
// Make newform 5491.2.a.s in Magma, downloaded from the LMFDB on 20 February 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5491_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_5491_2_a_s();".
// 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_5491_2_a_s();".
// 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 := [8, -12, -19, 26, 9, -10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0], [0, -5, 0, 1, 0, 0, 0], [4, 0, -6, 0, 1, 0, 0], [-10, -7, 26, 7, -10, -1, 1], [-11, 1, 26, 0, -10, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 2, 1];
    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_5491_a();"
function MakeCharacter_5491_a()
    N := 5491;
    order := 1;
    char_gens := [2604, 1446];
    v := [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_5491_a_Hecke( : Kf := Kf)
    return MakeCharacter_5491_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 := [[0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 0, 0], [0, 0, 1, -1, -1, 0, 1], [3, -1, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, 1, 0, 0], [-1, 0, 0, 0, 0, -1, -1], [-4, 2, -1, 1, 1, 1, 1], [-4, 0, 0, -1, 0, 2, 2], [-3, 1, -2, 0, -1, 4, 1], [2, -2, 2, 0, -1, -2, -1], [-3, 0, 0, 1, 1, 2, 0], [-2, 0, 0, 1, 2, -2, -2], [-1, 3, 0, -1, -1, 0, -1], [-1, -3, 3, -1, 0, -2, 2], [2, -1, -1, -2, -2, 4, 1], [1, 0, -1, 1, 1, 0, -2], [-5, 2, -3, -1, -1, 4, 2], [2, 2, -1, 0, 3, 0, 1], [0, -3, 0, -1, -2, -2, -1], [-3, 2, -1, 1, 1, 0, 2], [-3, -6, 1, 0, 1, -1, -2], [-1, 3, 0, 1, 2, 0, 0], [1, 0, 0, 3, 2, 0, 1], [-2, 2, -4, 1, 0, 5, 0], [-2, -1, 0, 1, -1, 1, -2], [0, 2, -1, 3, -1, -1, -3], [6, 3, -2, 1, 1, -2, -2], [5, -4, -1, -5, -1, -2, 0], [0, 4, 1, -1, 0, -2, 4], [0, -5, 2, 2, 1, 2, 0], [1, -1, 2, -1, -4, 4, 0], [0, -2, 1, 0, 1, 4, 5], [2, -1, 2, -4, 0, 0, 3], [9, -3, 0, 0, 1, 0, 1], [0, -1, 4, -2, -1, -1, 2], [-3, -2, 1, 1, -1, -3, 0], [-1, -3, -3, 3, 0, -2, 0], [-3, 0, -1, 1, 3, 3, 2], [-10, -3, 1, -1, 1, 4, -2], [-1, 1, 5, 0, 0, 1, 2], [-2, 5, -1, -2, 0, -4, 3], [1, -2, -1, 3, 5, 0, 0], [2, 1, -2, 5, -3, 2, -4], [-1, 0, 4, -2, 4, -6, 1], [-5, 1, -1, -3, -2, 2, -2], [10, -2, 3, -4, -1, -2, -5], [0, 3, 5, 1, 2, 2, -3], [0, -3, 2, -5, -3, 0, 2], [-7, -2, 2, -2, 0, -3, 