, -6, 14, -5, 0, 10, 0], [-14, -4, 0, 0, 0, -6, 0, 4], [0, 0, -7, -4, -3, 0, -4, 0], [1, 10, 0, 0, 0, 22, 0, 0], [0, 0, 29, -3, 3, 0, -9, 0], [0, 0, 3, -4, -17, 0, -14, 0], [0, 0, 19, -1, -15, 0, -6, 0], [0, 0, 2, -7, 2, 0, -36, 0], [15, 21, 0, 0, 0, 5, 0, 1], [-19, 9, 0, 0, 0, 9, 0, -1], [0, 0, 9, -2, 5, 0, 3, 0], [0, 0, 20, 4, 7, 0, 1, 0], [0, 0, 3, -9, 8, 0, 27, 0], [21, 2, 0, 0, 0, 5, 0, -11], [-1, 0, 0, 0, 0, -16, 0, 13], [16, -2, 0, 0, 0, 2, 0, 8], [-8, -15, 0, 0, 0, -11, 0, 2], [0, 0, -14, -4, 2, 0, -15, 0], [0, 0, 4, 9, -5, 0, -30, 0], [10, -16, 0, 0, 0, -6, 0, 4], [0, 0, -18, 4, 4, 0, 28, 0], [0, 0, -7, 1, -2, 0, -1, 0], [0, 0, 1, 8, -11, 0, -30, 0], [17, 6, 0, 0, 0, 9, 0, -1], [-34, 16, 0, 0, 0, -4, 0, -6], [0, 0, 5, 1, -7, 0, 0, 0], [0, 0, -15, -8, 1, 0, -18, 0], [20, 6, 0, 0, 0, 10, 0, -4], [13, 21, 0, 0, 0, 3, 0, -2], [0, 0, 1, -5, 6, 0, 35, 0], [0, 0, -16, -11, 7, 0, 44, 0], [24, -10, 0, 0, 0, -12, 0, -3], [0, 0, 5, 10, -11, 0, 5, 0], [-23, 9, 0, 0, 0, -3, 0, 9], [0, 0, -21, -4, -7, 0, -18, 0], [0, 0, 20, 0, -4, 0, -26, 0], [-12, -6, 0, 0, 0, -6, 0, 6], [48, -24, 0, 0, 0, -7, 0, -8], [14, 20, 0, 0, 0, -4, 0, 10], [0, 0, -4, -8, 5, 0, -12, 0], [0, 0, 1, -10, -5, 0, -22, 0], [0, 0, 13, -5, 4, 0, -7, 0], [7, 5, 0, 0, 0, 25, 0, 0], [0, 0, 17, -9, -7, 0, 5, 0], [3, 2, 0, 0, 0, -22, 0, 4], [11, -26, 0, 0, 0, -3, 0, 5], [0, 0, 7, 14, 1, 0, 4, 0], [0, 0, -14, 4, 6, 0, 20, 0], [8, 5, 0, 0, 0, -15, 0, 8], [-7, -4, 0, 0, 0, 4, 0, -11], [0, 0, 5, -15, 6, 0, -1, 0], [0, 0, 6, -7, 25, 0, 4, 0], [10, 39, 0, 0, 0, -7, 0, 1], [0, 0, -4, -4, -18, 0, -8, 0], [0, 0, 11, -3, 0, 0, 6, 0], [18, -10, 0, 0, 0, -12, 0, 6], [-6, -6, 0, 0, 0, -16, 0, 4], [0, 0, -20, -8, 1, 0, 32, 0], [29, -8, 0, 0, 0, -19, 0, -3], [0, 0, -3, -1, 15, 0, 25, 0], [0, 0, 0, 6, -5, 0, -2, 0], [16, -16, 0, 0, 0, -14, 0, 0], [-11, 16, 0, 0, 0, -4, 0, 1], [0, 0, -11, 9, -13, 0, -10, 0], [0, 0, -20, 4, -15, 0, -8, 0], [0, 0, -23, -4, -10, 0, 11, 0], [8, 29, 0, 0, 0, 3, 0, 3], [0, 0, 21, 3, -8, 0, 5, 0], [0, 0, -7, 8, 7, 0, 24, 0], [-29, -9, 0, 0, 0, -9, 0, -5], [-2, -4, 0, 0, 0, -8, 0, 6], [-32, -20, 0, 0, 0, 10, 0, -4], [2, -19, 0, 0, 0, 3, 0, -7], [0, 0, -13, 10, -2, 0, 25, 0], [3, -12, 0, 0, 0, 15, 0, -3], [0, 0, 11, 0, -5, 0, -6, 0], [0, 0, -43, 6, 8, 0, 6, 0], [-33, -2, 0, 0, 0, 6, 0, 0], [0, 0, 8, 0, 12, 0, 12, 0], [-2, 2, 0, 0, 0, 20, 0, -4], [0, 0, 24, 8, 8, 0, 3, 0], [-2, -3, 0, 0, 0, 27, 0, -8], [0, 0, -5, 1, -10, 0, -17, 0], [0, 0, -10, -11, -7, 0, 13, 0], [24, -14, 0, 0, 0, 8, 0, 4], [-54, 2, 0, 0, 0, -4, 0, -2], [-3, 2, 0, 0, 0, -11, 0, -1], [0, 0, -39, 1, 8, 0, 11, 0], [0, 0, 5, 5, -8, 0, 3, 0], [-13, 8, 0, 0, 0, -12, 0, -3], [-7, 11, 0, 0, 0, -11, 0, -4], [28, -16, 0, 0, 0, 0, 0, -2], [0, 0, -11, -1, 8, 0, 3, 0], [0, 0, -7, 12, -17, 0, 8, 0], [6, 6, 0, 