
// Make newform 1519.2.a.f in Magma, downloaded from the LMFDB on 25 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1519_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_1519_2_a_f();".
// 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_1519_2_a_f();".
// 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 := [3, -9, -3, 17, 0, -8, 0, 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], [2, -4, -1, 1, 0, 0, 0], [0, 6, -1, -6, 0, 1, 0], [-3, 1, 11, -1, -7, 0, 1], [-5, 1, 12, -1, -7, 0, 1], [-8, 8, 16, -7, -8, 1, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 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_1519_a();"
function MakeCharacter_1519_a()
    N := 1519;
    order := 1;
    char_gens := [1179, 344];
    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_1519_a_Hecke( : Kf := Kf)
    return MakeCharacter_1519_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, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [-2, 0, -1, 1, 1, 0, -1], [0, 0, 0, -1, 0, 0, -1], [1, 2, 1, -1, 1, -1, 2], [-1, 1, 0, -2, -2, 0, 1], [-2, -1, 0, 0, -1, -1, 0], [-4, 0, -1, 1, 0, 1, -2], [1, 0, 0, 0, 0, 0, 0], [-2, 0, 2, 0, -1, 0, -1], [0, 0, -2, 2, 1, 1, -2], [-1, 0, 1, 0, -2, 3, -1], [-1, 3, 0, 3, 0, 3, -1], [0, 1, 0, -2, 0, -4, 5], [0, 3, 2, 1, 3, 0, -1], [-4, -1, -3, -2, -1, -1, -1], [3, 0, 2, -3, -3, 1, 3], [-4, 0, 1, -5, 0, -2, -1], [-1, -1, 1, 0, -6, 6, 0], [1, 6, 0, 0, -2, 2, 5], [3, 1, -1, 2, 4, 1, -4], [0, -3, 2, -4, -1, -6, 2], [-1, -7, 0, -1, -2, 0, -1], [2, 1, 0, 2, 3, 2, 0], [-2, -1, 3, -3, 0, -3, 0], [1, 0, 1, 0, 2, -4, 7], [-2, 1, 2, 1, 3, -2, 0], [-3, -2, -1, 1, 2, -1, 3], [3, 0, 3, 1, 2, 4, 0], [-4, -6, -3, 3, -1, -1, -2], [-4, 6, 2, 1, -3, 6, 2], [6, -2, 0, 1, 2, 4, 4], [-3, -4, -4, -2, 1, 0, 3], [4, -1, -2, 7, 3, 3, -5], [3, -2, -3, -1, 2, -5, -3], [-6, -1, -4, 2, 0, 1, -6], [-2, -1, -2, -2, -5, 3, 4], [11, 2, 0, 2, 0, 5, 4], [2, -3, 0, -5, 0, -2, 6], [1, 2, 2, 0, 4, 0, 7], [-8, -3, -4, -3, -1, 3, -6], [-7, -4, -7, 0, 1, 1, -10], [-10, -6, -5, 0, -1, 4, -2], [2, 1, -3, -2, 0, -7, 0], [-2, 0, 4, -2, 0, -7, 5], [-2, 2, -2, 2, 2, -3, -4], [6, 2, 1, 1, -3, -2, 0], [-3, 1, 3, 4, -2, 4, -6], [4, 0, 3, 2, 3, 1, 0], [1, -2, 2, 1, 5, -5, 6], [4, 0, 3, 3, -5, 8, -2], [-5, 0, -2, 5, 7, -6, -5], [0, -2, -2, 3, 