
// Make newform 3380.2.a.r in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3380_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_3380_2_a_r();".
// 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_3380_2_a_r();".
// 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 := [1, -26, 149, -153, -87, 106, 16, -19, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0], [226, 1620, -3215, 214, 1964, -394, -241, 42, 5], [-79, 1873, -5103, 1512, 2683, -969, -256, 93, -1], [773, -6805, 6347, 3590, -4073, -543, 678, 27, -33], [-1645, 8288, -13452, -3976, 9369, 614, -1649, -38, 84], [82, 9473, -19082, -5450, 12070, 1055, -2096, -73, 108], [-1292, 13473, -16922, -7780, 11478, 1551, -2086, -107, 112], [3551, -44995, 53223, 25992, -35175, -4731, 6206, 295, -323]];
    Rf_basisdens := [1, 1, 338, 338, 338, 338, 338, 338, 338];
    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_3380_a();"
function MakeCharacter_3380_a()
    N := 3380;
    order := 1;
    char_gens := [1691, 677, 1861];
    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_3380_a_Hecke(Kf)
    return MakeCharacter_3380_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+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], [0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 1, -1, 0, 1, 0, 0, 0], [0, 1, 0, -1, 0, 1, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 1, -1, -1], [-1, 2, 1, 0, -2, 1, 0, 0, 0], [1, -1, -1, -1, 0, 0, 0, -2, 0], [2, -1, 0, 1, 1, 1, -1, 0, 0], [-3, 2, 2, 0, -2, 1, 1, -1, 1], [0, 0, -1, 0, -2, 0, 1, 1, -1], [2, -1, -2, 1, -2, -1, 1, -1, 2], [2, -1, 1, -1, 4, -1, -1, 3, 1], [3, -1, -1, 1, 2, 0, 0, 2, 2], [4, -1, -1, 1, 0, 1, 0, 1, -1], [0, 2, 1, 0, 4, 0, 1, 3, -1], [3, -3, -2, 1, -4, -2, 0, 0, 1], [-4, -2, 0, -2, -4, 2, -1, -6, -2], [1, -3, -3, 1, -2, 0, -1, 0, -2], [4, 0, 2, 2, 2, 0, 2, 0, 0], [2, 1, 2, -3, 2, 1, -1, 2, -2], [-6, -2, 1, -1, 2, -1, -2, -1, -1], [5, 1, -1, 3, -3, -1, 0, 0, 1], [3, 4, 1, 0, 4, -1, 2, 6, 2], [-2, -2, 1, 0, -3, -1, 0, -7, -2], [3, -1, 1, 3, 2, 1, 0, -2, -2], [-2, 1, 1, -1, 4, 1, 0, -5, -1], [1, 0, -2, 2, -4, -1, -1, 0, 1], [-1, 