0, 0, 30, 0, -6], [0, 0, 0, -8, 12, 0, 4, 0], [0, 0, -30, -3, 2, 0, -8, 0], [-41, -25, 0, 0, 0, 15, 0, -5], [23, -16, 0, 0, 0, 4, 0, 7], [0, 0, -21, -12, 5, 0, 18, 0], [17, -5, 0, 0, 0, -15, 0, 2], [39, -2, 0, 0, 0, -4, 0, 7], [0, 0, 22, 10, -3, 0, 2, 0], [-10, -7, 0, 0, 0, 11, 0, -5], [-8, 24, 0, 0, 0, -6, 0, 1], [4, 28, 0, 0, 0, 6, 0, 4], [0, 0, -35, 1, 8, 0, 25, 0], [0, 0, 15, -15, -8, 0, -8, 0], [-33, -12, 0, 0, 0, 4, 0, -7], [-20, 4, 0, 0, 0, 14, 0, 8], [38, 8, 0, 0, 0, -10, 0, 12], [0, 0, 27, 5, -10, 0, 3, 0], [0, 0, -10, -14, 2, 0, 20, 0], [-26, 22, 0, 0, 0, 6, 0, 0], [22, 38, 0, 0, 0, -10, 0, 2], [0, 0, -2, -6, -10, 0, -16, 0], [0, 0, -8, 10, 3, 0, 9, 0], [0, 0, -11, -3, 2, 0, 32, 0], [1, -8, 0, 0, 0, 16, 0, 6], [0, 0, 6, 1, 17, 0, 5, 0], [0, 0, -19, -12, 2, 0, 21, 0], [-11, -14, 0, 0, 0, 12, 0, 1], [-3, 11, 0, 0, 0, -16, 0, 7], [0, 0, -9, 0, 8, 0, -1, 0], [30, -18, 0, 0, 0, 6, 0, 0], [6, -10, 0, 0, 0, 0, 0, -12], [37, 13, 0, 0, 0, 21, 0, -11], [0, 0, 10, 15, 2, 0, -12, 0], [0, 0, 37, -7, 2, 0, -23, 0], [0, 0, -44, 4, 7, 0, 0, 0], [4, 20, 0, 0, 0, -8, 0, -6], [-8, 31, 0, 0, 0, -3, 0, -5], [43, 10, 0, 0, 0, 4, 0, -3], [0, 0, -41, 11, 10, 0, 17, 0], [0, 0, -2, -7, -8, 0, 3, 0], [0, 0, -2, 2, 0, 0, 28, 0], [0, 0, 5, 6, 0, 0, 14, 0], [0, 0, -2, -1, 11, 0, -6, 0], [20, 20, 0, 0, 0, 14, 0, 0], [20, -17, 0, 0, 0, 3, 0, 0], [-12, -8, 0, 0, 0, -2, 0, -7], [0, 0, 8, 6, 0, 0, -12, 0], [0, 0, -44, 2, 7, 0, 53, 0], [14, -13, 0, 0, 0, 13, 0, -2], [-26, -6, 0, 0, 0, -16, 0, -2], [0, 0, 1, 4, -7, 0, 20, 0], [33, -1, 0, 0, 0, 13, 0, -16], [0, 0, 37, 0, -5, 0, -25, 0], [4, 10, 0, 0, 0, 9, 0, -8], [6, 16, 0, 0, 0, -16, 0, 10], [16, -21, 0, 0, 0, 11, 0, -10], [0, 0, 12, -3, 11, 0, 10, 0], [0, 0, 1, -10, -11, 0, -9, 0], [0, 0, 24, 5, 7, 0, 2, 0], [6, -20, 0, 0, 0, -6, 0, 12], [9, -8, 0, 0, 0, 6, 0, 9], [-4, -26, 0, 0, 0, 8, 0, -6], [0, 0, -52, 3, -3, 0, 38, 0], [0, 0, -11, -3, 2, 0, 47, 0], [0, 0, -24, -9, 9, 0, 21, 0], [0, 0, 12, -1, 11, 0, 23, 0], [27, 32, 0, 0, 0, -14, 0, 5], [-9, -5, 0, 0, 0, 11, 0, -20], [9, -15, 0, 0, 0, 1, 0, 5], [-24, 8, 0, 0, 0, 12, 0, 0], [0, 0, 12, 4, 9, 0, 16, 0], [39, 6, 0, 0, 0, 2, 0, -8], [18, -27, 0, 0, 0, -9, 0, 3], [-16, 26, 0, 0, 0, -6, 0, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1058_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1058_2_a_n();".
// 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_1058_2_a_n(:prec:=8)
    chi := MakeCharacter_1058_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_1058_2_a_n();".
// 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_1058_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1058_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![22, 10, -9, -2, 1]>,<5,R![144, 0, -1584, 0, 417, 0, -36, 0, 1]>],Snew);
    return Vf;
end function;