
// Make newform 8280.2.a.bw in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8280_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_8280_2_a_bw();".
// 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_8280_2_a_bw();".
// 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 := [-7, -51, -76, 60, 46, -18, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [553, 264, -1172, -352, 238, 24, -11], [-133, -386, 212, 244, -84, -16, 5], [383, 984, -652, -608, 226, 40, -13], [-651, -2280, 1228, 1248, -434, -80, 25], [-4501, -9608, 6308, 5344, -1974, -344, 111], [2737, 8872, -4612, -5072, 1694, 328, -99]];
    Rf_basisdens := [1, 224, 14, 32, 56, 224, 112];
    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_8280_a();"
function MakeCharacter_8280_a()
    N := 8280;
    order := 1;
    char_gens := [2071, 4141, 4601, 1657, 3961];
    v := [1, 1, 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_8280_a_Hecke(Kf)
    return MakeCharacter_8280_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, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 0, 0, 0], [-2, 0, 1, 1, 0, 0, -1], [0, 0, 0, 0, -1, 0, 1], [-1, -1, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0], [1, 1, -1, 0, -1, 0, 0], [-1, -1, -1, 0, -1, 1, 0], [-3, 1, 1, 1, 1, 0, 0], [2, 2, -2, -1, -1, 1, 1], [-1, 1, -1, 1, 0, 1, 1], [-1, 0, -1, 1, -2, 1, 0], [-2, -2, 1, 0, -1, 0, 0], [-2, 1, -1, -2, -1, 1, 1], [1, 0, 0, 1, -2, -1, -1], [-2, 2, -1, 0, 1, -1, 0], [-3, -2, 2, 1, 1, -2, -2], [-1, -2, 1, 1, 0, 1, 0], [0, -1, 0, -1, 2, 0, 1], [-2, 0, 0, -2, -1, 0, 1], [-1, -1, 0, -1, 0, -1, 2], [1, -1, 1, -1, 0, -1, 1], [-1, 0, 0, 1, 1, 0, -2], [-2, 2, 1, 0, 2, 0, -1], [-2, 1, 1, -1, 1, -2, -1], [-1, -2, 0, 1, 0, 1, 1], [-6, -1, 1, 1, 1, 0, -1], [-10, -2, 1, 1, 0, 0, -1], [-8, 0, 0, 2, 0, 0, -2], [-4, 0, 1, 2, -2, 0, -1], [-1, 0, 0, -3, 1, 1, 2], [-3, 1, 2, -2, 2, -1, -2], [-2, 3, -1, -1, 0, 2, 2], [-7, -4, 2, 0, 1, -2, -2], [-2, 0, 0, 0, -2, 2, 0], [-5, 2, -1, 1, 2, 3, 2], [-4, 2, 0, 0, 2, 0, -2], [-8, -1, 1, 1, 0, 0, 0], [-2, 1, 2, 1, 0, 0, -1], [-9, 1, 2, 2, 0, -1, 0], [12, 2, -2, -4, 