
// Make newform 170.2.k.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_170_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_170_k_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_170_2_k_b();".
// 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_170_2_k_b();".
// 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 := [4, 0, 1072, 0, 3780, 0, 5264, 0, 3709, 0, 1412, 0, 286, 0, 28, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-150, 0, -1496, 0, -1415, 0, 1178, 0, 1599, 0, 553, 0, 72, 0, 3, 0], [0, 4512, 0, 23630, 0, 39204, 0, 29161, 0, 10990, 0, 2146, 0, 202, 0, 7], [27314, 795538, 1930112, 1813186, 5200301, 992877, 5426998, -268947, 2683097, -398393, 645475, -124847, 70430, -15156, 2681, -606], [27314, -795538, 1930112, -1813186, 5200301, -992877, 5426998, 268947, 2683097, 398393, 645475, 124847, 70430, 15156, 2681, 606], [-40002, 326578, 2271370, -33592, 6402955, -1165423, 6530061, -1312806, 3121007, -619755, 730079, -141535, 78142, -14870, 2940, -553], [40002, 326578, -2271370, -33592, -6402955, -1165423, -6530061, -1312806, -3121007, -619755, -730079, -141535, -78142, -14870, -2940, -553], [2424, 27314, 1445170, 1930112, 4103866, 5200301, 4182861, 5426998, 1978707, 2683097, 457279, 645475, 48469, 70430, 1812, 2681], [46328, -615366, 2458676, -3123580, 6300400, -6128857, 6124374, -5650018, 2828070, -2619768, 645754, -608090, 68038, -65025, 2536, -2448], [-46328, -615366, -2458676, -3123580, -6300400, -6128857, -6124374, -5650018, -2828070, -2619768, -645754, -608090, -68038, -65025, -2536, -2448], [-25102, 835540, -1010718, -458184, -3143553, -5410078, -3681429, -6799008, -1944826, -3519400, -484394, -854926, -53807, -93298, -2067, -3546], [-62288, 46328, -2008890, 2458676, -6129860, 6300400, -6757415, 6124374, -3429614, 2828070, -836808, 645754, -92038, 68038, -3519, 2536], [-9792, -46328, -2008890, -2458676, -6129860, -6300400, -6757415, -6124374, -3429614, -2828070, -836808, -645754, -92038, -68038, -3519, -2536], [29526, -835540, 2849506, 458184, 7257049, 5410078, 7172567, 6799008, 3421368, 3519400, 806556, 854926, 87053, 93298, 3295, 3546], [-77580, -27314, 5987910, -1930112, 16909776, -5200301, 17242983, -5426998, 8220721, -2683097, 1917437, -645475, 204753, -70430, 7692, -2681]];
