
// Make newform 840.2.f.b in Magma, downloaded from the LMFDB on 12 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_840_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_840_f_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_840_2_f_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_840_2_f_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 := [6561, -4374, 729, -972, 810, -864, 639, -210, 74, -70, 71, -32, 10, -4, 1, -2, 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, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-37179, -12393, 23814, 5346, 8640, -1296, 627, -2863, 1025, -517, 160, -176, 74, -50, 55, 13], [9477, 11907, 23166, -4806, -1944, -1296, 763, -1483, 497, 135, -152, -48, 114, 22, -17, 17], [8019, -10287, 4158, 3474, -3372, 1092, -799, 67, -473, 333, -220, 84, -30, 14, -19, 7], [111537, -83835, 486, -39690, 18576, -12744, 12159, -4333, 2741, -1687, 1072, -392, 218, -182, -5, -17], [-102789, -5103, 10854, -10098, 9756, 4104, 1393, -1977, 2127, -683, -276, -184, 10, -126, 45, 19], [-6561, -1458, 486, -972, 189, 648, -198, -62, 73, 46, -34, 8, -5, -4, -1, 2], [-181521, 12393, -31590, 65610, -9936, 24300, -7839, 5251, -3605, 1957, -1936, 284, -218, 182, 5, 71], [-280665, -8019, 33534, 25866, -16596, 29940, -3835, -1129, -421, 1849, -1892, 68, -62, 22, -47, 75], [194643, -86751, 71928, -60588, -1242, -36666, 19959, -9199, 5975, -1291, 2446, -914, 704, -140, 19, -131], [-60021, -4617, 2592, 12096, -3570, 4982, -1305, 483, -577, 651, -362, -34, -72, 24, -13, 15], [-26244, 486, -1296, 1917, -2520, 2748, -884, 217, -68, 215, -184, 28, -16, -1, -4, 6], [-305451, 124659, -28674, 51786, -16992, 32376, -17941, 5717, -2839, 1951, -2048, 440, -302, 70, 7, 57], [-131220, 20412, -23571, 20817, -3699, 13437, -6489, 2371, -824, 940, -775, 185, -209, 11, -13, 35], [-1047573, 44469, -118098, 124578, -111564, 156204, -38103, 7943, -10273, 15569, -9980, 2332, -1102, -2, -491, 427]];
    Rf_basisdens := [1, 1, 46656, 46656, 15552, 34992, 23328, 2187, 69984, 46656, 69984, 7776, 2916, 23328, 11664, 139968];
    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_840_f();"
function MakeCharacter_840_f()
    N := 840;
    order := 2;
    char_gens := [631, 421, 281, 337, 241];
    v := [2, 2, 1, 2, 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_840_f_Hecke();"
function MakeCharacter_840_f_Hecke( : Kf := false)
    N := 840;
    order := 2;
    char_gens := [631, 421, 281, 337, 241];