0, 1, -4, -2, 1, -2, -2, -2], [8, 2, 0, 0, 4, 1, 1, 4, 2], [3, 1, 0, -1, 0, -2, 2, 9, 1], [-10, 1, -1, -1, -4, 1, 0, -3, -1], [-1, -1, -2, 5, -2, 0, -2, 3, -1], [1, 5, 3, -3, -1, 0, -1, 2, 0], [3, 0, 3, -2, 4, 1, 0, 4, 0], [-2, -2, -1, 0, -8, 2, -3, -3, -1], [7, -2, 2, 1, 6, 0, 0, 7, 1], [4, -2, -3, 2, 4, 1, -2, 5, -1], [3, 6, 0, 0, 0, 1, 1, 7, -1], [-3, 2, 1, 2, -6, -1, 0, -8, 2], [8, -2, -2, 0, 3, -3, -1, -2, -1], [-1, 4, 3, -2, -4, 1, 0, -2, 2], [0, -3, 0, -3, 0, 3, -3, 0, -2], [-8, 0, 2, 2, -2, -2, -2, 2, 2], [10, 0, -3, 0, 2, 2, 3, 1, -1], [-4, 2, 2, -6, -4, 0, -2, -4, 2], [1, -2, 3, -1, -4, 0, 1, -4, 2], [-1, -5, -4, 0, 4, 1, -1, 7, -1], [9, 3, 3, -5, 3, 0, -3, 0, 2], [2, 4, 1, 2, 4, 0, -1, 3, 3], [5, 4, -3, 0, 0, -1, 2, 8, 0], [-2, -3, 0, -1, 5, 2, 0, 0, -3], [-9, 5, 3, -1, 0, 0, 1, 0, 0], [4, 2, -2, 4, 2, -2, 0, 6, 0], [-5, 2, 0, -6, -2, 2, -2, -5, -5], [7, 6, -3, 4, 1, -1, 6, 4, 2], [-1, 3, 4, 1, 4, 0, 2, 1, -1], [11, 7, 1, -3, 4, 0, -1, 2, 4], [2, 1, 2, 3, -1, 1, 3, -8, 0], [0, -4, -4, 5, 6, -1, 3, -2, -4], [-5, 3, -1, -1, 2, 2, 1, 0, 0], [-14, -4, 0, 1, -6, 1, 1, 0, -2], [-2, 0, -1, 4, 0, 2, 1, 3, 1], [7, -4, 0, 0, 10, -5, -1, 1, -1], [-11, 5, 5, -1, 4, 0, -1, -2, 2], [-9, 1, -2, -3, -8, -2, 2, 1, 3], [11, 1, 2, 3, 0, 0, 0, -1, -3], [-4, -1, 1, 0, 0, 0, -5, -5, 1], [1, 1, -1, 5, 0, 0, 1, -2, 0], [0, -4, -4, 2, -2, 0, 2, 0, 2], [12, 2, -2, 4, 6, -2, 2, 12, 2], [-5, 7, 2, 4, -8, 1, 1, -3, 5], [16, -3, 1, 1, 4, -3, 2, 9, 3], [7, 1, 3, -3, 4, -6, -1, 8, 2], [7, -2, -5, 3, -8, -2, -1, -6, 4], [-18, -6, -1, -2, 1, 1, -4, -9, -6], [-7, -1, -4, -1, -4, -4, -2, -11, -1], [3, -6, -2, 0, 2, -3, -1, 0, -1], [0, 7, 2, -3, -6, -3, -1, -5, 4], [2, -3, -3, 7, -4, -7, 0, 3, 3], [6, -3, 1, 3, -5, -3, -2, -3, 1], [-5, -6, -4, 0, 0, -1, -3, -9, -5], [7, -2, -5, 6, -6, -5, 2, 2, -2], [3, -5, -2, -3, 14, 0, -6, -5, -3], [-3, 1, 3, -2, 18, 1, -2, 6, 2], [6, 1, 3, 3, 3, -3, 0, 7, 3], [6, 4, -5, 0, 6, 2, 3, 15, -1], [11, -1, 2, 1, 3, -5, 3, 7, 4], [8, -5, -4, 1, -2, 5, -1, -2, -6], [1, -3, -2, 3, 4, 0, -5, 5, -3], [7, -5, -3, 5, 6, -4, -3, -12, 0], [-18, 0, 1, 0, 6, 2, -2, -9, -1], [-4, 