    Rf_basisdens := [1, 1, 136, 272, 52496, 52496, 52496, 52496, 52496, 52496, 52496, 52496, 52496, 52496, 52496, 52496];
    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_170_k();"
function MakeCharacter_170_k()
    N := 170;
    order := 8;
    char_gens := [137, 71];
    v := [8, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_170_k_Hecke();"
function MakeCharacter_170_k_Hecke(Kf)
    N := 170;
    order := 8;
    char_gens := [137, 71];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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+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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, -1, -1, 1, 0, 0, 1, -1, -1, 0, 0, 1], [-1, -1, -1, 0, 0, 0, -1, 0, 0, 1, 0, -1, -1, -1, 0, -1], [1, 1, 0, 1, 0, 2, 0, 2, -1, -1, -1, 1, 1, -1, -1, 1], [1, -1, 1, 0, 1, -2, 0, 0, 1, -1, 0, -1, 1, 0, -1, 0], [0, 1, 1, 0, 1, 0, 0, -2, 0, 1, 0, 0, -1, 0, 1, -1], [-1, 1, -1, 1, 0, 2, 2, 2, -2, 1, 1, 0, -1, 1, 0, 0], [0, 1, -1, 1, 0, -4, 0, 0, 0, -1, 0, -1, 1, 1, 0, 0], [1, 2, 1, 0, 2, 1, -3, 2, -1, 0, -2, 0, -2, 1, -1, 1], [-1, 1, -2, 1, 3, 0, -1, -4, 0, 1, -2, 0, -1, -2, 0, -2], [-2, 0, 0, 1, 4, 0, 2, 0, 0, 2, -1, 2, 0, 0, 0, -2], [-4, 2, -2, -2, -4, 0, -1, -2, -1, 0, 0, 1, 0, 4, 2, 2], [0, 1, 0, -1, -2, -2, 3, 1, 1, -3, -3, -2, 0, 0, 2, -1], [-1, 0, 0, 4, 2, 5, -4, -1, 1, -3, 0, 1, 3, 0, -1, 1], [0, 1, -1, 1, 0, 3, 3, -1, 0, 0, -1, 0, 0, 1, -1, -1], [-3, -3, 1, -4, -3, -3, -1, 0, 0, 3, 4, -1, -1, 3, 3, 1], [1, 0, 0, 0, -4, -4, 1, -1, 3, 0, 0, 1, 1, 1, 1, 3], [1, -1, 0, 2, 3, 0, -4, -4, 0, 2, 0, 0, 1, 0, 0, 1], [4, 0, 3, 0, -4, 3, -4, 0, 1, 0, 1, -1, 0, -3, -1, 0], [4, -1, 1, -2, 0, -2, 1, -2, 2, 2, -2, 1, 1, -1, 0, 1], [2, -2, -2, -2, -1, -3, 5, 0, 2, 2, 0, 2, 2, 0, 1, -1], [0, 3, 0, 3, 4, 0, 7, 7, 0, 1, 1, 2, 0, 0, -2, 0], [2, 1, 0, -2, 4, -3, 4, 0, 1, 2, 1, 1, 1, 0, -1, 0], [4, 0, 0, 0, -1, -1, 2, -2, -1, -4, 4, -1, 0, 0, -1, -1], [-2, 0, 0, 0, 4, 4, 2, -2, -2, 4, -4, 0, -2, -2, 0, -2], [0, 1, 2, -3, -5, 0, -5, 0, 0, 3, -3, -2, 1, -2, 0, 0], [0, -1, -1, -1, 1, -1, 0, -2, 0, 0, 