3], [3, 1, 5, -3, -5, -2, -1], [-2, 1, 3, -1, 0, 4, 3], [-3, -1, 1, 1, 5, -2, 1], [3, 4, 2, 3, 4, -3, -5], [2, 1, 0, 2, -2, 0, 1], [-4, -5, 0, -3, 1, -6, -2], [-13, -2, -5, -2, 1, 6, 4], [1, 6, 1, 4, 3, 0, -6], [-3, -8, -3, -1, 1, 0, 0], [-5, 5, -3, 5, 0, 4, -4], [11, 1, -4, 2, -1, 0, -3], [-3, 8, 1, -4, -3, 0, 0], [-5, 2, 0, -2, -2, -2, 3], [-2, -1, -2, 0, -1, 0, 8], [-14, 2, 0, 0, 0, 0, 6], [1, 7, 5, 0, 0, 0, 0], [8, -2, 1, 0, 2, 0, 8], [9, -4, 2, 2, 1, 0, -6], [-5, 3, -1, -1, 0, -4, -6], [17, -3, 0, 1, -3, 0, -3], [11, -5, 3, 4, 2, -10, -2], [3, -8, -3, -2, 1, -7, 0], [-10, -1, -3, -1, 2, 0, -3], [-5, 3, 1, 0, 1, -4, -3], [-8, 5, 0, -3, 3, 2, 6], [2, 7, -6, 0, 3, -3, 2], [2, 1, 4, 2, -1, -8, 0], [10, -4, -2, -2, -6, 6, 0], [0, 0, -4, 2, 4, 2, 2], [-2, -5, -4, 2, 3, -7, -4], [10, 5, 4, -4, -3, 4, 2], [-4, 2, 2, 0, 4, -7, -6], [1, -3, -1, 4, 4, -6, -6], [-6, 1, 6, -5, -5, 3, 0], [-3, -1, -3, 3, -2, 12, 0], [3, 2, 7, 2, 0, -2, -1], [-13, 0, -2, -1, -1, -2, -4], [9, 1, -5, -6, -4, -2, -2], [9, 2, 0, 0, -2, -2, 7], [-5, -10, 1, 1, -3, 3, -6], [-5, 1, 2, -6, -4, 2, 2], [8, 5, 3, -1, 2, -4, 5], [-6, -2, 7, -2, -5, 0, -3], [-1, 2, 0, -7, -10, -4, -1], [1, 5, -2, -2, 3, -5, 5], [21, -2, -3, -2, 3, 2, 2], [-2, -4, -3, 3, -1, -10, -7], [-10, 6, -2, -3, -2, 1, -2], [4, -1, 2, 1, 4, 6, 9], [-10, 6, 8, 2, 2, -2, -2], [-5, 5, -6, 2, -1, 2, 3], [0, 0, -6, 6, -2, 6, -6], [-12, 2, 2, -3, -3, 4, 1], [-11, 11, 0, 1, -2, 4, -4], [-10, 1, -2, 1, -4, 10, -3], [-5, 1, 8, 0, -1, 0, -7], [16, -3, 3, 7, 2, -2, 3], [-2, -2, -5, -1, 1, -1, 3], [7, -2, -3, -5, -8, -4, -1], [-6, -2, -3, 4, -1, 0, -1], [7, 7, 11, -4, -8, -2, 6], [-9, 3, -9, 1, 4, 0, -6], [-11, 4, -3, -1, -3, 5, 0], [-6, 9, -2, -5, 1, -6, 0], [10, -6, -1, 1, -3, -8, -5], [7, 0, -7, 2, -7, 6, -2], [0, 11, -9, 1, 2, 8, -1], [12, -5, -2, 8, 3, -8, -4], [7, 7, -4, 5, 5, -7, 3], [-4, -1, -1, -7, -4, 2, -1], [-5, 1, -3, 4, 2, -1, -6], [1, 8, 4, 2, -2, -2, -7], [1, 0, 9, 2, 5, -8, -4], [-3, 2, 4, 2, 0, -4, 3], [-2, -11, 1, 0, -2, 4, -1], [-3, 4, 2, 0, 1, 4, 4], [9, 7, -1, 4, -1, 12, 5], [1, 0, 7, 1, 6, -4, 7], [-3, -8, -5, -5, 5, -8, -2], [14, 4, 6, -1, 0, -13, -6], [-2, 10, 6, 1, -3, 0, 3], [0, -2, -3, 0, 2, -4, -12], [-9, 8, 2, 7, -5, 6, -2], [14, 3, 