
// Make newform 1015.2.a.k in Magma, downloaded from the LMFDB on 16 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1015_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_1015_2_a_k();".
// 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_1015_2_a_k();".
// 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, 14, -31, 3, 21, -6, -3, 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], [-5, 10, 5, -7, -1, 1, 0], [2, -17, 10, 13, -7, -2, 1], [5, -22, 9, 14, -7, -2, 1], [-6, 30, -10, -21, 7, 3, -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_1015_a();"
function MakeCharacter_1015_a()
    N := 1015;
    order := 1;
    char_gens := [407, 871, 176];
    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_1015_a_Hecke( : Kf := Kf)
    return MakeCharacter_1015_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, 0], [-1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, -1], [0, 0, 0, 0, -1, 0, -1], [2, 0, 0, 0, 1, -1, -1], [-1, 0, -1, 1, 1, 0, -1], [1, 0, 0, -1, 0, -1, -1], [1, 0, 0, 0, 0, 0, 0], [-2, 0, 1, 0, -2, 0, -1], [3, 2, -1, 1, 0, -1, 0], [0, 2, 1, 0, -3, 1, 0], [4, 0, 1, 0, 0, 0, -1], [1, -2, 0, 1, -1, 0, 2], [4, -2, 0, 0, -1, 2, 1], [0, 0, 0, -2, 1, 3, 1], [-1, 0, -1, -1, 0, 0, 0], [1, 2, 0, 3, -2, 2, -1], [2, -4, 0, 2, 2, -3, 0], [1, 0, 4, -1, 0, 1, 1], [0, 2, 0, 0, 1, -1, 1], [-1, -2, 2, 3, -3, -2, -4], [-1, -2, -1, -1, -2, 1, 0], [2, 0, 0, 0, 3, -1, 1], [4, -4, 1, 0, 3, 0, 0], [5, -4, 0, 1, 3, -5, 0], [7, 0, -2, 3, 3, 0, 2], [-1, -4, 0, -1, 2, 0, 1], [4, -4, 1, -2, -1, -2, 0], [3, -2, 1, -1, -1, -1, 1], [1, -2, 1, 1, 2, -2, -4], [11, -2, -1, 3, 3, -2, 1], [2, -2, -2, 0, 1, -2, -3], [8, 0, 2, 2, 2, -5, 0], [-1, 0, -2, 1, 0, 1, 1], [-2, 6, 0, -2, -4, 3, 0], [5, -2, 3, -1, 2, 0, 0], [-8, 2, 4, 0, -4, 2, -2], [-1, 2, -4, 1, -3, -1, -4], [2, 0, 0, -2, -4, 5, 2], [-1, 4, 0, 1, 5, 0, 2], [-4, 8, 0, -6, -2, 5, 4], [3, -2, -5, -5, 5, -4, 3], [5, 2, 2, 1, -4, 2, 3], [-4, -2, 6, -4, 2, 3, 2], [-3, 0, 0, -1, 1, 1, 4], [-7, 2, -4, -1, -2, 0, 1], [-2, 2, 0, 0, -3, 5, -3], [-1, -4, 5, 3, -2, -2, -2], [6, 2, 2, 4, 2, -3, -2], [-6, 4, 0, -2, 2, 1, 4], [-9, 6, -4, -3, -5, 7, 2], [-6, 2, 3, 2, 1, -4, -2], [10, -4, 0, 2, 3, 1, 3], [7, 0, -1, -1, 7, 0, 1], [-1, 0, -5, 1, 3, -3, -1], [-1, 0, -1, 1, -1, -2, 5], [2, -4, 0, 2, 5, -3, 3], [4, 6, -4, -4, 5, -1, 3], [-7, -4, 2, 3, -4, 1, -5], [2, 6, -2, -8, -1, 0, 5], [-3, 0, 0, -3, 1, -3, 2], [0, 2, 5, 0, -5, 0, 2], [-2, 0, -4, 4, 0, -1, 6], [14, 2, -3, 0, -5, 1, -2], [-7, 0, -4, -1, -1, 1, -6], [9, -10, 3, 3, -2, -3, -2], [-6, 0, -2, 0, -7, 7, 3], [-5, -6, 0, -1, 2, -4, 7], [0, 6, -2, -4, 1, -1, 5], [-7, -2, 0, -3, 2, 1, 3], [-11, 8, 0, 1, 1, -1, 2], [3, -6, -2, 1, 3, 3, -4], [-5, -4, 4, 5, -3, 5, 2], [-1, 