9, -2, 1, -8, 5, 1, 2, 4], [1, 2, 7, -2, -4, 1, -2, 2, 0], [-12, -1, -3, -1, 2, 4, -1, -7, -1], [1, -3, -2, 1, 5, 2, 2, -3, -5], [-14, 1, 5, -7, -9, 4, -1, -17, -2], [-8, -4, -4, 4, -16, 0, 1, -6, -2], [6, -4, -1, 4, -13, -6, 1, 7, 7], [-13, 5, -2, 1, -8, 3, 3, -7, 1], [3, 9, -1, 3, 8, 2, 3, 12, -2], [-2, 2, -2, 0, 0, 3, 1, 0, 2], [-7, -2, -7, -4, -18, 1, 0, -11, -2], [-16, 5, 4, 1, -2, 5, 1, -8, 0], [-8, -1, 5, -1, -2, -1, 4, -17, -1], [5, 5, -3, 3, 0, 1, 4, 2, -2], [-7, -7, -6, 5, -8, -4, 0, 1, 1], [6, -3, -2, 5, 10, -5, 3, 0, 0], [3, 2, 3, 0, -10, 7, -4, -1, -2], [7, 0, 0, 6, -10, -5, 4, 5, 3], [3, 0, 2, 4, -4, -5, 3, 11, 3], [10, 5, 0, -1, 16, -1, 5, 6, 0], [13, -3, -1, 7, -6, -2, 1, 8, 0], [5, 7, 5, -3, 6, 0, 5, -4, -4], [-7, 1, 1, 1, -10, 4, 1, 1, 4], [-7, 5, 3, 1, -2, 4, 1, -8, -6], [1, -2, 4, -4, 12, 5, 1, 13, -5], [-1, 3, 7, -1, 2, 4, -1, 2, 0], [-9, 3, -3, 3, 4, 2, 3, 8, 0], [0, -11, -8, 7, -1, -2, 0, -2, -3], [16, -4, -9, 4, -4, -4, 3, 1, -3], [-24, -8, -1, -4, -14, 0, -3, -17, -1], [0, 6, 6, 0, -2, -8, 1, 10, 0], [4, 1, -4, 1, -12, 7, -1, 6, 2], [6, 2, -4, 2, -6, -2, 4, -1, 6], [16, -5, -2, -3, 12, -5, 3, -5, -4], [-9, -1, -3, 1, -6, -2, -5, -4, 2], [7, 6, 5, -2, 6, -5, 1, 10, 2], [7, 9, -1, -3, 0, -2, -3, 6, -2], [-8, -4, 1, -6, 0, -4, 1, -3, 5], [12, 11, 2, -3, 18, 4, 4, 6, 0], [19, 9, 2, 5, 18, -4, 0, 13, 1], [2, -11, -5, -1, 10, 1, -4, -9, -9], [2, -3, 5, 3, -3, 3, -2, 3, -1], [23, 6, -1, 0, 8, 1, 4, 13, 6], [-10, -4, -1, -6, 4, -4, -5, -7, 1], [-4, 5, -2, -5, 2, 3, -1, -8, 0], [-13, 3, 0, -5, 4, -2, 2, -9, -3], [7, 7, -5, -1, -4, 2, -1, 13, 0], [-1, -5, -6, 13, -4, 0, 6, -7, -5], [16, -1, 5, -3, 5, 0, 7, 9, 4], [9, -7, -1, -3, 6, -8, -3, 0, 2], [-16, -4, -5, 5, -12, 3, -2, -5, 1], [2, 1, 7, 3, -5, 1, 0, 7, 7], [6, -3, 2, -3, 0, 5, 1, 2, -4], [-1, 5, -1, 1, -14, -6, 3, -2, -2], [12, -8, -10, 10, -2, -4, 8, 4, 0], [6, -1, 0, 5, 20, 1, 1, 6, 2], [7, -1, -2, -2, 2, -1, -3, 13, -1], [8, -2, -7, 0, 10, -4, 1, -1, 3], [-10, -1, 0, -3, 8, 3, -3, 2, -10], [-20, 8, -1, -8, -16, 6, -7, -19, -3], [-9, 7, 0, -1, 12, 3, 1, -1, -5], [6, 6, 3, 2, 16, -5, 2, 29, 7], [2, 