2, -1, 1, 0, -4], [-4, -9, -1, 4, 4, -7, -7], [-2, 7, -8, 8, -1, 8, -10], [-1, 0, 3, -2, 1, 4, -2], [-11, 4, -9, -4, 3, -6, 0], [5, 6, -3, 5, 1, -2, -14], [-7, -1, 3, -2, -2, 6, 6], [-2, -11, -2, 2, 7, -8, -6], [-4, 3, 11, 3, 2, -6, -3], [-17, 8, -7, -1, -1, 7, 2], [-13, 4, -2, -1, 2, -4, -7], [6, -10, 10, 7, 0, 3, -2], [-6, -6, -3, -8, 2, 4, 0], [-4, -4, 10, 2, 0, -6, -4], [-10, -2, -7, 3, -3, 3, 1], [12, -1, 2, 1, 1, -14, 2], [-5, 9, -1, -4, -7, 12, 5], [-1, -5, 1, -7, -2, 7, 4], [-16, 4, -2, -7, -4, 4, 0], [-19, -6, 6, 0, 6, 7, -1], [-11, -5, -8, 4, 6, 10, -4], [4, -6, 4, 2, -4, 0, 0], [5, 2, 7, 4, 4, 2, -3], [-1, 0, 5, -5, -3, 3, 6], [-10, 3, 9, 2, -4, -4, -1], [15, 7, 9, -4, -2, -14, 2], [-1, -8, 0, -1, -1, -6, -4], [15, 7, 2, -2, -5, -8, 7], [-12, 0, -3, 0, 3, 0, -9], [0, -5, 6, -3, -5, -1, 2], [5, 1, 16, 0, -5, -9, 7], [-19, 4, -2, 2, 10, -2, 3], [18, 10, 0, -2, -8, 1, 6], [-12, 6, 9, 2, -3, -10, 7], [2, -13, 2, -1, -7, 6, -2], [15, -15, 3, 1, 4, -8, -10], [-9, -7, -7, 4, 2, -10, -6], [-1, 0, 0, -1, -11, 2, 4], [-10, -12, -7, -2, 9, 2, -3], [15, -3, 11, -4, -2, -8, 0], [1, -4, -2, 3, 0, 10, 7], [-18, 3, -2, 8, 1, 8, -4], [28, -2, -1, -2, 3, 6, 7], [7, -11, 15, -1, -3, -6, -1], [-24, -7, -1, 8, 6, 8, 1], [8, 1, 2, 1, 5, -3, -2], [-10, -9, 2, -11, -7, 5, 12], [-21, -13, -1, 3, 0, 2, -4], [-21, 11, -1, 10, 3, 0, -3], [-19, -10, -2, 2, -2, 6, 5], [-9, -19, 3, -1, -4, -11, -2], [5, 0, 2, -1, -6, 10, 5], [21, -8, 6, -3, 4, -19, 1], [-10, -1, 1, -1, -2, 16, -3], [5, 11, 5, -1, 4, -11, -2], [-3, -6, -10, 1, -4, -4, -3], [-19, 7, -7, -9, -4, 6, 0], [-10, 4, 0, -2, -8, 8, -4], [18, -9, -6, 2, -5, 2, -4], [14, 1, -16, -3, -3, -4, 2], [-12, 6, -6, -7, 0, -2, 0], [-1, -2, 6, 5, -8, 10, -3], [-4, 7, 0, -4, -11, -4, -2], [20, 4, 4, 6, 4, -12, -4], [1, 1, 8, -3, -5, 1, 5], [-19, -9, -7, 0, -4, 5, 4], [-6, 4, -3, -4, 11, -2, 9], [-13, 9, 1, 5, 4, 4, 10], [-7, 14, -8, 3, -6, 12, -5], [-29, -3, -6, 3, 2, 16, 2], [-7, 1, 0, 2, -2, -2, -2], [13, 6, 6, -12, -12, 2, 9], [-2, -13, -5, -8, 4, 2, 1], [-14, 4, -7, -3, 3, 6, 11], [10, -4, 4, -1, 0, 8, 12], [0, -4, 7, -7, -1, 8, 7], [-3, 6, 2, -14, -4, 4, -7], [-1, -4, 6, -6, 2, 0, 19], [-2, 19, 12, -10, -1, 11, 8], [-4, 9, 2, 2, 5, 6, -4], [-20, 7, 