4, -7, -2], [-8, 1, 0, 2, -1, -1, -3], [10, -3, 0, -5, 3, -5, 11], [5, 0, 1, 2, 1, 0, 6], [7, 8, -2, 5, -1, 1, 2], [-10, 4, -4, 9, 4, 0, -6], [5, 4, -1, 1, -8, -2, 5], [-13, 4, 1, -6, -10, -3, 5], [-10, 3, -2, 5, 2, 1, -1], [0, -3, -3, 4, 14, 2, 0], [-1, -6, -8, -2, 3, 0, 1], [2, -7, 3, -2, -1, 7, -1], [4, 1, 3, -2, -9, 0, -3], [-11, -10, -1, 4, -2, -1, -6], [4, 2, 4, 4, 4, -7, 4], [-5, -3, 4, -8, 1, -3, 1], [-6, 1, -3, 4, 10, -2, -3], [1, 6, 4, 4, 3, 5, -5], [15, -7, 4, 3, 5, -1, 4], [-1, -1, -6, -1, -13, 2, -4], [0, -1, 4, -2, -9, 9, -8], [8, -7, -7, -4, 0, -5, 2], [-6, 5, 3, 4, 2, -1, 1], [-2, -4, 3, -2, 0, 2, 2], [-13, 9, 0, 0, 4, -2, 4], [-9, -2, -1, 4, 0, 11, -16], [9, 4, 7, -1, -4, 1, 1], [-6, 7, 1, -5, -5, 10, 1], [0, -7, -5, 0, 3, -10, -4], [0, -5, -6, 0, 7, -1, -11], [-17, -4, -8, -1, 1, -2, -11], [3, 4, 6, -10, -4, -10, 7], [-5, 4, 5, 3, 6, -1, -1], [-2, 8, 1, 7, -1, -5, -2], [-9, -3, 6, -2, -7, 0, 3], [-3, 10, -7, 6, 8, -3, -6], [-2, -4, -3, -2, 10, -2, -10], [-15, -12, -6, 4, -3, 1, -13], [8, -12, -6, 13, 3, 2, -5], [-3, 10, 4, -3, 3, 1, 5], [-6, -8, -1, -11, -4, -14, 0], [1, 7, 12, -7, -16, 1, 4], [14, -2, 9, -2, -12, 5, 1], [-9, -3, -6, -2, -6, 6, -2], [5, 11, 9, 3, 1, 9, 9], [-7, -12, -2, 5, 9, 4, -7], [-5, -4, -4, -3, 0, 4, 1], [-16, 13, 3, -10, -7, 0, 4], [-7, -8, 3, -10, 2, -3, 4], [-6, -13, 3, 2, -4, 5, -5], [2, -1, 5, 5, -7, 5, -2], [-19, -7, -8, 4, -2, -9, -9], [-12, 1, -4, 8, -6, 2, -9], [5, 11, 7, 5, 7, -2, 0], [-8, -8, 1, -3, -4, 3, 8], [-10, 2, 1, 2, 10, -3, -2], [-11, -2, 5, 0, 4, -6, -4], [-10, -9, -6, -6, 2, 2, -12], [-14, -7, 5, 1, -11, 5, 2], [-5, 9, -6, 7, -7, 4, -10], [11, -6, -6, 6, 16, -4, -1], [-27, -4, 6, -6, -15, -1, 10], [10, -4, -6, -2, 3, 1, 1], [10, -2, 3, -8, 0, -6, 14], [7, 2, -2, 14, 13, 9, -4], [5, 13, 0, 0, 0, 8, 11], [-3, 1, -5, -6, -5, -8, -7], [9, 3, 6, -6, 2, 2, 2], [1, -1, 11, -2, 1, -4, 11], [4, 0, 4, -1, 0, -10, 5], [5, 3, 3, 0, 10, -9, 14], [-13, 12, 3, 4, -11, 4, 1], [-6, -1, 8, -6, 2, -5, 0], [5, -6, -2, -3, 2, -12, -1], [-12, 7, 3, -10, 3, -10, 6], [-7, -7, 0, -3, -7, -1, -1], [10, 9, -8, 5, 10, 11, -5], [18, 3, 5, 3, -5, 1, -1], [5, -8, 7, 0, 6, 3, 9], [-5, -16, 3, -1, -5, -3, -9], [8, 6, 1, 3, -1, 15, -4], [-3, 1, -5, 3, 0, -4, 6], [-12, 3, 1, 10, -14, 3, -7], [13, 5, 7, -8, -4, -15, 