10, 0, 0, 8, 2, 0, 8, -6], [-2, 8, 10, -10, 4, 6, 0, 6, -2], [19, 0, -6, -2, 1, 0, 6, 1, 2], [-8, -1, 3, -5, -2, -7, -4, -3, 5], [1, -3, 1, -11, 4, 2, -3, -9, -8], [-16, -5, -5, 3, -10, 2, -1, 3, 1], [-3, 7, 1, -7, -14, 6, -1, 2, 4], [-12, -7, 1, -1, -8, -5, -2, -7, 5], [15, -5, -9, 3, 12, -4, -3, 8, 4], [-21, 8, 8, -8, -4, 5, -1, 3, -3], [1, 15, 5, 4, 6, -1, 6, 12, 4], [-3, 4, -5, -2, -18, -1, 0, -6, -2], [-4, 3, 9, -13, -6, 7, -4, -3, -3], [8, -4, -2, 10, -3, -7, 3, 2, -3], [-2, 3, -6, -5, 8, -1, 7, 14, 0], [16, -1, -7, -3, 0, 1, 0, -1, 1], [9, 0, 1, 10, 4, -3, -2, 1, 2], [-7, 4, -3, 8, -18, -5, 8, -16, 4], [-19, 1, 4, 5, -12, -4, 2, -3, 3], [-7, 2, -2, -4, 16, 1, -3, -7, -3], [-22, 1, 8, -9, 11, -1, 1, -8, -4], [-12, -10, -6, -2, 8, -2, -2, 0, -4], [1, -8, -5, -4, 13, -2, -7, -18, -1], [21, 2, 2, 4, 16, 1, -1, 15, 5], [7, -3, 1, -7, -9, -3, 0, 0, 1], [2, 2, -5, 6, 12, -5, -2, 17, -1], [-19, -5, -5, -1, -16, -4, -5, -16, -2], [31, 11, 0, 9, 12, 0, 8, 19, 3], [0, -8, -6, -4, -2, -12, -4, -2, 6], [-6, -10, -4, -1, -10, 7, -5, -28, -10], [-1, -7, -5, 9, -4, -4, 1, 11, -4], [13, 4, 5, -8, 0, 11, -6, -4, 2], [-5, -2, 2, -4, -8, -4, -2, -29, -1], [-9, 0, 0, -2, 11, -3, -1, -7, -9], [20, 0, -6, 10, -14, -8, 8, 6, 6], [31, 11, 3, 1, 6, 0, 5, 18, 2], [18, 6, 5, 4, 18, -7, -2, 13, -1], [13, 10, 0, 0, 16, -1, 7, 11, 1], [8, -12, -4, 0, -11, 6, -4, 2, -2], [-8, 13, 2, -1, -2, 7, -1, 8, 4], [-17, -1, 8, -3, -10, 2, 0, -29, -1], [12, 2, -3, -2, -8, 0, 3, -8, 11], [12, -7, -3, 5, 0, -1, -2, -1, 3], [-1, -1, -4, 3, 4, -4, -6, -5, 5], [-16, -7, 9, 0, -10, 0, -7, -17, -1], [-5, -2, 2, -10, 1, 8, -2, 1, -8], [30, 0, -8, 12, -4, -8, 2, 14, 2], [10, 9, 4, 9, 0, 3, 1, 6, -2], [26, -15, -5, 7, 10, -1, -4, 0, 1], [10, -7, -10, -5, -4, -1, -3, -14, -6], [-18, -5, 8, -1, 0, 9, -7, 4, -2], [-4, -2, 1, 0, -4, 4, 3, -19, -13], [-31, -5, 1, 1, -26, -2, -2, -14, 2], [5, -8, -3, -2, 13, -3, -10, -4, -2], [3, -2, 2, 2, -10, 1, -3, 21, 7], [23, -7, -1, -1, 14, -2, -1, 18, 0], [9, 5, -3, 1, -14, -8, 1, 3, 8], [11, -10, -8, 1, -2, -8, -6, 17, -1], [-14, -5, -6, -3, 12, 5, -2, -12, -12], [11, 5, -3, -5, -2, 