0, -2, -1, 3, -4, 0], [7, -2, 2, -1, 0, -4, 1], [3, -2, 4, -1, 8, 0, 3], [18, -6, 6, 0, -2, 3, -6], [4, -6, 5, 6, 0, -2, -1], [-2, 8, 0, 2, -6, 6, 6], [-13, 0, 0, 3, -2, -3, -9], [2, -2, -2, -4, -5, -4, 1], [18, -8, -2, 6, 2, -4, -2], [3, 6, 0, -7, -2, 10, 9], [5, -2, 1, 1, 3, 7, 1], [8, -4, 0, -6, 2, 0, 4], [8, 2, 2, -6, 2, 3, 2], [18, -10, -1, 8, 4, -2, 1], [-8, 2, 2, -6, -1, 0, -5], [-4, 0, 0, -4, 6, -1, -2], [-18, 4, -9, -4, 0, 4, 1], [-6, 2, -2, 4, 3, -4, -5], [11, -8, 2, 1, -1, -11, -2], [-7, 0, -2, 1, -11, 0, -4], [9, 4, -4, -3, 9, -5, 6], [-11, 2, -6, 1, 2, -4, -9], [6, -6, 2, -4, -4, -3, -2], [-12, 10, -2, -4, -2, 7, 6], [-9, -4, 2, 1, -5, 8, 6], [8, -2, 6, -4, 0, 5, 2], [1, -4, 0, -1, -3, 0, -4], [-1, -2, -2, -5, -1, 4, 10], [22, -6, 0, 6, 9, -2, 3], [-5, 10, -4, -5, -7, 6, 6], [-9, 2, -2, -3, 2, 4, 5], [-9, -2, -4, 5, 4, -8, -3], [8, 8, -4, 2, 4, 4, 6], [-6, -2, -4, 4, -4, -4, -6], [4, 2, -7, -4, -3, 1, 2], [0, 2, -2, 8, -6, -6, -2], [14, 0, 2, 8, -2, -10, -4], [3, -8, 3, -1, -1, 7, 1], [-6, 2, 3, 6, 3, -4, -8], [-1, -4, -2, 3, 1, -5, 2], [-1, 8, -6, 1, 3, -2, 6], [3, -10, 4, 3, 1, -3, -2], [-1, 8, -4, 1, 12, 4, -1], [-1, 4, 5, -3, -6, 6, -2], [11, 6, -6, 1, 3, 0, 0], [-1, -8, 2, -3, 9, 0, 6], [3, -10, -4, -3, 9, 3, 0], [7, 6, -6, 9, 0, 4, 3], [6, -2, -4, -2, 12, -7, 2], [-5, -6, 6, 3, 7, -1, 4], [0, 6, 4, -2, 4, 0, 6], [-2, -6, 0, -4, 4, -12, 0], [-7, 12, 2, 5, 1, -4, 0], [-11, 0, -6, 5, 7, -9, 2], [4, -2, -4, -6, 15, 4, 5], [-11, -2, 0, 1, 0, 3, -7], [-2, 0, 3, 4, -15, -1, 0], [6, -12, 4, -4, 0, 5, 4], [-2, -10, 3, 4, -3, -1, -10], [13, 2, 0, -5, 7, 5, 6], [3, -2, 3, -5, -10, 1, 4], [-9, 14, -6, -9, -3, 3, 0], [-9, 8, 0, -3, 0, 6, 3], [-4, 10, 6, -8, -4, 9, 4], [13, -10, -6, 3, -5, -1, 2], [-3, 0, -2, -3, 5, -2, 0], [9, 2, 4, -1, -5, 1, -8], [24, -14, -9, 0, 6, -3, -3], [-11, -8, -3, -9, 2, 3, 6], [1, -6, 13, -1, 1, 6, 5], [4, 6, -5, -2, 6, -5, 3], [14, -10, -6, 0, -5, 6, 5], [13, 0, 2, 1, -8, -3, -3], [-21, 4, 1, -1, -6, 1, 6], [-19, 4, 0, 1, -1, 0, -6], [-6, 6, -4, -4, -8, 13, 8], [1, 8, 9, 5, -16, 6, -2], [-1, -2, 2, 9, -6, -2, -7], [-23, -4, 2, -1, -2, -6, -3], [15, -2, 1, -7, 4, -2, 6], [7, -4, 0, -11, -1, -3, 4], [-9, 2, 8, -3, 4, -3, -3], [-28, 14, 4, -2, -8, 9, 2], [17, -10, 12, 1, 2, -2, -5], [11, -4, 0, 7, -4, -11, -11], [5, -10, 5, 3, 10, -20, -6], [15, -12, 4, 1, 1, 16, -2], [5, -8, 13, -3, -5, 0, 7], [-3, -8, -11, 7, 1, -4, -5], [-9, -6, 4, 1, -3, -3, -6], [-17, 0, -2, 1, -2, 2, 3], [-22, 8, -4, -4, 1, 7, -5], [-10, -2, 2, -10, 2, 1, 0], [-4, -12, 6, 4, 1, -6, -1], [11, -10, 4, 3, -1, 5, -4], [8, -2, 6, -4, -6, 5, -6], [3, -6, -2, 3, -4, -2, 5], [27, -8, -2, 9, 2, 3, 3], [-8, 0, 1, -4, 2, 5, -5], [-20, -4, -6, -4, 6, -4, 2], [16, -4, 4, -8, 