    char_values := [[1, 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, 0], [-1, 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, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 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, 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, 1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, 0], [0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, 0], [0, 0, -1, -1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0], [2, 1, 0, 0, -2, 0, -1, -1, 0, -2, -1, 1, 2, -1, 0, 0], [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [2, 0, 1, -1, 0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, -1, -1, 2, 1, 0, 1, 1, -1, 0, 1, 0, 0, -1, -1], [-2, -1, 1, -1, 1, 0, 0, 1, 0, 1, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, -1, 2, -1, 1, 0, 0, 0, 1, 0], [0, 1, -2, 2, -1, -1, 1, -1, 0, -1, -1, 1, 0, -1, 0, 0], [0, 1, -2, -2, -1, -1, 0, 0, 1, 1, 0, -1, 0, 0, 1, 0], [0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 0, 0, 2, -1], [0, 2, 1, -1, -1, 0, 0, -2, 0, -1, 1, -1, 2, 0, 0, 0], [2, 1, 1, -1, -2, 0, 0, -1, 0, -2, 0, 0, 2, 0, 0, 0], [-2, 2, 0, 0, 0, 0, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0], [0, 1, 2, 2, 0, 1, 0, 2, 1, 1, -1, -1, 0, 0, 0, -1], [2, 0, 0, 0, 2, -1, 0, 0, 0, 2, 0, 0, -2, -2, 0, 0], [0, 0, 1, 1, 0, -1, 0, 2, 0, 0, -1, 1, 0, 0, -2, 0], [0, -2, 0, 0, -2, -1, 0, -1, -4, 2, -1, 0, 0, 0, -1, 0], [2, 0, 1, -1, -1, -1, 2, 0, 0, -1, -1, 1, 0, 0, 0, 0], [0, 1, 3, 3, 0, 0, 0, 1, 1, 1, 0, -1, 0, 0, 1, -1], [-2, 1, 1, -1, 0, -1, 2, -1, 0, 0, -1, 1, 2, 0, 0, 0], [2, -1, -1, 1, -1, -1, 0, 1, 0, -1, 0, 0, 0, -2, 0, 0], [0, -1, 1, 1, 2, 2, 0, 1, -1, 1, 0, -1, 0, 0, 1, -3], [0, 1, 2, 2, 0, 1, 0, -1, -1, -2, 0, 0, 0, 0, 0, 2], [0, 0, 2, 2, -1, -2, 0, -1, -2, -1, -2, -1, 0, 0, -1, 2], [0, 0, -3, 3, 1, 0, 2, 0, 0, 1, 1, -1, -2, 2, 0, 0], [0, -1, 1, 1, -1, 0, 0, 0, -3, 0, -1, 1, 0, 0, -2, 1], [-2, 0, 3, -3, 0, 2, -3, 0, 0, 0, -2, 2, 2, 1, 0, 0], [-6, 2, -2, 2, 1, -1, 3, -2, 0, 1, 1, -1, -2, 1, 0, 0], [-2, 2, 1, -1, 1, 0, 0, -2, 0, 1, -1, 1, 2, 0, 0, 0], [0, 0, 3, 3, 1, 2, 0, 2, 1, 1, -1, -1, 0, 0, 0, -2], [0, 1, 1, 1, 2, 2, 0, 0, 2, -2, 2, 1, 0, 0, 1, 0], [0, 1, -3, -3, -4, -1, 0, 0, -1, 2, -1, 0, 0, 0, -1, 2], [-2, 2, -2, 2, 0, -1, 2, -2, 0, 0, 0, 0, -2, 0, 0, 0], [0, -1, 3, 3, 2, 0, 0, -1, -3, -1, 0, -1, 0, 0, 1, -1], [0, 0, 1, 1, 2, 1, 0, 3, 1, 0, 0, 1, 0, 0, -1, -2], [4, 0, 3, -3, -1, 2, -2, 0, 0, -1, 