4, 3, 2, 0], [16, -16, -4, 0, 14, -6, -4, 21, -4], [4, -9, -13, 6, -12, 2, -5, -7, -9], [-3, -8, -4, -6, 14, -3, -7, -2, 3], [-9, -6, 0, -12, -10, 1, -4, -19, -1], [1, 12, 1, -10, 2, 13, 0, 8, -2], [22, -1, 9, 5, -7, -5, 4, -7, 1], [5, -14, -2, 6, 2, 3, -1, -3, -1], [-3, 10, 13, 2, 13, -4, 1, 12, 5], [-2, 1, 8, -3, -10, -3, -4, 8, 6], [3, -1, -2, -9, -6, -2, -3, -3, -5], [13, 4, 7, 6, -7, 3, 8, -16, -6], [-16, -10, -2, 2, 0, 8, 0, -26, -8], [11, 11, 5, 3, 10, 0, 7, -2, 6], [20, -4, 1, -1, -6, -3, 8, 1, 5], [22, -1, -1, -1, -2, -1, 2, -13, -7], [13, 0, 6, -8, 6, -1, 1, 1, 1], [-18, 1, 1, 5, -16, 9, 4, -27, -5], [-9, 5, -8, 3, -16, -8, 0, -13, 5], [3, 3, 4, -5, 7, 8, -8, -5, -3], [3, -8, -2, -2, 0, -7, -1, -7, -5], [4, 7, 4, -11, 0, -5, 4, 0, 0], [11, 6, -1, -8, 15, 14, -3, 4, 1], [-10, 5, 2, -1, -8, 5, -1, -16, 0], [2, -10, -4, -2, 6, 0, 0, -12, -10], [18, -5, 1, -1, 6, 3, 6, 13, -3], [17, 10, 9, -12, 20, -1, -1, 22, 8], [4, -16, -12, 2, 4, -6, -2, -18, 2], [3, -10, -5, -4, -2, 4, -5, 2, -2], [11, -6, -2, 6, 9, 0, 2, 1, 0], [-18, -1, -4, 7, 2, 5, -1, -8, -4], [10, 0, 13, -4, 4, 2, 1, 1, 7], [1, 0, -2, -6, 12, -4, 2, 5, 1], [5, 11, 13, -7, 14, 0, -5, 16, -4], [7, -11, 0, 1, 26, -12, 2, 11, 3], [11, -7, 1, -2, 4, -1, 4, -10, -4], [-11, 10, 3, -6, -2, -1, -6, 6, -4], [10, -13, -12, 9, -12, -1, -3, -18, -6], [6, -1, -2, 5, 22, 8, 4, 4, -10], [19, -2, -2, -4, -1, -8, 6, 13, 0], [-10, -12, -6, 0, 6, -4, 0, 18, 2], [-9, -16, -6, -2, -8, 3, -7, -29, -9], [2, -19, -14, 11, -13, -9, -1, 6, -2], [-19, -6, 4, 1, -4, -6, -6, 3, 3], [6, 4, 8, -4, 8, 2, -10, 2, -4], [-2, 3, -3, 5, -8, -3, -2, -3, 1], [25, 10, 7, 4, 6, -3, 4, 2, 4], [11, -2, -11, 12, 16, -3, 1, 4, 10], [-6, -10, 1, -8, 5, -1, -4, -15, -2], [16, 4, -4, 6, -2, -2, 2, -4, 6], [21, 4, -2, -2, 8, -7, 5, 37, 7], [9, -4, -1, 2, -8, -5, 2, 14, -2], [-3, -2, 4, 0, -7, 7, -9, -3, -1], [3, -14, -2, 8, 6, -3, 3, -12, -1], [30, -11, -2, 3, 22, -3, 1, 2, -6], [16, 5, -3, 11, -14, -1, 4, 13, 1], [7, -11, -6, -7, -6, -4, 0, 2, -3], [-4, 13, 1, -1, -4, 2, -3, -13, 1], [-14, -10, 5, -2, 26, 6, -1, 9, -9], [29, -2, 8, 2, 10, -9, 1, 17, 