11, -7, -1, 4, 4], [-12, 2, 3, -1, -3, -10, 3], [-17, -7, -2, -2, 1, 5, -1], [10, 3, 0, 3, -11, 0, -8], [14, 18, -1, -3, -3, -4, 5], [6, 2, -15, 0, -3, 4, 3], [-12, 12, -11, 5, 7, 10, 3], [1, -4, 10, -8, 0, 2, 15], [3, -3, 4, -3, -7, -8, -1], [10, 4, 4, -7, 7, -16, 7], [19, 8, -13, -1, -1, 1, -4], [-40, 9, -2, 6, -1, 11, 0], [-10, -5, -11, 3, -2, 10, -3], [-1, 4, 6, 0, 0, -16, -3], [4, -11, -6, 6, 7, 10, -14], [27, -13, 5, 2, 0, -8, -14], [-17, 9, -7, 13, 4, 6, -4], [-12, -5, -1, -6, -6, -8, -3], [-9, 9, -8, 0, 6, 2, 12], [13, 2, -14, 4, 1, -4, 10], [8, -1, -2, 0, 7, -20, 0], [-21, -20, -6, 7, 10, 6, -7], [9, -9, 1, 11, -4, 6, -8], [12, 1, -3, 10, 6, -4, 9], [-8, 0, 9, -8, -9, 0, 5], [-23, -7, -5, 7, 7, -10, -7], [13, 10, -8, 6, 0, -2, 1], [0, -10, 3, 10, 1, -6, -1], [17, -2, 3, 5, 5, -2, -6], [-23, 0, -10, 4, -2, 8, 17], [11, -3, 10, 1, 1, 8, -5], [-33, 3, -7, -2, -4, 8, -6], [-26, 4, 9, 4, 9, 2, 1], [-13, 19, -2, -8, 0, 2, 12], [5, -23, -12, -7, 8, 0, -2], [-6, 6, -11, 7, -3, 14, 9], [-28, -8, 3, 4, -3, -2, 3], [5, -9, 4, -2, -9, -1, -5], [-14, 4, -5, -5, 0, -6, 2], [3, -5, 2, 1, 5, -18, -3], [3, -18, -2, -5, -2, 2, 7], [-2, -12, -8, -4, 4, 3, 10], [16, 0, -8, 7, 0, 5, -10], [15, 9, 7, -16, -3, -4, 3], [14, 14, 0, 2, 2, 1, -2], [24, -21, -3, 4, -4, 4, -7], [-21, 1, 0, -8, -5, -4, -9], [-27, 19, 6, -10, -7, 8, 1], [-14, -1, -4, 5, 5, 2, -4], [-6, 1, 8, 0, 7, -18, 2], [-9, 4, 6, 4, 2, -6, -3], [-6, -5, -5, -5, -3, 4, 2], [-12, 11, 6, 1, 3, -19, 2], [-15, 5, -4, -4, -9, 0, 5], [8, 1, 18, 4, -3, 1, 4], [13, 3, -7, 9, -2, -12, -10], [-3, -4, 10, -3, 3, -2, 12], [-13, -6, 5, -3, -4, -16, 5], [7, 3, 15, -10, -8, 0, 4], [11, -4, 3, 10, 4, -18, -1], [-31, -13, 0, -2, -2, 6, 4], [20, -2, 2, -4, -4, 4, -8], [21, -4, 1, -12, -1, -2, -6], [10, 2, -9, 5, 9, 4, 3], [11, 2, -6, 7, -2, 14, 1], [1, -4, -9, -1, 5, 10, -10], [20, -6, -1, 7, -1, 2, 7], [-21, -16, -6, -3, 10, 2, -7], [20, 20, -6, 2, 0, -4, -4], [-14, 10, -2, -3, 14, -10, 6], [12, -10, -7, 6, -3, -2, -5], [-18, 12, -14, 3, 6, 9, 14], [-1, 7, 5, -10, -15, 0, 15], [10, -3, -9, 2, 4, 10, -13], [5, -6, 3, 0, -9, 8, -4], [-5, 6, 9, 0, 3, -2, -8], [-18, 3, -3, -5, 6, 10, 7], [31, -3, -6, 6, -7, -7, -9], [-19, 3, -12, 7, 13, -1, -3], [-21, 8, 