2, -2, 0], [0, 2, -2, -2, 0, 0, 2], [4, 3, -4, -3, 0, 0, 3], [-3, -2, 2, 1, -1, -1, -2], [-1, 0, 2, 4, 2, -1, -3], [-10, -1, 4, 3, 2, -2, -1], [4, -1, 2, 1, 4, -2, -1], [-8, 2, -2, -2, -2, 2, 0], [-5, 2, -2, -1, 1, 0, 0], [3, -3, 4, 2, 2, -3, -2], [-3, -1, -2, -1, 0, 1, 0], [-3, 2, -3, -1, 0, -1, 2], [-4, -4, 3, 0, -1, 0, 0], [3, 1, -1, 0, 3, -2, -2], [-7, -2, 4, 5, -1, -1, -4], [-1, 0, 0, 2, 4, 1, -1], [-8, -4, 0, -1, -2, 0, 0], [0, -1, 2, -1, -1, -3, -2], [-10, -1, 2, 1, 3, -2, -4], [3, 2, -5, -5, -2, 1, 2], [-5, 0, 2, -2, 4, -1, 1], [4, -2, 3, 4, 3, -1, -2], [1, 0, -3, -1, -4, 1, 0], [7, -1, -1, 2, -3, 1, 2], [-3, -6, 0, 0, -4, 1, 1], [-2, 4, -2, -4, 4, 2, 0], [9, -2, 2, 1, 1, -3, -2], [-9, 4, 1, 1, 0, -1, 2], [-3, -5, 2, 0, 0, -5, 0], [-9, 1, 1, 1, 5, -2, 0], [12, 0, -3, -2, -2, 2, 5], [0, 2, -2, -2, -4, 0, 4], [2, -1, -2, -3, 1, 0, 0], [-2, 2, -2, 0, 0, 4, 0], [-3, 2, -2, 4, -2, 1, -1], [-13, -3, 2, -1, 2, -5, 0], [15, 2, 0, 3, 1, 3, -2], [-10, 4, 0, 1, 0, 2, 2], [7, 2, -4, 3, -4, 3, 1], [6, 2, -2, 2, 0, 4, -2], [2, -1, 0, -3, 7, -1, 0], [4, -3, 1, 3, 4, 0, 0], [1, -2, 1, -1, 2, -1, 2], [4, -1, -3, -2, 1, 1, -1], [-2, 2, 1, 2, 1, 3, 0], [-9, 4, 4, 2, 4, -1, 1], [-6, 0, 3, -1, 6, -2, 1], [-12, -3, 3, 1, -1, 0, 1], [-7, 1, -2, 2, -4, -1, -2], [-6, 2, -1, 1, 0, -2, 1], [-6, -4, -2, 1, -5, 1, 1], [5, 1, 1, 2, -1, -1, -2], [8, 3, -2, 3, -5, 2, 0], [-11, 1, -1, 3, 0, 1, 1], [-8, 0, 0, -3, -6, 2, 0], [-7, 0, -2, 0, 0, 1, 3], [-6, -5, 5, 1, 0, -4, -6], [-6, 2, 2, -2, -4, 4, 2], [2, -4, 1, -4, -7, 0, 0], [0, 1, -1, -1, -1, 0, -3], [-3, 3, 2, 3, 2, -3, -4], [-9, 1, -6, 2, -2, 1, 2], [-6, 4, -2, 0, 6, 2, 0], [0, -2, 0, 0, -2, 0, -4], [9, 4, -1, -3, 4, 3, 2], [-5, -8, 2, 0, -4, -1, 1], [-7, -1, 1, 4, 1, 5, 0], [-16, -2, -1, 3, -4, 0, 3], [2, -1, 0, -3, 2, -2, 3], [-6, -2, -2, 2, -5, 2, -1], [12, 0, -4, -4, 0, 2, 0], [-3, -4, 0, -3, 4, -5, -1], [6, 4, -8, 3, -6, 4, 2], [-2, -4, -1, -2, 3, 1, 0], [3, 0, 1, -3, 4, 1, 0], [-11, 4, -1, 1, 6, -1, 0], [4, 0, -2, -7, 0, 2, 2], [-2, 5, -2, -3, 6, 0, -3], [5, -2, 7, -3, 2, -3, -2], [12, 2, -5, -8, 1, 2, 6], [-5, -2, 5, 5, 6, 