1, -1, -2, 2, 0, 0], [0, -4, -1, -1, 0, 1, 0, -2, -2, 0, -1, 1, 0, 0, -2, 0], [-6, 3, 0, 0, -3, 1, -2, -3, 0, -3, 1, -1, 2, 0, 0, 0], [0, -1, 3, 3, -2, -3, 0, -2, -2, 2, 0, -1, 0, 0, 1, 0], [0, 1, -1, -1, 0, 1, 0, 1, -1, -1, 2, 3, 0, 0, -1, 1], [0, -1, -3, -3, 0, 1, 0, 0, -3, 2, 1, 0, 0, 0, 1, -2], [0, -3, -1, -1, 1, -2, 0, -1, 0, -1, -1, 1, 0, 0, -2, 0], [2, -1, 1, -1, -1, 0, -2, 1, 0, -1, 0, 0, -2, -2, 0, 0], [-4, -2, 1, -1, 1, -1, 1, 2, 0, 1, -1, 1, -2, -1, 0, 0], [0, 4, 4, 4, 2, 3, 0, 3, 4, 0, 3, 0, 0, 0, 3, -2], [4, -1, -3, 3, 3, -1, 4, 1, 0, 3, 1, -1, 0, 2, 0, 0], [0, 0, 1, 1, 0, -2, 0, -1, 1, 0, 2, 1, 0, 0, 1, 0], [-2, -1, 4, -4, 0, -1, 1, 1, 0, 0, 1, -1, -4, -1, 0, 0], [0, 1, 2, 2, 0, -2, 0, 1, 2, -2, -2, 0, 0, 0, -2, 2], [0, 0, 1, 1, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 2, 0], [2, 0, 1, -1, 1, 0, 0, 0, 0, 1, -1, 1, 2, 0, 0, 0], [0, 2, -1, -1, 4, 4, 0, 4, 2, 0, 2, 0, 0, 0, 2, -4], [-4, -2, -2, 2, 2, -1, 2, 2, 0, 2, 2, -2, -2, 0, 0, 0], [0, -1, -4, -4, -2, -3, 0, -2, -1, 1, -1, -1, 0, 0, 0, 1], [0, -1, 3, 3, 2, -1, 0, -5, -1, -3, 2, -1, 0, 0, 3, 1], [0, -2, 2, -2, -2, 2, -4, 2, 0, -2, 0, 0, 0, 0, 0, 0], [2, 2, -6, 6, -2, -2, 2, -2, 0, -2, 2, -2, 0, -2, 0, 0], [0, -4, 2, 2, 2, 2, 0, 1, 0, 0, -1, 2, 0, 0, -3, -2], [0, -1, -5, -5, -6, -1, 0, -2, -6, 6, 0, -1, 0, 0, 1, 0], [8, 0, 2, -2, 2, -1, 3, 0, 0, 2, 0, 0, 0, 1, 0, 0], [0, 3, 2, 2, -1, -1, 0, -4, 3, -3, 0, -3, 0, 0, 3, 4], [0, 4, -1, -1, -4, -5, 0, -2, -2, 0, -1, -3, 0, 0, 2, 4], [-2, -5, -2, 2, 4, 0, -1, 5, 0, 4, -1, 1, -2, -1, 0, 0], [8, 0, 6, -6, 0, 0, -2, 0, 0, 0, -2, 2, 0, -2, 0, 0], [10, 0, -1, 1, -2, 2, -3, 0, 0, -2, -2, 2, 2, 1, 0, 0], [0, 0, -5, -5, 4, 1, 0, 1, 2, -2, 1, 0, 0, 0, 1, -2], [0, -1, 0, 0, 0, -2, 0, 4, 3, 1, -2, 2, 0, 0, -4, -1], [0, -4, -3, -3, -3, -1, 0, -6, -4, 3, 1, -1, 0, 0, 2, 0], [0, 3, -1, -1, -1, -1, 0, 1, 4, 1, -1, -3, 0, 0, 2, 0], [-2, -3, 1, -1, 1, 4, -6, 3, 0, 1, 0, 0, 2, 2, 0, 0], [-4, -1, -4, 4, 4, 0, 0, 1, 0, 4, 2, -2, -4, 0, 0, 0], [0, -5, 1, 1, 0, -3, 0, -2, -1, 2, -1, 0, 0, 0, -1, -2], [0, 1, 1, 1, -3, 1, 0, 0, 1, 0, 0, 2, 0, 0, -2, 3], [0, -2, -7, -7, 0, -3, 0, -1, -3, -2, -2, 1, 0, 0, -3, 2], [0, -4, 0, 0, 0, 2, 0, 1, 0, 2, 1, 4, 0, 0, -3, -2], [0, -1, 4, 4, -2, 0, 0, 1, 0, 6, 2, 0, 0, 0, 2, -4], [2, 0, 2, -2, 0, 1, -2, 0, 0, 0, 2, -2, -2, 0, 0, 0], [-4, -3, -3, 3, 7, -1, 2, 3, 0, 7, 1, -1, -4, 0, 0, 0], [2, 1, 3, -3, -2, 2, -2, -1, 0, -2, 3, -3, 0, 2, 0, 0], [-2, -1, 0, 0, 5, 1, 0, 