1, 0, 1, 1, 3, 0], [-7, 0, -1, -4, 0, -1, -5, -1, -1, -5, -4, 0, -1, 0, 0, 0], [3, 1, -1, 4, 0, 1, -3, 1, -4, 0, -5, 4, 0, -1, 1, 1], [-7, 2, -2, 4, -1, -1, -1, 0, -1, -4, -7, 4, 2, 2, 1, 0], [-4, 0, 4, 0, 0, 0, 2, -2, 1, -1, 1, 2, 0, 0, 2, 1], [-1, 0, 2, -6, -1, -3, 5, -4, 3, -6, 3, 0, 0, 2, 3, -4], [-2, 2, 0, 0, 7, 3, 0, 8, -4, -1, -1, 5, -2, 2, -5, 4], [-4, 0, 0, 2, -4, 4, -6, 8, -2, -2, 0, -2, -2, 0, -4, 4], [-1, 2, 0, 4, -1, 1, -7, 5, -6, -3, -3, 0, -1, 1, 0, 6], [5, 1, 0, 3, 4, -1, -2, 5, 0, -5, -3, -2, 0, -1, -4, 2], [-4, 2, 2, 0, 6, 2, -2, -6, -2, 0, 4, 0, -2, 2, -2, -4], [0, 4, 0, 4, 6, 2, -2, -2, -2, 4, -4, 0, -4, 0, -2, -4], [0, 0, 0, 0, -8, 2, 4, -8, 2, 0, 4, -2, 0, 2, 0, 0], [5, 0, 4, 7, 0, -4, 1, -4, 0, -3, 7, 1, 4, 0, 0, 1], [1, 2, 0, 6, 2, -4, 7, 7, 0, -4, -4, -1, 1, -1, 1, 0], [2, -2, 1, 4, 0, -3, 1, -3, 2, 0, 4, -3, 1, -2, 0, -3], [0, -3, -2, -1, -4, -1, 4, -5, 0, 0, 1, 4, 2, 3, 6, -4], [3, -3, 4, -2, 0, -1, 0, 0, -3, 2, 2, -5, -3, -4, 3, 0], [-6, -1, -2, -2, -4, 5, 0, -1, 0, 6, 2, 0, 2, 1, -4, 0], [-4, -3, -3, 0, 0, 3, -4, 2, -1, 1, 0, -1, -1, 0, -2, 2], [2, -2, 2, 0, 0, 4, -1, 4, 3, -2, 0, 0, 2, -2, 0, 0], [7, -4, 4, 3, 5, -5, -1, 4, 4, 0, 2, -4, 0, -4, -1, -1], [2, -5, 0, -4, -2, -3, -2, 0, -3, 4, 7, -1, -5, 0, 3, 0], [1, 0, 4, 0, 5, 5, 3, -3, 1, 0, 0, 2, 1, 1, 2, 1], [-5, 0, 0, 6, -2, 2, -2, 4, -2, 6, 5, 0, 0, 0, -2, 4], [-3, 0, 0, -2, -1, 1, -3, 9, -6, 8, 8, 0, -3, 3, 0, 6], [4, 5, 5, 3, 1, 7, -4, 1, 3, -10, 0, 3, 2, 0, -2, 2], [-1, 3, -3, 2, 0, 3, 5, -1, 2, 0, -3, -2, 0, 1, -1, -1], [-3, 5, -3, -2, -2, 2, -1, 4, -2, -2, -5, 0, -5, -3, -2, 3], [0, 0, 4, 0, -4, -4, 2, -2, -2, -2, 2, -2, -4, -4, -2, -2], [-2, 2, -4, -6, 4, 8, 4, 0, 2, 6, 0, 0, 2, 4, -2, 0], [-6, -1, 1, -7, -7, -8, -6, -1, 2, 0, 1, -2, 0, 5, 6, 6], [0, -1, 1, 0, 0, -6, 11, -6, -1, -2, 0, -2, 1, -1, 0, -2], [0, 5, 0, -7, 5, 1, 5, 3, 1, 2, 2, 3, 0, 0, -3, -1], [-17, 0, -2, 0, -4, -4, -3, 3, 2, -2, 2, -1, 1, 1, -1, 2], [8, 1, 5, 14, 1, -5, 1, 0, 3, -14, 2, -2, 1, -5, -3, 0], [4, 1, -4, 4, -4, 7, 5, -11, 0, -4, -4, 5, 4, -1, 4, -5], [9, -1, 1, -1, 0, -2, -6, -2, -2, 7, -1, -4, 1, -1, 0, -4], [-2, -2, 2, -4, -6, -11, -6, -6, 5, 0, 17, -5, 0, 5, 0, 