1, -4], [14, 5, -5, -3, 0, 2, 4], [7, -1, 6, 2, 2, -5, -2], [-15, 2, 0, 5, 6, 1, -3], [-7, -3, 3, 8, 1, -2, -4], [-7, 6, -2, -2, 1, 2, -2], [3, -3, 1, 5, -5, 0, 0], [-9, 3, 5, 6, 7, 1, -6], [8, -4, 0, -4, -8, 0, 4], [-5, -4, -2, -5, -2, 3, 1], [-3, -4, 0, -4, 7, 0, 2], [0, 7, -7, -6, -1, -1, 1], [17, 1, -2, 0, 0, 1, 4], [6, 4, 0, 2, 0, 4, 0], [-10, -4, -1, -6, -5, 1, 0], [-10, 4, 3, 0, -1, 0, 0], [6, -2, 2, -3, 3, -7, -3], [-11, -2, 4, 1, -3, 3, 2], [7, 1, -9, -1, -8, 3, 3], [5, 0, 0, 2, -4, 3, 3], [20, 3, -3, -3, 1, -4, -1], [15, -1, -1, -4, 1, -1, 4], [0, 0, -4, -6, 2, 0, 2], [-5, 0, 0, -4, 0, -1, 7], [4, -6, 6, 0, 8, -4, -2], [5, 4, -2, -5, 1, 3, 2], [6, 0, 4, 6, -2, 0, 0], [-23, 2, 0, 2, 0, -1, -1], [-16, 4, 0, 1, 8, -2, 2], [3, 2, 3, -1, 4, -1, 0], [0, 0, -4, -6, -2, 0, 6], [-10, -2, -3, -2, -5, 1, 4], [4, -4, -6, -6, 0, 4, 2], [10, 5, -8, -1, -6, 4, 1], [16, -3, -1, -2, 3, 3, 1], [1, -3, 5, -7, 4, -5, -1], [1, -3, 0, -2, 6, -1, -4], [-8, -2, -2, -4, -4, 2, 4], [8, -3, -4, -7, 6, -2, 5], [-20, -4, 1, 7, -4, -2, -1], [15, -1, 0, -7, 2, 3, 6], [-4, 4, -4, 0, -7, 8, -1], [28, 1, -4, -3, -5, 2, 4], [-27, 2, 2, 1, 7, -1, -2], [-11, 4, -6, -6, 2, 5, 5], [16, -4, -2, 2, -6, 0, 2], [-13, -6, 1, -3, 8, -3, -4], [3, 5, -6, 1, -2, 1, -2], [-3, 0, -4, -1, -3, -3, 2], [-8, 2, 2, 4, 2, -2, -6], [32, -1, 0, -1, -5, -3, -2], [10, 2, 0, 10, 0, 2, 0], [10, 0, 0, -5, 4, -6, 4], [12, -1, -5, 1, -8, 0, 0], [-13, 1, 3, 6, 1, 0, -4], [3, -1, -1, -3, -7, -2, 0], [11, 2, -8, -7, -4, 5, 1], [-2, -2, 3, 7, -6, 0, -1], [14, 2, -6, -1, -10, 4, 0], [3, 6, 0, -2, -8, 3, 5], [17, 6, 3, -3, 6, -3, 2], [4, -2, 7, 6, 3, 0, -6], [3, 5, 6, 6, 0, 1, -6], [-10, -4, -5, -1, -8, 2, 1], [-16, 6, 5, 2, 5, 1, 0], [17, -5, 9, 4, -1, -7, -6], [25, -2, 2, 1, 3, -4, 0], [-9, 2, 7, 3, 12, -3, -2], [0, 6, 0, -4, 6, 2, 4], [7, 9, -4, 0, 8, 1, -2], [13, -4, -2, 3, -3, -2, -2], [4, 2, 5, 6, 2, -2, -1], [6, 3, -7, -5, -3, 8, 7], [23, 1, -8, 0, -6, 3, 6], [-6, -2, 3, -2, -2, 0, 7], [-2, 3, 4, 3, -10, 4, 1], [12, 4, 4, 6, 6, 2, -6], [13, 3, 2, 0, 6, 5, 0], [-12, 3, 3, 3, 0, -4, -8], [-6, -6, 1, -1, -8, -2, -7], [13, -3, 