3, 3, -5, 2, 12], [6, -6, -12, 3, -4, 16, 12], [-2, 3, 3, 5, -9, -8, -4], [4, -16, 15, -3, -1, -6, 1], [-17, 3, -8, 10, 0, 6, 0], [-19, -4, 8, -6, 7, -12, -8], [-11, -3, -13, 4, 6, 8, -14], [24, -16, 3, -1, 1, -2, 11], [2, 1, 1, 2, -8, 1, 13], [22, 10, -8, -6, -8, 6, 4], [-33, -22, 3, -2, 7, 0, -10], [3, -1, 2, 1, -5, -14, -9], [21, 2, -9, -3, 1, -2, -6], [26, 6, -5, 2, -4, 4, 6], [-4, 10, 6, -3, 6, -2, 10], [-3, 6, 11, -12, 1, 12, 12], [-26, 8, -10, 14, 6, 2, 6], [-23, -1, 7, 5, 14, -8, 4], [-1, 4, 4, -6, -6, 18, 7], [9, 9, 17, -7, 0, -4, -10], [-9, -9, 4, 0, 5, -22, -5], [17, -28, 10, -6, -5, -24, -18], [-2, -6, -3, -3, -5, -7, 13], [-22, 8, 3, -13, -9, 14, 17], [-37, 22, 3, -3, 0, 16, 11], [-23, -16, -3, 4, 5, -9, -10], [-4, -2, 4, 6, 2, 6, -14], [11, 6, -3, 3, 3, -2, -14], [-23, -3, -6, -7, -9, 10, 15], [15, 3, 11, -5, 0, -16, -20], [-11, 2, -1, 7, 7, 8, -8], [10, 11, 6, 0, -11, 3, 2], [10, 12, -4, 0, 8, -6, 2], [7, -7, 16, 1, 5, -13, 19], [-5, -3, 0, 15, 5, -10, -7], [7, -23, -7, -3, 2, -6, -12], [38, 3, -4, -6, -7, 2, 8], [10, -1, 12, -4, 3, -6, 12], [14, -5, 14, -12, -14, -4, 3], [4, 18, -20, 3, 0, 16, -6], [-6, -2, -10, 8, -6, -4, -6], [-6, -9, 3, -1, 0, -2, -15], [-9, -14, 10, -11, -2, -11, 11], [-33, 10, 1, -10, -15, 0, -4], [-13, 16, -7, 10, 12, 2, 7], [-7, 17, -9, -6, -4, 10, 8], [-9, 2, -8, 1, -13, 10, 4], [-16, 10, -23, 1, 9, 7, 7], [7, -5, 6, -3, 5, 0, 5], [-11, -7, -2, -11, -3, 2, -5], [13, 4, 19, -1, -11, -19, -10], [-37, 9, 0, -8, -3, 18, 9], [-11, 6, 7, 6, 13, -14, -16], [6, 8, 0, -1, 6, -4, 0], [1, -14, -9, -6, 3, 14, 12], [-2, -6, -13, 8, -3, 8, -9], [2, -1, 1, -9, -6, -1, 15], [-17, -11, 4, -4, -5, -16, 11], [-4, 4, 4, -11, -6, 8, 2], [1, -4, 3, 6, 3, -28, -6], [-34, 11, 11, -2, 4, -4, 5], [-13, 19, -9, 8, 7, 8, 25], [-1, -1, 1, -9, 2, -2, -16], [-32, 18, 18, 3, 0, -6, 8], [15, 28, -2, -4, -6, -6, -1], [-5, -4, 12, -2, 2, -4, -7], [5, -8, 4, -6, 2, -8, 7], [-37, -7, -14, 5, 7, 22, 1], [10, -2, 1, 12, 17, 1, 3], [0, -18, 3, 4, -5, -6, -9], [-10, -9, 4, 9, 1, -8, -8], [7, -2, 8, 17, 4, 0, 5], [8, 10, 2, 8, 2, 14, 0], [-14, 11, -4, 13, 11, 19, 6], [22, 6, 2, 1, 3, 20, 11], [-1, 7, -3, 11, 6, -15, 4], [3, 18, -9, 8, -1, 22, 8], [-19, 16, 6, 11, 10, 2, -1], [20, -6, -8, 