11], [8, -2, -3, 3, 1, 1, 19], [-16, -5, 6, -5, -2, 12, -5], [6, 13, 4, 4, 0, -1, 7], [-5, 12, -4, 0, -9, -6, 0], [-1, 0, -3, 1, -14, 7, 3], [19, -1, 3, -1, 1, -9, 13], [-6, 7, 5, -3, 9, -8, 13], [23, 3, -6, 3, 6, -1, -10], [3, 6, -11, 7, 3, 2, -11], [2, 10, 3, 0, 1, -5, 9], [0, 9, 11, -7, -6, -2, 6], [-6, 1, 0, -9, -12, -7, -2], [-3, 13, 0, 6, 10, 6, 2], [20, -2, -6, -1, 2, -6, 10], [13, -2, -1, -6, 2, -4, 3], [-7, 7, 3, -8, 1, -1, 2], [22, 8, 0, 12, 19, 3, -11], [-8, -11, 2, -3, -10, 2, -1], [7, 7, 1, -6, 0, -5, 11], [3, -4, 3, 3, -18, 9, -7], [-8, -3, -3, 1, -14, -4, -9], [-3, -7, -8, 10, 11, 0, -15], [-7, 2, 1, 2, 1, -11, -7], [2, 4, 8, -13, -14, -11, 7], [6, 6, -2, -3, 10, -8, -8], [0, 4, -3, 3, -4, 18, -7], [-23, -10, 2, -7, -5, -1, -3], [-7, -7, -10, -8, -2, -3, 7], [-7, -4, 0, -1, -12, 4, -9], [-2, -11, 3, 4, 0, 12, 4], [-4, -1, -15, -2, 19, -11, -2], [18, 19, -5, 4, 8, -2, -8], [5, 5, -3, -5, 13, -10, -5], [-14, 12, 8, -4, -7, 14, 1], [-21, 1, 3, 4, -21, 6, -3], [-22, -7, -5, -3, -3, -10, -2], [13, -6, 2, 3, 6, 18, -6], [7, 3, 12, 0, 8, -9, 10], [0, -8, -3, -5, -1, -19, 7], [-13, -6, -5, 6, -4, 1, -14], [23, -2, 7, 3, 9, -2, 0], [32, -4, 0, 3, 8, -7, 0], [-5, 3, 2, -9, -3, 13, -7], [10, -11, 5, 0, -20, 13, -3], [10, -5, -9, 5, 10, 6, -6], [-14, -2, -1, 7, -10, -3, 3], [-1, -1, -2, 9, -10, 28, -6], [19, 2, 14, 8, 9, -5, 3], [0, -13, -10, 6, 14, 3, -5], [-21, 12, -5, 9, -4, 10, -15], [-8, -11, 5, 8, 14, -6, -12], [-31, -11, -11, 10, -6, 4, -17], [22, -1, -6, 19, 13, 8, -16], [1, 0, -13, 3, 0, 9, -17], [13, -5, -9, 3, 3, -3, 3], [2, -8, 10, -1, -2, -4, 0], [-1, 8, -16, 7, 14, -1, -6], [-12, -2, 16, -9, -8, -2, 13], [-16, 6, 1, 5, 17, 3, 3], [20, 3, 10, 12, 5, -9, 9], [6, 15, 7, 3, 4, 1, 21], [-12, 3, 2, 4, -6, 18, -4], [20, 1, 9, 9, 9, 4, 6], [-4, -17, 4, 2, -18, 12, -6], [-17, 6, 0, 9, 0, 15, -10], [11, 19, 3, 7, 2, 17, 9], [13, -3, -8, -8, 12, -10, -3], [21, -7, -12, 0, 6, -8, 1], [12, 1, 3, 5, 3, 7, 19], [-5, 5, -2, 6, -11, 7, -1], [-35, 3, -4, 3, -4, -1, -13], [2, -7, -7, 0, -5, -8, 13], [-23, -3, -10, 19, 7, 5, -28], [3, 17, -3, -4, -2, -7, 7], [2, -4, -11, 1, 13, -13, 10], [19, -1, 6, 2, -3, 4, 11], [-11, 1, 27, -9, -25, 6, 12], [2, 17, 3, 15, 4, 4, -3], [17, -6, -5, -12, 10, -6, 1], [5, 23, 6, -6, 1, 1, -4], [21, -6, -7, 3, 