8, 0, 6, -1, -4], [-8, -2, 12, 9, 2, -2, -8], [-6, 0, 8, 6, 8, -2, -4], [11, -6, -2, -1, 0, -1, -3], [3, 10, -3, 1, 0, 11, 2], [14, 6, 0, -1, 9, 3, 1], [-14, 1, -1, 1, 12, -2, -4], [2, -2, 2, -2, 4, 2, 0], [7, -2, 2, -4, -4, -1, -3], [-10, -8, 9, 6, -2, -4, -7], [-2, -4, -3, 4, -6, 4, -1], [-9, 1, 3, 4, 5, 0, -2], [9, -5, 4, 0, -8, 1, 0], [-17, -6, 6, 4, -6, 3, -1], [-10, -4, 8, 1, 5, -5, 3], [10, 2, 9, 4, 1, -6, -4], [-11, 0, -3, -3, 6, 1, 6], [-9, -5, 3, -4, 11, -5, -6], [16, -1, 0, -7, 4, -4, 3], [4, -1, 5, 0, -7, 1, -1], [-25, -3, 5, 3, -7, 2, -4], [1, 0, 1, 5, -4, 1, 6], [-7, 2, -6, -7, 7, -3, 4], [2, 3, -8, -3, -6, 0, 1], [-15, 4, 0, -3, -3, 2, -2], [20, 1, -2, 1, -2, -4, -1], [0, 0, 0, 5, 7, 3, 1], [6, -4, 11, 9, 0, -6, -3], [19, -1, 2, -4, 4, -1, -2], [-4, 1, -2, -5, -4, 4, 3], [-2, 3, 5, 2, 3, -7, -5], [-14, -12, 3, -2, -1, -3, -4], [4, -1, 7, 7, 4, 2, -4], [23, -1, 2, -4, 8, 5, 0], [4, 2, -6, -2, -12, 2, 2], [-24, -1, 1, 2, -1, -1, -5], [13, -9, 3, 0, -1, 0, -4], [-12, -2, 0, 0, 9, -2, -5], [-11, -3, 1, -6, 7, -1, 2], [17, 0, 0, -4, -2, 1, 1], [14, 3, -1, -3, -4, -4, 0], [-8, 4, 2, -2, -2, 4, 6], [3, -4, -10, -4, -4, -1, 5], [-26, 4, -1, -1, -4, 4, 3], [12, 2, 4, 1, 0, -4, -6], [-14, 0, -6, 6, -10, 6, -2], [-8, 3, 8, 3, 5, -2, 0], [-20, 2, 1, 6, 2, 2, -1], [-9, -5, -2, 1, -2, 1, -6], [-6, 1, 2, 3, 1, 0, -10], [8, 0, 4, 2, 4, 0, -2], [-7, 2, -3, -7, 2, 7, 8], [4, 3, -3, 0, -5, 3, 3], [-13, -8, -2, 5, -6, 3, 3], [-23, -2, 0, -2, 8, 1, -1], [23, -2, -13, -7, -10, 7, 4], [8, -4, 2, -6, 4, -6, 4], [3, 5, -1, 1, -11, 6, 2], [-12, 1, 5, 7, 1, -4, -3], [-17, -1, 3, -2, -1, 3, -2], [5, 8, -10, -10, 1, 2, 2], [10, -2, 4, 4, -2, -4, 0], [-3, -1, -2, 0, 2, -7, 2], [7, -5, 0, -6, 6, -5, 0], [-1, -5, 4, -5, -2, -1, 4], [10, -1, 2, 5, 9, 1, 2], [0, -4, -6, -4, -2, -2, 2], [-11, -3, 5, 12, -1, 5, 0], [-26, 0, 5, 0, -3, 3, 4], [0, -4, 5, 2, -11, 1, -4], [-12, 1, 6, 7, 0, -6, -9], [-9, 8, 1, 3, -2, 1, 0], [-10, 2, -1, -1, -6, -2, 7], [5, 8, 1, 7, -14, 3, 4], [25, 1, -3, -12, 1, 1, 6], [37, -1, -2, 4, -4, 1, 2], [20, 2, -2, 3, -14, 4, 2], [10, -4, 4, 2, -10, 