5], [12, -4, -11, 0, -6, -14, 1, -3, 3], [-7, -5, -5, 4, -18, -1, -4, -10, 2], [29, 8, 0, 10, -16, -2, 6, 15, 3], [13, 8, 5, -2, 8, -1, 12, 14, 4], [8, -12, 1, -4, -2, -10, 3, -7, 1], [-26, 2, 6, 2, 0, -6, -2, 12, -2], [36, -19, -13, 10, 0, -6, -5, 13, -1], [15, 1, 3, -7, 16, -10, 5, 24, 4], [25, 5, 5, -3, 2, 3, 6, 20, 8], [0, -5, 7, -1, 27, 0, 1, 11, -10], [-1, -8, 4, 8, -16, -11, -2, -17, 3], [39, -7, -12, 7, 12, -6, 6, -1, 1], [17, -6, 6, 6, 2, 1, -3, 25, -3], [-2, -5, -6, -9, 6, 13, -9, -12, -4], [8, -7, -3, 13, 2, -3, 0, 25, 5], [7, 4, 6, -6, -9, 6, 0, -15, 8], [-27, 8, 7, -10, -1, 7, 4, 4, 2], [15, 15, -2, 7, 8, -8, 8, -1, 1], [-14, -8, 0, 4, 14, 0, 4, -12, -2], [-19, -3, 0, 3, 6, 2, 4, -17, 3], [20, -7, -2, -5, -2, 3, -5, -8, -6], [6, -11, -10, 7, 7, -4, 4, 6, -7], [-4, -4, -2, 8, -8, -4, 0, -6, -6], [5, 15, 1, -3, 0, -4, 7, -18, 4], [19, -8, -4, -4, 22, -1, 3, -1, 5], [-42, 12, 5, -12, -10, 2, -4, -5, -1], [-13, -1, -4, -7, -8, 14, -2, -23, -9], [7, -1, 6, 5, 24, -8, -4, 15, 1], [18, 0, 1, -6, 12, 6, 7, -13, -7], [-2, 9, -3, 1, 14, -2, 5, 3, 1], [6, -1, 11, -11, -10, 5, -10, 1, 7], [9, 22, 13, 0, 4, -1, 10, 4, 0], [-44, 8, 14, -6, -28, 5, 7, -34, 0], [-14, -2, 2, -6, 1, 8, -2, -4, -12], [-6, -8, 1, 4, 4, -2, -3, 11, 9], [-12, -7, 3, -7, -34, 3, -2, -21, 1], [-25, -7, 2, -2, -18, -1, -3, -11, -1], [-7, 5, 3, 5, -20, -10, -1, -17, 6], [8, -7, 3, -9, 3, 5, -2, 7, -5], [-9, 10, -7, 0, 10, -1, -4, 6, 0], [4, 0, 5, 6, 6, -3, -10, -9, 1], [6, -9, -6, 7, -17, -4, -4, -2, 7], [7, -9, -2, 5, 8, 0, -2, 31, 5], [-1, -12, -8, 12, -4, -13, 3, 15, -5], [0, -9, -4, -11, 12, -3, -7, -22, -10], [9, 4, 7, -2, 14, -9, -8, 19, 8], [5, 0, -11, 8, 12, -1, 4, 6, 8], [-8, 10, -5, -8, 4, 4, 3, 7, -7], [-9, 23, 8, -5, 2, 8, 6, -17, 9], [7, -10, 7, -12, 18, 2, -11, 14, -10], [-12, -21, -1, 1, 4, -5, 0, 1, -13], [-10, -6, 10, -10, -5, 5, -9, 4, -1], [-2, 4, -2, -12, -10, 4, -4, 2, -10], [20, -2, 2, -6, 32, 6, -9, 20, 4], [7, -19, -12, 6, -20, -13, -3, -5, 3], [-24, 2, 9, -12, -6, -2, -7, -23, -5], [-37, -3, 8, -3, -12, 2, 2, -16, -5], [8, 22, -2, 8, -10, 6, 4, 0, 8], [8, 2, 1, 2, 0, 8, -3, 