0], [0, -8, -3, -4, 3, 2, 0, 4, -2, -4, 5, 0, 8, -3, -2, -3], [-4, -2, 0, -10, -7, -5, 3, -2, 0, 4, 10, 3, 0, 2, 7, -3], [-6, 6, -6, 0, 15, 7, 4, 12, -3, 0, -6, 3, 0, 2, -3, -3], [1, -4, 0, 6, -6, 0, -3, -5, 1, 0, 0, -3, 1, -1, 3, -1], [-5, -3, -3, 0, -2, -2, -2, -1, -4, 4, 0, -4, -2, 0, 3, -3], [3, -3, 3, -1, 2, -10, 4, 8, 0, -3, 1, 4, -3, 3, 2, -4], [1, 0, 2, 0, -1, -1, 5, -5, 2, -7, 7, 2, 11, 11, 2, 2], [2, 2, 5, -3, 7, -10, 7, 0, 4, 3, -5, 0, 2, -5, -4, 0], [1, 3, -3, 4, -4, 5, 1, -3, -4, 0, -9, 4, 0, -5, 1, 1], [-6, -2, -2, 2, -2, -2, -2, 6, -4, 8, 0, -4, -2, 0, -4, 4], [-1, 3, -5, 3, 0, -10, 2, -10, -2, 9, 3, 0, -5, 3, 0, 0], [-1, 1, 4, -3, 0, 5, 2, 3, 0, 1, 3, 2, -4, -1, -8, -2], [3, 0, 3, 0, -8, -8, -9, 9, 2, 4, -4, -2, -5, -5, -2, 2], [3, -2, -1, -3, 0, 9, 0, 9, 2, 5, -3, 6, -1, -2, 0, 6], [4, -2, 0, 4, -7, 1, 6, -10, 8, 9, 9, -3, 4, -4, 3, -8], [2, -2, -6, -8, 0, -10, 6, -10, 4, 14, -8, -8, -6, -2, 0, -8], [-5, 6, -6, 1, -2, 6, 1, 0, -5, 0, 2, 5, 0, 8, 2, 2], [0, -4, 6, -12, 4, -4, -6, -6, 4, -12, 10, 0, 4, 6, 4, 2], [-1, 0, -4, 0, 5, 5, -11, 11, 0, 4, -4, -8, -3, -3, -8, 0], [9, 6, 0, 6, 6, -4, -4, 14, 0, -9, -6, -4, 0, -6, -10, 4], [-5, -2, -2, 3, 0, 12, -13, 4, -1, 16, 0, -1, 0, 0, -4, 4], [8, 2, 2, -6, 3, 5, -4, -4, 1, 4, 0, 1, 0, 0, 1, -1], [5, -7, 0, 5, -6, 12, -10, -6, -2, 4, 4, 3, 5, -5, -3, 2], [11, -7, 7, 4, -11, -3, 4, -7, 7, 0, 4, -7, 0, 0, 4, 4], [1, 0, -3, 2, 4, 3, -1, 2, -3, 2, -4, 0, 0, -3, -3, -3], [-2, -2, -3, -5, 0, 11, -3, 11, -6, 4, -5, 7, -3, -2, 0, 7], [-2, -4, -4, -5, 11, -4, 5, -11, 1, 1, 0, 1, -3, 0, 0, 0], [-3, -5, -5, 0, 0, -8, -9, -8, -2, 7, 0, -3, -5, -5, 0, -3], [7, -3, 3, 1, -8, -8, -2, 8, 0, -7, -1, -2, -3, 3, 0, 2], [10, 0, 10, 0, 8, 8, 2, -2, 6, -18, 18, 0, 4, 4, 0, 6], [-4, -2, 3, 2, 0, 5, -10, 5, 4, -10, 2, 2, 3, -2, 0, 2], [-3, 6, 0, 0, -13, 9, -11, 9, -10, 4, 4, -2, -3, 3, 2, 10], [7, 0, -8, 5, -3, 3, 1, 10, -3, 5, -15, 0, 0, -8, -3, 2], [10, 3, 1, -6, -5, 3, 8, 2, -3, -6, -12, 0, -3, 1, -3, -3], [1, 7, 0, -5, 0, 14, -8, -6, -1, -11, -11, 7, 1, -1, -7, 1], [-1, 3, 3, 2, 17, -5, 7, -15, 2, 4, 0, 2, -2, 0, -2, 2], [7, 5, -5, 12, 7, 22, 14, 5, -8, 0, -15, 8, 0, -3, -2, -2], [-9, -6, 