2, -4], [15, 2, -4, 1, -4, 5, 11], [-2, 4, 4, 10, 4, -2, -10], [-6, -13, 12, 1, 8, -6, -9], [-17, -4, 5, 5, 10, -1, -2], [-5, -7, -5, -4, 5, -4, 2], [-2, -5, 6, 3, 3, -5, -10], [-4, 8, 0, 4, 12, 6, -6], [32, 3, -5, 1, -4, 4, 8], [16, -10, 2, 0, -4, -4, 4], [-2, 6, 4, 2, -4, 0, -2], [-20, -4, 6, 4, -1, -2, -7], [-36, -2, 0, 2, 6, 2, -2], [22, 4, -6, 1, -6, 0, 2], [-10, 0, 3, 3, 2, -4, -1], [-4, -4, 2, -2, -10, 2, -6], [-9, 0, 4, -5, 2, 3, -1], [-13, 2, 3, 5, 10, 5, -4], [30, 5, -5, -10, -1, 1, 9], [16, 0, 2, 6, -10, -2, -4], [16, 5, -2, 1, 1, 3, 0], [24, -7, -2, -1, -1, 0, 0], [10, -5, 8, 9, -2, 0, -1], [3, 10, -3, -1, -10, 1, 4], [0, 6, -2, 0, -4, 2, 8], [21, -2, -5, -11, -6, 1, 14], [19, 2, 5, -7, 4, -3, -2], [-16, -7, 2, 7, -6, -2, -5], [15, 15, -10, -6, 0, 3, 6], [4, -4, 1, 3, -2, 4, -1], [-4, 6, -2, -2, 8, -2, 0], [15, -1, 2, 4, -8, -5, -4], [-2, 4, -2, 9, 0, 2, 6], [-12, -4, 0, 2, 8, -2, -8], [52, 7, -8, -17, 1, 2, 12], [-13, 1, 0, 6, -8, 5, -4], [4, 3, -3, -10, 17, -3, -3], [31, 4, -6, -7, -1, 6, 12], [-18, 1, -8, -3, -3, -3, 4], [-12, -5, 3, 9, 8, 0, -8], [17, -5, 0, 0, -10, -1, 0], [16, 10, 1, 4, 3, 7, 0], [11, -4, 2, 0, 4, 3, -1], [21, -8, 6, 0, 6, -7, 7], [12, -5, 5, 1, -11, 2, 1], [21, 11, -2, 6, 4, 7, 2], [17, 1, 0, -2, 2, 1, 6], [21, -8, -1, -1, -14, -1, 6], [-4, 4, -2, 8, 4, 8, -2], [-8, -7, 8, 7, 3, 0, -2], [-11, 6, -4, 1, -5, -3, 6], [-7, -2, 1, -1, -6, -7, 0], [5, -1, 5, 2, 3, 3, -8], [23, 2, -5, -7, 0, 7, 4], [-36, -2, 2, -2, -4, -2, -4], [16, 6, -12, -2, -4, 2, 6], [-20, -3, 8, 5, 10, -4, -11], [12, -9, 0, -3, 2, -6, -5], [12, 1, 3, -5, 8, -2, -4], [-17, 3, -1, 7, -5, 0, -10], [-9, 0, -6, -10, 4, -5, 5], [-12, 1, 4, 13, 2, 6, -3], [3, 4, -4, 6, -8, 11, 5], [17, 12, -4, -3, -8, 3, 7], [-44, -2, 5, 12, 3, -6, -10], [-18, -14, 8, 1, -1, -7, -3], [7, 5, 1, -2, -1, 3, 2], [3, 10, -1, -1, 2, 7, -2], [16, 2, -8, -4, 6, -8, 0], [6, 5, -4, -7, 7, 2, -4], [-9, -1, 1, -2, 11, -7, -2], [-11, -6, -3, 7, -6, 1, -4], [11, 12, -4, -1, 7, 6, 8], [15, -5, 1, -5, 3, -8, -4], [-11, 2, -4, 5, 2, 5, 5], [-10, 2, -10, -6, -4, 6, 8], [-7, 3, -8, -4, -2, 1, 8], [-5, 3, 