11, -10, 0], [20, -3, 1, -6, 8, -18, 14], [29, 7, 11, -3, 3, 11, 9], [-27, -7, 4, 5, -18, -2, -3], [-7, 13, 4, 9, -2, 11, 12], [18, -3, 1, -15, 9, -3, 9], [-4, 4, 12, -23, -10, -4, 13], [7, 0, 5, -21, 11, -15, 9], [1, 8, -5, 3, 20, -23, -2], [-14, 13, 0, 0, 2, 0, -5], [24, -14, 1, -14, 16, -14, 10], [-4, -10, 4, 7, -21, 21, -9], [23, 6, -4, -7, -2, 14, -12], [20, 1, 7, 5, 5, -14, 6], [-1, 1, 9, 3, -11, -5, 14], [-14, 6, -12, 19, 16, 8, -5], [-25, 4, 0, -2, 6, -1, 3], [7, -5, 4, -15, -26, 1, 5], [-5, 5, -6, -7, -13, -7, 13], [-12, -8, -5, -9, 2, -16, -7], [5, -1, 4, -2, 15, 3, -4], [5, 3, 7, -3, 10, 17, 0], [-5, 4, -9, 11, -11, 4, -10], [-7, -20, 2, -14, -18, 0, -1], [11, -4, -15, 7, 21, -9, -2], [33, 12, -4, -4, 12, 2, -3], [9, 11, -1, 15, 9, 30, -17], [0, -6, -2, -9, 5, 12, -6], [-7, -6, -11, 12, 13, -9, -6], [21, 9, 8, 1, 2, 17, -3], [4, 6, -7, 6, 5, 14, -19], [29, 15, -12, -2, 8, 13, 1], [-8, 15, -2, 19, 10, 4, -7], [4, 6, 6, -5, -18, 0, 6], [24, -11, -1, -13, 13, -17, 7], [-15, -3, 9, -20, -19, -14, 24], [-28, -5, 3, 0, -16, 4, 3], [-2, -25, -1, 9, 12, 6, 1], [1, -6, 7, -5, 4, -22, -6], [-7, -3, -13, 24, 25, 1, -14], [-21, -14, 15, 0, -2, -2, -10], [11, 4, 13, -11, -2, -6, 18], [3, -20, 1, 14, 5, -4, -5], [-5, 6, -5, 0, 9, -5, -18], [-5, -3, 2, -25, -9, 11, -1], [-37, 6, 0, -1, -3, 11, -6], [-9, -3, -16, 20, 30, -1, -15], [-6, -9, -2, -5, -6, 19, -17], [5, -7, -7, -2, 22, -8, 2], [4, -7, -20, -2, 16, -9, -21], [14, 18, 9, 11, 20, -9, 1], [17, -8, -8, -9, 15, -10, 3], [14, 22, 8, -11, 5, 5, 3], [-29, -16, -1, -3, -20, 1, -3], [-20, -8, -12, -10, -8, 3, 9], [-17, -2, -13, 18, 6, 15, -27], [35, 5, 6, 18, 2, 15, 2], [-2, -1, -6, 18, 6, 25, -30], [-18, -26, -10, 6, 8, 6, -25], [5, -17, 7, -22, -7, -5, 9], [-20, -11, -10, 2, -12, 11, 8], [-6, -12, 0, 8, 1, -4, 16], [6, 8, 15, -15, 19, -13, 24], [15, 12, 20, -14, -13, -6, 33], [23, -4, -3, -3, 19, -9, 1], [-30, 2, 0, -3, -9, -12, 0], [-28, -16, 7, -6, -16, -4, 17], [13, 22, 6, 8, -3, -11, 3], [-8, -17, -11, 6, 27, 8, -10], [12, 2, 13, 5, -7, 15, 15], [-17, -7, 14, -1, 12, -18, -11], [-1, 1, 5, -21, -28, 0, 0], [-9, 4, 4, -5, -20, 8, 8], [-19, -11, -14, -7, 15, -9, -4], [-18, 6, 17, -12, -10, -5, -2], [8, 1, -21, 4, 16, -19, -5], [7, -4, -1, 7, 20, -3, -15], [20, -7, -14, -4, 6, -1, 21], [7, 