0, -10, 0, -6, 1, -2, 0, 9, 10, 1, 0, 6, 8, -1], [14, 0, 2, 0, 9, 9, -2, 2, -5, 6, -6, -9, -6, -6, -9, -5], [6, 2, -2, 8, -14, -12, -6, -10, 6, 0, 10, -6, 0, 2, 4, 4], [-1, -3, -3, -2, 5, -3, 9, -7, 6, 0, 0, 6, 0, 0, 2, -2], [-6, 4, 0, -2, -3, 9, -8, 10, -9, -2, -2, 3, -6, 6, -3, 9], [-6, 7, -2, -6, 0, -5, 0, -5, -8, -2, -6, 6, -2, 7, 0, 6], [0, 3, -6, -2, -2, 15, -2, 0, -3, 2, 3, 2, 3, 6, 3, 0], [0, 0, -5, 0, -10, -10, -8, 8, -5, -2, 2, -1, 0, 0, -1, -5], [-15, -6, -1, -19, 1, 6, 1, 0, 0, 19, -8, 1, -6, 1, 0, 0], [6, -2, 7, 4, -22, -3, -4, -1, 0, -6, -4, -4, -7, 2, 4, 4], [16, 8, 8, -6, 3, -5, 10, 2, 5, 0, 0, 5, 2, 0, -5, 5], [-12, -4, -3, 3, 1, -10, 1, 0, 4, -3, -5, -6, -4, 3, -4, 0], [-2, 8, 5, 3, 0, 1, 4, -8, -1, 3, -1, 0, -8, 5, -1, -4], [-18, 0, -9, 0, 10, 10, -6, 6, -8, 5, -5, 1, 0, 0, 1, -8], [2, 1, 8, 3, 2, -11, -4, 15, 0, -2, -3, -4, -8, -1, -4, 4], [5, 5, 0, -5, -5, -3, 6, -12, 9, -4, -4, -4, 5, -5, 4, -9], [6, 2, 2, 6, 6, -6, 8, -10, 2, -16, 0, 2, 10, 0, 4, -4], [-9, 4, -5, 6, -3, -6, 3, 5, 0, 9, -6, 3, 5, -4, 1, -3], [-7, -3, -2, 1, -21, 0, 17, 0, 0, 1, 8, 0, 3, -2, 0, 4], [0, -5, -3, -6, 8, 0, -9, 6, 0, -6, 2, 0, 5, -3, 0, 1], [-7, 1, -5, -2, 0, 18, 3, 18, -16, 3, -2, -1, -5, 1, 0, -1], [-2, -2, 0, 6, 1, 5, 2, 8, -3, 2, 2, 3, -2, 2, -3, 3], [9, 3, 1, 4, 1, 13, 1, 0, 1, -4, 5, 5, 3, -1, -1, 0], [-5, -2, -5, 4, -3, -4, -3, 0, 0, -4, 2, -3, -2, 5, 0, 0], [2, -1, 0, -11, 0, 4, -1, -3, 1, -5, -5, 2, 2, -2, -2, -1], [-4, -3, -3, -3, -1, 3, -5, 7, -2, 6, 0, -2, -4, 0, -6, 6], [-5, 3, -3, -2, -7, -12, -13, -3, -1, 0, -5, 1, 0, 5, 4, 4], [0, 3, -4, -6, -4, -11, -4, 0, 7, 6, 1, 2, 3, 4, -7, 0], [-5, -2, -3, 14, -2, -3, 3, -6, 3, 14, 4, 0, 2, -3, 3, -1], [14, -8, 8, 6, 5, -6, -5, 10, 1, 0, 19, -1, 0, -1, 5, 5], [1, -4, 0, 4, -13, 13, -7, -11, 2, 13, 13, 0, 1, -1, 0, -2], [0, -1, 0, 3, 6, -6, -1, -9, 4, 14, 14, 0, 0, 0, 0, -4], [12, -11, 11, 1, 3, 7, 9, 1, 2, 0, 10, -2, 0, -4, -2, -2], [0, 6, -4, -8, 0, -14, 0, 0, -2, 8, -2, 5, 6, 4, 2, 0], [6, 7, 7, -3, 13, 5, 1, -1, 6, -14, 0, 6, -4, 0, -12, 12], [-4, 5, 9, 7, -18, 6, -2, -6, 0, 4, -7, -2, -9, -5, 0, 2], [-4, -2, -2, 1, -7, 9, -7, 0, -11, -1, 0, 6, -2, 2, 11, 0], [19, 9, 3, 