2, 7, 0, -5, -6], [1, -8, -7, -5, 2, 5, 10], [6, 2, -4, -10, 10, 0, 2], [-23, 5, 1, -3, 3, 2, 2], [-14, 8, -6, -7, -1, 1, -5], [10, 2, -10, 4, -14, 4, 0], [-27, 1, -6, -1, 10, 7, 4], [42, 0, -12, -4, -14, 4, 8], [-14, 6, -4, 0, -12, 0, 2], [-41, -20, 13, 3, 4, -9, -10], [-11, -1, 5, 5, -3, 4, 4], [12, -12, -8, -2, -6, 2, 8], [22, 3, 5, 0, 1, -1, -5], [19, 1, 9, 8, 1, -3, -6], [-32, -4, 6, 8, -6, 2, -4], [16, -4, 4, -4, 2, -6, 0], [14, -3, 6, 7, -2, 4, -1], [24, 8, -13, -1, -4, 4, 7], [0, 11, -5, 2, -3, -3, -9], [6, 0, 2, 0, -2, -6, 2], [8, -1, -2, -9, 1, -1, 6], [31, -3, 4, 9, 8, -1, -4], [10, -6, -8, -12, 4, 0, 2], [14, -4, 0, 0, 8, -6, 0], [-2, -7, 0, 3, -18, 0, -1], [-7, -6, 4, 4, 8, -11, -7], [-1, 0, 1, 1, 14, -1, 4], [42, 2, 0, -8, 4, -2, 8], [30, 0, -6, 4, 6, -6, -6], [25, -8, -6, -3, -7, -5, 2], [-24, 4, 1, 1, 10, -8, -1], [-44, 3, -2, -3, 7, 9, 10], [-24, 7, 6, -1, 13, -2, -2], [25, 3, 2, 6, 6, 1, 0], [-10, -6, 8, 0, -8, 2, 6], [-10, 6, -2, -3, 1, 3, 1], [15, 4, -5, 3, 8, -5, -2], [-11, -17, 14, 10, 2, 1, -6], [40, 1, -4, 1, 5, -1, 6], [-26, 9, 2, -1, 4, 4, 5], [3, -15, 2, 3, -16, -5, -2], [0, 4, 0, 12, -8, 2, -4], [1, 2, 0, 3, -17, 0, 0], [-7, 12, -8, -7, 1, -3, -2], [-20, 8, 0, 0, 16, 2, -2], [7, -6, -2, -10, -17, 2, 6], [-6, 10, -16, -8, -1, 10, 5], [23, -4, 0, 3, -1, -6, 0], [29, -5, -1, -2, -1, 3, -2], [-17, -3, 1, -1, -9, -8, 4], [-2, 4, 4, 14, 0, -6, -10], [17, 9, -7, -11, 0, 7, 13], [-34, 2, 2, 0, 0, -10, 4], [13, 0, -3, 7, -6, 3, 2], [-27, -10, 5, -3, 10, -5, -4], [0, 1, 2, 9, -2, 4, 5], [42, 2, 0, -4, -12, 2, 8], [2, 5, -8, -13, -7, 1, 4], [7, -2, 0, 2, 12, 3, 7], [-27, 2, 12, 14, 0, 1, -1], [-8, -10, -7, 3, 0, 10, 5], [0, 0, 4, -12, -3, 4, 7], [22, -6, -12, -8, -12, 2, 12], [-9, -1, -2, 10, -4, 7, 4], [-14, 8, -1, -8, -2, 0, 5], [5, -4, 1, -11, 4, -1, -2], [-38, -6, 8, 8, 4, -10, -10], [3, -10, 9, -1, -2, -3, -2], [-6, 8, -10, -16, -2, 0, 4], [-14, 11, -10, -7, 14, 2, 1], [36, 8, 0, 5, 7, 7, -1], [44, -1, -5, -7, 4, -4, -4], [-19, -1, 3, 2, -7, 2, -6], [-10, 0, 6, 6, -6, -6, 4], [15, 9, -10, -9, 0, 3, 12], [-20, 1, 9, 3, 8, -8, -4], [20, 