11, -5], [-31, -3, -6, 7, -18, 3, 7, 1, 5], [-20, -7, 3, -11, -18, 3, 4, -9, -7], [15, 0, -6, 7, 28, -6, 8, 35, -1], [-6, 6, -7, -2, -27, 6, 1, -25, -1], [0, 2, 10, 6, -4, 2, 6, -4, 16], [-9, -4, 6, -10, -17, 18, -8, -19, 0], [11, 1, 4, 8, 26, -11, 1, 19, 1], [-1, 9, 14, -1, 18, -6, 2, 21, 13], [-18, -7, -3, 7, 8, -11, 8, 11, -1], [-29, 13, 0, -11, -9, 7, -3, -11, -4], [2, -1, 4, 11, 2, -5, 3, -6, 0], [-36, 17, 3, -3, -4, -1, -2, 7, 5], [-13, 3, 5, -15, -16, -4, -3, 6, -2], [4, -19, -8, 3, -3, -12, -2, -2, 3], [-3, 28, 4, 0, 4, 3, 0, 13, 13], [-5, -7, 7, -7, 26, 0, 1, 10, -2], [-14, 9, 5, 13, -26, -9, 14, -5, 11], [34, 11, 0, -1, -12, 7, 1, 0, 6], [8, 2, -1, -8, -10, 4, 5, -21, 5], [5, 12, -3, -2, -21, 14, 3, -4, 5], [10, 3, 0, 17, 20, -7, 2, 2, 6], [19, 2, 7, 12, 12, 1, 4, 29, -2], [-15, 0, -9, 14, 8, 3, 8, 6, -8], [-10, -15, -11, 9, -14, -7, 2, -3, -5], [0, -18, -4, -6, -8, 2, 2, -8, -4], [5, -12, -8, 6, -30, -9, -1, -7, 11], [-13, -9, -5, -5, 12, 10, -8, -12, -18], [-10, -4, 6, -4, -14, -2, 0, -6, -6], [-21, 4, -3, 6, 12, 7, -4, 19, 4], [32, 22, 2, -2, 6, 3, 5, 32, 6], [34, 25, 11, -3, 26, 1, 6, 33, 3], [-16, -3, -2, -9, -8, 5, 0, -26, -2], [13, -15, -6, -5, 12, 6, -6, -1, -7], [36, 2, -5, 12, 14, -4, 1, 11, -3], [0, -5, -4, -2, 6, 2, -4, 20, -10], [22, 2, -7, 14, 3, -2, 5, 7, 3], [-2, 0, 5, 2, 18, 10, 1, 3, -15], [-10, -16, -11, -6, -18, 6, -11, -7, 1], [-12, 1, 1, 7, -6, 1, 1, 19, -5], [-12, 20, 13, -8, 7, 10, -7, 5, -7], [-20, -1, 6, -15, -18, 3, -3, -22, -2], [-20, 11, 10, -5, -4, -7, -1, -10, 6], [-8, 8, 15, -14, -10, -4, 7, -11, -3], [-32, 6, 5, -2, -38, 12, -5, -47, -7], [-8, -6, -4, -4, 0, 0, 10, 22, 4], [1, -6, 1, 16, 12, 1, 2, -2, -8], [-5, 11, 18, -17, 0, 12, 6, 9, 3], [6, 20, 7, -4, -2, -2, -11, -8, 3], [-33, -5, 10, -15, -2, -4, -2, -17, 5], [27, -19, -10, 7, 28, -14, 5, 31, 3], [20, -10, 9, 10, -4, -4, 3, -3, -5], [-13, -16, -8, 4, -20, -5, -3, -39, 1], [-17, 1, 0, -5, 8, -10, 5, -23, 1], [-11, -16, -10, 16, 18, -7, 9, -7, -1], [-30, -1, -4, -15, -14, 5, 1, -2, 0], [11, -12, -10, -10, -6, -9, 0, -17, 1], [29, 6, 6, -10, 18, 10, -8, 7, 7], [-8, -2, -14, 6, -14, 8, 