3, 5, 8, 3], [22, -9, -3, 1, 0, -18, -3], [-3, -22, -2, 5, 0, -11, -7], [-39, 1, 0, -2, 5, -8, -19], [1, 8, -3, 12, -3, 4, 16], [0, -28, 0, 3, -7, 0, -1], [17, -4, -6, -4, -4, -16, -7], [-3, 23, 1, 9, 4, -4, -10], [55, -1, 7, 6, 1, -12, -11], [33, -7, -9, 2, -2, 8, 0], [25, -20, -3, 0, -9, 6, -22], [-17, 2, 5, -10, -1, 3, -20], [2, -7, 8, 2, 5, 12, 16], [-28, 26, -3, -2, 4, 0, 4], [11, -12, 9, 0, -12, -6, -7], [-12, 11, 16, -1, -7, 2, 12], [6, -10, -4, -8, -10, 8, 6], [13, 6, 2, -6, 8, 9, 15], [14, -12, 2, 3, 10, -15, -6], [4, 2, -3, -5, 12, -6, 4], [-40, 13, 1, 4, 7, -18, 2], [0, 4, -9, -1, -11, 18, -9], [-14, 3, 2, -9, -15, 13, 0], [-26, -2, -3, 9, 11, 14, -3], [1, 10, -5, 3, -3, 4, -14], [-17, -13, -11, 4, 8, 21, 6], [-18, 13, 13, 7, -2, -6, -3], [22, 0, 7, -15, -11, -17, 7], [16, 4, 19, -5, -9, 4, 19], [3, -10, 3, -23, -3, 4, 10], [-2, -5, 11, 7, 8, -1, 9], [0, -20, 1, -7, 7, -6, 3], [14, 13, 1, 17, 4, -6, -7], [-6, -15, 3, -14, -1, 10, 18], [-3, -26, -4, 8, 14, -2, -11], [-37, 3, 9, -12, -16, 12, 6], [-25, -16, -21, -6, 11, 10, 2], [20, -15, -6, 0, 9, -18, 0], [13, -18, 5, -12, -2, -10, -5], [-9, 3, 3, 1, -2, 4, 16], [23, 17, 0, 4, 3, 12, 17], [7, -1, 2, 18, 2, 6, 8], [14, 17, 19, 3, -14, -2, 5], [27, -10, 14, -8, -2, -7, -3], [-2, 15, 2, 13, 7, 6, -4], [-15, -3, 0, 7, 13, 6, -1], [7, -1, 12, 0, -3, 24, 1], [-12, -11, -10, 14, -7, -4, -12], [-20, -4, 4, -2, 10, -19, -12], [-15, 19, 11, 16, 4, 12, -8], [22, -5, 0, 12, -5, -4, -20], [33, -16, 0, -5, -2, 7, -5], [-4, -17, -2, 11, -1, -4, -8], [13, 12, -6, 2, 8, 9, 15], [7, -12, 5, 13, -3, 14, -2], [15, 22, 4, 7, 6, -12, -7], [-4, 2, -3, -9, -19, -8, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5491_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5491_2_a_s();".
// 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_5491_2_a_s(:prec:=7)
    chi := MakeCharacter_5491_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_5491_2_a_s();".
// 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_5491_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5491_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![8, -12, -19, 26, 9, -10, -1, 1]>,<3,R![-8, 7, 68, 17, -29, -9, 3, 1]>],Snew);
    return Vf;
end function;