1, 0, 5, 3, -3, -6, 2, 0, 0], [-4, -2, 0, 0, -2, -1, 2, 2, 0, -2, 0, 0, -2, 0, 0, 0], [2, -1, -3, 3, 4, -1, 2, 1, 0, 4, -1, 1, -2, 0, 0, 0], [0, -2, -4, -4, 0, 0, 0, -2, -3, 0, -2, -2, 0, 0, 0, 0], [4, -2, 3, -3, 1, 0, 0, 2, 0, 1, 3, -3, -2, 0, 0, 0], [-2, 0, 8, -8, 1, 3, -5, 0, 0, 1, -3, 3, 2, 1, 0, 0], [-6, 0, -8, 8, 0, -2, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0], [8, -1, -1, 1, -3, 1, 0, 1, 0, -3, 2, -2, 0, 2, 0, 0], [0, 2, 4, 4, 1, -3, 0, -2, 2, -1, 3, -1, 0, 0, 4, 0], [-2, 4, -5, 5, 1, -1, 0, -4, 0, 1, -1, 1, 0, -2, 0, 0], [-4, -3, 0, 0, 3, 0, 1, 3, 0, 3, 1, -1, -2, 1, 0, 0], [0, 5, -9, -9, 0, 1, 0, -1, 3, -5, 0, -1, 0, 0, 1, 5], [-4, 4, -2, 2, 1, -1, 2, -4, 0, 1, 1, -1, 0, 0, 0, 0], [0, 1, -1, -1, -1, -1, 0, 3, 0, 1, -3, -1, 0, 0, -2, 0], [-4, 4, -4, 4, -4, -2, 0, -4, 0, -4, 0, 0, 0, -4, 0, 0], [0, -1, -2, -2, -4, -1, 0, -4, 1, 1, 1, 1, 0, 0, 0, 3], [4, -4, -6, 6, -1, -3, 6, 4, 0, -1, 1, -1, -4, 0, 0, 0], [-4, 0, 6, -6, -6, -1, 1, 0, 0, -6, -2, 2, 4, -1, 0, 0], [0, -1, 1, 1, 2, -1, 0, -2, 1, -4, -1, 0, 0, 0, -1, 2], [0, -1, -1, -1, 1, 1, 0, -3, -6, -3, -1, -1, 0, 0, 0, 2], [0, -4, -1, -1, 0, -5, 0, -2, 2, 4, -1, -3, 0, 0, 2, -4], [6, 1, 10, -10, 2, 1, -1, -1, 0, 2, -1, 1, 0, 1, 0, 0], [0, 3, 5, 5, -2, 3, 0, 1, 1, -1, 0, 1, 0, 0, -1, 3], [0, -7, -2, -2, 4, -2, 0, -1, -3, 0, -2, 0, 0, 0, -2, -4], [-4, 2, 5, -5, -3, -2, 2, -2, 0, -3, -1, 1, 2, -2, 0, 0], [0, 1, 3, 3, -3, -5, 0, -1, -6, 1, -1, -1, 0, 0, 0, 2], [0, 2, -5, -5, -4, 0, 0, 0, -6, 4, 0, -2, 0, 0, 2, 0], [-14, -2, -1, 1, 4, 0, -1, 2, 0, 4, 0, 0, 2, -1, 0, 0], [0, 1, 5, 5, 0, 0, 0, 3, -1, 3, 2, 1, 0, 0, 1, -3], [0, 0, -4, -4, 0, 2, 0, -2, -1, 0, 2, 0, 0, 0, 2, 0], [0, 0, 0, 0, 3, 1, 0, -3, -4, -1, 1, -4, 0, 0, 5, -2], [-6, -2, -4, 4, 2, 1, 0, 2, 0, 2, 2, -2, -2, 2, 0, 0], [-6, -2, -1, 1, 3, -2, 2, 2, 0, 3, 1, -1, -2, -2, 0, 0], [0, 4, 2, 2, 0, 0, 0, 7, 0, 2, 1, 2, 0, 0, -1, -2], [0, -5, 2, 2, -6, -3, 0, 1, -7, 8, -6, -2, 0, 0, -4, -2], [0, -4, 3, 3, 0, -1, 0, -1, -8, 4, -1, -2, 0, 0, 1, -4], [2, 0, -7, 7, -5, -1, 2, 0, 0, -5, 1, -1, 0, 0, 0, 0], [16, 2, 6, -6, 0, -1, 0, -2, 0, 0, -2, 2, 2, -2, 0, 0], [0, -4, -2, -2, 7, 2, 0, 0, 6, -3, 0, 0, 0, 0, 0, -4], [0, -1, -4, -4, 2, -4, 0, -4, -3, -3, 2, 0, 0, 0, 2, 1], [-8, 0, 3, -3, 1, 0, 0, 0, 0, 1, 1, -1, 2, 0, 0, 0], [0, 4, 0, 0, 4, 3, 0, 1, 0, -2, 3, -2, 0, 0, 5, -2], [-12, -4, 1, -1, 1, -2, 4, 4, 0, 1, -1, 1, -2, 0, 0, 0], [14, -3, 0, 0, -4, 5, -5, 3, 0, -4, -3, 3, 4, 5, 0, 