4, -11, 6], [-28, -11, -10, 5, -2, -9, 1, 8, -14], [-26, -4, -10, 4, 0, -10, 0, -8, 2], [-27, -13, -17, 11, -30, -1, -8, -14, -6], [-6, 7, 6, 3, -2, -1, 1, -38, 6], [4, -11, -12, -5, 18, -1, -3, -16, -10], [-49, 0, 2, -10, -14, 7, -1, -25, -7], [-2, 15, -8, -9, -13, 8, 2, -8, 5], [-26, -7, 3, -7, 8, 5, 6, -17, -3], [15, -16, -2, -10, 29, -8, -14, 5, 2], [-5, 0, -5, 6, -4, 9, 4, -30, -4], [7, -5, 0, -5, 0, 16, 2, -7, -9], [-7, -17, 4, 0, -6, 11, -1, -23, -5], [11, 3, 12, -1, 0, -6, 12, -5, -1], [15, 3, -14, -5, -5, 1, -7, 11, 2], [19, -26, -7, 4, 6, -7, 4, 18, 0], [19, -20, -3, 2, 30, -13, -2, 2, -4], [38, -12, -8, 16, 4, -4, 0, 48, 4], [-7, 9, 11, -3, -18, -8, -3, -24, 10], [7, 9, 0, 11, -12, 0, 16, 7, -1], [-9, 25, 17, -11, -6, 2, 1, -6, 6], [9, -2, 5, 2, -6, -3, -10, 2, 8], [-15, -13, 1, -1, -12, -4, 0, -28, -10], [-15, -8, 0, 12, -12, -7, 5, 5, -9], [13, -18, -16, 4, 8, -7, -9, -1, -11], [-20, 10, 1, 8, -35, 9, -6, -9, 0], [0, -2, 4, 10, 16, -16, 2, 10, 0], [-21, 0, 9, -6, -4, 7, 9, -36, -10], [2, -8, -6, -6, 34, -6, -8, -28, -6], [26, -21, 1, 2, 14, -4, -7, 17, 1], [-31, 8, 9, -6, 15, 10, -1, 16, -7], [37, 0, 11, -16, 12, -9, 0, 14, 2], [25, 1, 13, 14, 4, 5, 2, 12, -6], [-6, 13, 18, -9, 4, 7, 9, 12, 6], [11, 0, 1, 16, 12, 3, -4, 2, -2], [-8, -6, 3, -10, -32, 6, -7, -11, 5], [-7, 10, 6, 6, -26, 2, -4, -1, -9], [5, 6, 2, 8, 12, -3, -7, 6, -7], [10, -11, 1, -3, -6, 1, 0, -15, 7], [6, -9, -3, -1, 16, 3, 0, 3, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3380_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3380_2_a_r();".
// 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_3380_2_a_r(:prec:=9)
    chi := MakeCharacter_3380_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_3380_2_a_r();".
// 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_3380_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3380_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -26, -149, -153, 87, 106, -16, -19, 1, 1]>,<7,R![3121, 1126, -3909, -1877, 951, 518, -60, -41, 1, 1]>,<19,R![64, -320, -3600, -7568, -3412, 2116, 253, -87, -4, 1]>],Snew);
    return Vf;
end function;