11, -8, 3, 3, 2, 2], [-44, -6, -3, 15, -14, 0, -3], [-20, -10, 2, -1, -4, -2, -10], [-13, 5, 1, 6, 7, -5, 6], [2, -10, -6, -10, -12, 4, 0], [36, 6, -6, -8, 2, 4, 2], [-16, -1, 8, 3, 0, -10, -5], [36, 10, -2, -6, 2, -2, 6], [-28, -16, 4, 5, -6, 0, -4], [19, 3, -11, -6, -5, 11, 8], [8, -6, -10, 0, 0, 0, 4], [-5, -3, 11, 2, 11, -5, 4], [11, 0, 3, -3, -4, -5, 4], [2, 2, -12, -16, -10, 0, 14], [15, 4, 1, -9, -6, -3, 10], [33, 0, 4, 7, 7, -3, 4], [6, -11, 6, 3, -10, -6, -7], [-13, 10, 3, 7, 0, 5, 0], [1, -11, 4, 6, 10, -9, -6], [43, -8, 3, 1, 0, 3, -2], [12, -14, -4, 0, -20, 0, 6], [-5, 3, 5, -5, 4, 1, 3], [-15, -1, -13, -6, 1, -2, 12], [-23, -13, 9, 8, -1, 1, -2], [-46, 6, 4, -2, 8, -2, -10], [-27, -2, 10, 6, -1, -4, -18], [14, -3, -1, -3, 1, 4, -1], [-5, -6, 8, 7, 7, -7, -2], [17, 0, 6, 11, -2, -9, -9], [-5, 1, -6, -5, -10, 7, 2], [-17, 0, -14, -11, 3, 3, 8], [18, 4, -2, -10, 4, -8, 4], [43, 8, -11, -1, -14, 7, 8], [17, -4, 0, -3, 4, -3, 7], [14, 9, -3, 9, -5, 6, -3], [-2, -4, 10, 1, 18, 2, 0], [6, 2, -2, -6, -6, -8, -4], [31, 2, -8, 2, -12, 5, -1], [-29, 0, 1, -7, 0, 3, 0], [33, 15, -9, -7, -12, 5, 13], [-14, 3, -7, -1, 4, 0, -2], [2, 7, 3, -6, 5, -5, -7], [-59, 4, -4, -8, -9, 4, -2], [11, 6, -2, -1, 13, -4, -6], [4, -11, -6, 1, -10, 0, 5], [27, 0, -5, 5, -2, -3, -8], [-22, -10, 4, -4, 0, 0, -6], [19, 0, 1, 7, 6, 1, 0], [1, 5, 4, 1, -4, 3, 2], [4, 5, -4, -11, 9, 3, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8280_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8280_2_a_bw();".
// 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_8280_2_a_bw(:prec:=7)
    chi := MakeCharacter_8280_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(3457) - 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_8280_2_a_bw();".
// 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_8280_2_a_bw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![256, 732, 516, -77, -134, -16, 6, 1]>,<11,R![384, 1728, 1928, 556, -128, -50, 2, 1]>,<13,R![14336, -14368, 2072, 1412, -248, -58, 6, 1]>,<17,R![-2688, 6564, 7840, 1599, -364, -88, 4, 1]>],Snew);
    return Vf;
end function;