0], [-10, 0, 4, -4, -4, 2, -2, 0, 0, -4, 1, -1, 0, 2, 0, 0], [0, -4, -2, -2, -4, 0, 0, -8, -6, 2, 0, -2, 0, 0, 2, 2], [18, 4, 1, -1, -3, -2, 0, -4, 0, -3, -3, 3, 2, -4, 0, 0], [0, -6, 1, 1, 0, 0, 0, -6, -6, 0, 2, 2, 0, 0, 0, 0], [14, 2, -3, 3, -3, -2, 4, -2, 0, -3, -1, 1, -2, 0, 0, 0], [0, 0, 3, 3, -7, 1, 0, 2, -2, 5, -3, 1, 0, 0, -4, 2], [0, 3, 2, 2, -2, 3, 0, 5, 1, 0, -2, 2, 0, 0, -4, 2], [0, -4, 5, 5, 4, 5, 0, 1, -4, -4, -3, 2, 0, 0, -5, 0], [-2, -1, -1, 1, -2, 6, -6, 1, 0, -2, 3, -3, 4, 6, 0, 0], [0, 6, -6, -6, -2, -1, 0, 1, 4, -2, -1, -2, 0, 0, 1, 4], [0, 2, 4, 4, 5, 4, 0, 3, 4, -3, 5, 4, 0, 0, 1, -2], [-4, 4, 0, 0, -2, -3, 0, -4, 0, -2, 0, 0, 6, -6, 0, 0], [0, 1, 7, 7, -1, 0, 0, -6, -5, -2, -1, -5, 0, 0, 4, 3], [16, -2, 7, -7, -1, 1, -5, 2, 0, -1, 1, -1, 2, -3, 0, 0], [0, -3, -6, -6, 2, -5, 0, -3, -1, -4, 0, 2, 0, 0, -2, 2], [0, 0, -2, -2, 4, 4, 0, -3, 8, -4, 3, 0, 0, 0, 3, 0], [2, 1, -6, 6, 0, 4, -1, -1, 0, 0, 3, -3, -2, 7, 0, 0], [-4, 3, 1, -1, -7, 0, 0, -3, 0, -7, 0, 0, 2, 0, 0, 0], [8, -1, 2, -2, 3, -2, 5, 1, 0, 3, -1, 1, 2, 1, 0, 0], [20, 5, -4, 4, -1, -1, 1, -5, 0, -1, -1, 1, 4, -1, 0, 0], [4, 1, 0, 0, 7, 4, -3, -1, 0, 7, 5, -5, -2, 5, 0, 0], [0, -5, 0, 0, -3, -3, 0, -2, -7, 3, -2, 1, 0, 0, -3, 0], [12, 4, -3, 3, -5, 0, 0, -4, 0, -5, -3, 3, 6, 0, 0, 0], [-8, -5, -7, 7, 1, 2, -2, 5, 0, 1, 0, 0, 2, 2, 0, 0], [0, 3, 6, 6, 6, 7, 0, 6, 13, -1, 3, 1, 0, 0, 2, -5], [0, -5, 1, -1, 1, 2, 0, 5, 0, 1, -1, 1, -2, 4, 0, 0], [0, 0, -4, -4, 0, -2, 0, -3, 0, 2, 1, -4, 0, 0, 5, -2], [0, -3, 5, 5, -4, 2, 0, -1, -3, 5, 0, 1, 0, 0, -1, -1], [-16, 0, 6, -6, 7, -2, 0, 0, 0, 7, 0, 0, -4, -4, 0, 0], [-2, -1, -5, 5, -1, 3, -2, 1, 0, -1, -2, 2, 4, 4, 0, 0], [0, 5, 5, 5, -6, 2, 0, 3, 3, 5, 2, 1, 0, 0, 1, 1], [-8, -3, 3, -3, 2, 0, -1, 3, 0, 2, -2, 2, -4, -1, 0, 0], [4, 4, 2, -2, -6, 0, -2, -4, 0, -6, -6, 6, 8, -2, 0, 0], [0, -1, 2, 2, -6, -6, 0, -8, -5, 3, 2, -2, 0, 0, 4, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_840_f_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_840_2_f_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_840_2_f_b(:prec:=16)
    chi := MakeCharacter_840_f();
    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_840_2_f_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_840_2_f_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_840_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<17,R![1856, 4640, -1232, -4048, 1612, 98, -79, 0, 1]>],Snew);
    return Vf;
end function;