20, 1, 13, 1, 0, 5, -20, 7, 1, 9, -3, -5, 0], [2, 0, 0, -12, -10, -14, -2, 6, 0, -2, 12, -2, 0, 0, 8, 2], [15, -6, 5, 7, 0, 5, -2, 5, 8, 5, 7, 2, 5, -6, 0, 2], [-16, 6, 0, -2, -17, 7, -8, 2, -5, 2, 2, -5, -16, 16, 5, 5], [16, -3, 8, 2, 0, 5, 13, 5, 7, 0, 2, 2, 8, -3, 0, 2], [-3, 3, 2, -5, -3, -2, 3, -8, 2, -5, 2, 0, -3, 2, 2, 0], [2, 0, -3, -17, 7, -2, 2, 3, 0, -2, 17, 2, 3, 0, -1, -2], [-10, -3, 3, -13, 6, -3, -5, 7, -2, 0, -9, 2, 0, 1, 1, 1], [-2, -6, 0, -10, -4, -12, -4, -8, 2, 10, 10, -8, -2, 2, 8, -2], [-2, -5, 2, -7, -12, -5, -14, -1, 0, 2, 7, -14, -2, 5, 6, 14], [1, -2, 7, 4, -5, -2, -1, 8, 2, 4, 8, 0, 2, 7, 2, 6], [20, 0, 14, 0, -8, -8, -2, 2, 7, 0, 0, 0, -2, -2, 0, 7], [-9, 4, -1, -7, 10, -3, -4, -6, 3, -7, 4, 0, -4, -1, 3, -6], [-2, -1, 1, 10, 0, 6, -11, 6, -13, -4, 10, 0, 1, -1, 0, 0], [0, 2, -4, -2, -22, -16, -2, 4, 0, 0, 2, -2, 4, -2, 12, 2], [-5, 0, -18, 0, -6, -6, 9, -9, -12, 18, -18, 5, 1, 1, 5, -12], [16, -4, 4, -1, 0, 12, -4, 12, -8, 8, -1, 0, 4, -4, 0, 0], [3, 6, 6, 3, -11, -15, 15, 8, 0, -12, 0, 0, 0, 0, 3, -3], [2, -3, 5, 1, 0, 10, 16, 10, 7, -8, 1, 7, 5, -3, 0, 7], [2, 2, 4, 12, -8, -14, -8, 0, 6, -12, -4, -7, 2, -4, -6, 0], [-7, 0, 0, 0, 3, 3, -1, 1, -3, 5, -5, 8, 1, 1, 8, -3], [0, -7, -7, -7, 10, 6, -1, -13, 5, 14, 0, 5, 0, 0, 3, -3], [3, -6, -6, -6, 6, -11, 9, -15, -2, 9, 0, -2, 3, 0, 9, -9], [4, 9, -9, 13, 15, 11, 10, 11, -1, 0, -10, 1, 0, -8, -4, -4], [10, -11, -10, 4, 6, -1, 10, -12, 1, 4, -9, 0, 11, -10, 1, -16], [-9, 2, -1, 6, 17, -4, 17, 0, 8, -6, -10, -5, 2, 1, -8, 0], [2, 3, 6, 5, 0, -7, -2, -7, 12, -10, 5, -2, 6, 3, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_170_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_170_2_k_b();".
// 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_170_2_k_b(:prec:=16)
    chi := MakeCharacter_170_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_170_2_k_b();".
// 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_170_2_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_170_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 48, 776, 1552, 2888, -152, -2892, -304, 1461, 24, -60, -56, 0, 8, 0, 0, 1]>],Snew);
    return Vf;
end function;