8, 7, 13, -1, -1, 15], [0, 20, 6, -8, -4, -9, 21], [-7, -12, 0, 6, -5, -1, 3], [31, 8, 10, 11, 0, 5, 22], [8, -25, -6, -14, -3, -11, 9], [-10, -17, -9, -11, 0, -6, 10], [-33, 2, -8, 10, -7, 27, -10], [-7, -10, -12, -15, 6, -4, -12], [-14, 17, 4, -16, -4, -11, 19], [13, -4, -3, -5, -6, -7, -4], [3, 2, -9, 20, 18, 12, -30], [-21, -1, 11, -6, 0, 8, 4], [-11, 4, 2, 1, 17, -27, 10], [24, 16, 16, 7, 3, 1, 8], [-31, -26, -12, -5, -8, 1, 3], [-15, 14, -3, 12, -13, 13, -7], [-39, -2, -11, -1, 3, 0, 0], [-1, 1, 13, 1, -2, -3, 6], [29, 0, 12, -21, -19, -1, 14], [-6, 4, 1, 0, -7, 6, -7], [-1, -12, -4, 2, 25, -11, -13], [-2, 2, -4, -11, 9, -12, 10], [-1, -14, 0, 10, -2, 3, -20], [-8, -1, -21, -2, 19, -30, -5], [18, -14, 6, -18, -13, -2, 28], [-26, 11, 10, -14, -17, -2, 22], [33, -8, -10, -5, 28, -16, -12], [-14, 11, 2, 5, 9, 7, -13], [9, -5, -2, 3, 5, -18, 6], [-22, 13, -4, -13, -19, 3, -7], [-1, -33, -1, 12, 5, 2, -10], [12, 19, -9, 2, 3, 0, -12], [-6, -7, -9, 1, -6, 12, -6], [-3, -5, -9, 22, 21, 10, -21], [2, -1, 12, 10, -28, 19, 1], [17, -6, -1, 12, 12, 9, -9], [-13, 4, -9, -12, -15, 11, 3], [-25, 15, 1, -6, -17, 9, -13], [-6, -4, -1, -8, -12, -5, 11], [-3, 0, 15, 3, -24, 6, 17], [13, 1, 1, -6, -18, -11, 10], [7, 2, 14, -15, -22, -6, 13], [-8, -6, -5, 3, -6, -12, 4], [-20, -12, -14, -6, 6, -16, -19], [14, 23, 14, -12, 2, -4, 14], [4, 2, -4, -18, -2, -19, 20], [37, 8, 1, 5, -7, 4, 15], [-16, -7, 6, 11, -2, 0, 1], [-11, 1, 12, -15, -14, -11, 10], [-9, 20, 12, 5, -8, 9, 4], [-2, 0, 5, -15, -13, -4, 9], [-33, -9, -8, 2, -17, 29, -12], [2, 8, -2, -4, -3, 10, 7], [-44, -2, 7, 4, 5, 7, 3], [-15, 11, -4, 16, 9, 10, -14], [-12, 5, 13, 11, -25, 23, -1], [-34, -5, -16, 12, 7, 1, -29], [6, -13, 18, 2, -8, 5, -5], [18, -14, 2, -2, 8, -27, 3], [31, 1, 4, -4, 2, -12, 9], [-7, 13, 15, -21, -25, 1, 24], [27, 14, 17, 7, 8, 1, 17], [-17, 6, -7, -24, -6, 8, 0], [13, 27, 7, 12, 14, 8, -6], [-9, -23, -13, -10, -22, 1, 20], [-3, -3, -6, -7, 22, 3, 5], [32, -5, -9, -1, -8, 2, 14], [-21, 0, -16, 0, 5, -1, -14], [-3, 8, 5, -10, -4, 6, 11], [-21, 22, 0, 6, 6, 0, -12], [-20, -9, -2, -14, -1, -24, 12], [30, -12, 9, -10, 12, -12, 20], [2, -10, -9, 14, 9, -2, 4], [17, -9, -6, -1, -11, 5, -25], [27, -25, 15, 8, 6, -8, 4], [14, 7, 8, 5, -7, 19, 20], [-40, -16, -16, 19, 3, -9, -18], [18, -8, 1, 20, -9, 2, 1], [37, -24, -10, 7, 17, -16, 3], [9, 0, -13, 7, 12, 7, -7], [-6, 10, 21, -11, -8, 4, 7], [-41, 15, 5, -5, 0, 2, -9], [-28, 20, 9, 1, -8, 19, 3], [20, 7, -20, 24, 16, -3, -15], [-28, 19, -14, 20, 22, -11, -17], [-1, 34, 19, 3, 12, 10, 15], [11, 22, 21, -1, -2, 26, 6], [9, 9, -21, 20, 16, 1, -13], [-19, -12, -1, 6, -3, 4, -32], [-47, 11, 5, 3, -9, 24, -8], [-22, -12, 7, 7, 2, -21, -6], [28, 11, 0, 11, -5, 13, -7], [9, -10, -14, 19, 24, 5, -16], [2, 26, -10, -1, -4, 1, -6], [-9, -10, -2, -4, -33, 17, 3], [-24, 17, 1, -2, 5, -3, -2], [-27, -7, 17, -11, -23, -1, 6], [4, 2, 7, -12, -4, -9, 9], [-13, 5, 2, -7, -8, -22, -9], [-11, -12, -8, 2, 3, -4, 23], [38, 22, 19, 6, -5, -7, 14], [2, 3, -20, 29, 18, 23, -15], [-5, -25, -1, -2, 14, 14, -10], [-1, -16, -3, -27, 1, 1, 10], [24, 9, 5, -4, 1, 14, 13], [19, 9, 9, -14, 7, 4, 13], [-38, 16, -3, 8, 13, -5, 15], [15, 3, -9, -20, -8, 15, 13], [11, 16, -2, 15, -4, 7, -12], [10, -19, 3, 6, -17, 11, -24], [32, 4, 6, -14, -5, 6, 15], [-33, -17, 2, -8, -26, 15, 11], [-1, -2, -32, 2, 15, -1, -11], [-25, -6, -8, 12, 6, -4, -14], [-8, 41, -6, 4, -1, 5, -6], [14, -4, -3, 6, -2, -4, 0], [10, -11, -19, 13, 13, 3, 9], [-9, 5, 2, -7, -15, -4, 10], [0, -13, 7, -12, -20, 0, 5], [-20, 6, 5, 14, -9, 9, -11], [-21, 0, 19, -14, -5, -2, 12], [-24, -12, -13, 6, 12, 19, -31], [7, -10, 10, -7, -24, 22, -9], [9, 1, -1, -11, -28, 15, -12], [4, 11, 3, -4, -30, -1, -2], [28, 16, 1, -10, 37, -15, 7], [16, -11, -8, -15, 6, -32, -1], [-27, -12, -2, 4, -26, 39, -5], [-22, -7, -9, 0, -5, 23, -23], [-11, -22, -24, -6, -12, -8, -2], [23, 24, 18, -6, -10, -6, 16], [-13, -2, 7, -3, -18, 11, 11], [-13, 5, -24, 18, 0, 4, -31], [-15, -12, -10, -23, -16, 6, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1519_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1519_2_a_f();".
// 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_1519_2_a_f(:prec:=7)
    chi := MakeCharacter_1519_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_1519_2_a_f();".
// 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_1519_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1519_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 6, 24, 4, -17, -5, 3, 1]>,<3,R![-3, -9, 3, 17, 0, -8, 0, 1]>],Snew);
    return Vf;
end function;