9, -1, -7], [23, -14, -4, 3, -3, -11, -2], [0, -14, -8, 4, 11, -4, -9], [9, -22, 2, 1, -1, 0, -8], [-31, 6, -6, 1, 0, 4, -5], [-3, 4, -17, 1, 1, -4, -5], [-7, 8, -7, -3, -4, -1, 6], [8, 0, 2, -2, -8, -2, 0], [9, -16, 6, -1, -3, 9, 12], [-1, 0, 0, 15, -3, 0, -8], [10, 6, -2, 6, 3, 4, 11], [-17, 8, 9, 3, -11, 15, 3], [-22, 8, 0, -4, -11, 8, -9], [5, -12, -4, -5, -1, 10, 6], [21, -4, -6, -1, 20, -3, 9], [3, -6, 16, -1, -12, 6, -1], [4, 4, 1, -6, 7, 1, -4], [21, -14, -4, 3, 5, -18, 2], [10, 2, 3, -2, 9, -5, 8], [11, 6, 3, -17, -5, 1, 11], [-2, 0, 0, -6, 1, -8, -1], [-6, 10, -6, 4, -2, -3, -18], [16, 2, -14, -4, 21, -4, 11], [8, -12, -4, -4, 19, 1, 1], [-8, -6, 8, -4, 2, -2, 4], [13, -6, 10, 1, -6, 10, 13], [33, -6, 6, 5, 9, -9, -4], [-3, 20, -6, -3, -1, -2, 2], [10, -8, 5, 2, -7, -8, 0], [-5, -14, 7, 3, 6, -16, -12], [0, 0, 0, 4, -5, 4, -9], [-24, -4, 9, -8, 1, -7, -4], [-5, 0, 16, 3, -7, 8, 2], [12, 18, -6, -2, 4, -5, 2], [15, 0, 3, -1, 8, -5, 8], [23, 2, -3, 3, 13, -10, 3], [15, -10, -10, 1, 19, -11, 6], [20, -2, -3, -6, -7, 2, 0], [-26, 6, 2, -4, -9, -9, 1], [-6, 0, 5, 2, -7, -7, -6], [-1, 8, 4, -1, -18, 13, -7], [-16, 2, 8, -4, 6, -1, 2], [5, 4, 1, 7, -11, -1, 5], [-3, 14, -2, 1, 3, -7, -8], [14, -12, 1, 4, 7, -4, 4], [-13, 2, 0, -3, -19, -7, 4], [14, 6, 0, -2, 2, 7, 10], [5, 0, -12, -1, 15, -4, -6], [-12, 22, 5, -12, -6, 5, 15], [4, -4, 8, -2, -1, 1, 15], [-23, 10, -6, -3, -8, 16, 3], [19, -4, 14, 3, 3, -4, -8], [-11, 4, 4, -3, 2, 0, 17], [-1, 10, -8, -1, 7, -6, 4], [-1, 0, -2, 5, 8, 0, -11], [13, 12, -5, 5, -7, 13, 7], [-1, 8, 4, 9, -6, 3, 1], [6, 14, -8, 2, -9, -5, 1], [1, 0, 4, -9, 1, 6, 12], [-4, -6, -2, 0, 4, -3, 12], [-2, -2, 8, -8, -9, 13, 3], [20, 4, -3, 0, 19, -14, 6], [6, 14, -1, 2, 6, -6, 7], [-15, 10, -12, -5, 0, -10, 3], [5, 6, -5, -1, 7, -9, 13], [-8, 24, -5, -2, 7, 1, 8], [8, 8, 6, -8, 0, -4, 4], [16, 6, -12, 0, 4, -8, -6], [-12, 0, 10, 2, -14, 10, -6], [43, -8, -2, 1, 7, -3, 8], [-8, -4, 2, 2, -1, -5, 7], [-23, 20, -4, -3, -5, 5, -4], [-3, 12, -4, -3, -9, 8, 0], [-8, -12, -8, 4, 12, -5, -8], [15, 8, 3, -3, -20, 9, -4], [-14, 12, -1, -2, -4, 7, -7], [12, -8, -2, 10, 10, -6, -8], [-7, -30, 5, 3, 4, -4, -4], [3, 8, -9, 5, -5, 3, -3], [-22, 12, -10, -8, 1, 1, -3], [-4, 16, -4, 0, 4, -1, 8], [-11, -2, 6, -5, 3, 1, -6], [3, 16, -14, -5, -5, 9, 4], [-13, 2, 8, 7, -12, 19, -3], [29, -10, 4, 11, 9, 5, -2], [30, -2, -3, -8, 11, -16, 6], [12, 2, -1, 0, 0, -13, -3], [-9, 6, 0, 5, 11, -5, -6], [2, 4, 2, -4, -15, 19, -3], [-11, -12, 0, 3, 18, -5, -1], [12, 12, -1, -2, -8, 6, -7], [4, 12, -2, 8, -1, 17, -5], [-42, 0, -5, 2, 1, -6, 0], [23, -8, 3, 5, 2, 9, 8], [-8, 0, 8, -12, 5, 7, -1], [-23, 16, -6, 3, 5, 10, -8], [0, -8, 10, -2, 3, 11, 7], [13, 10, -4, 3, -5, 1, 0], [-9, -2, 10, -7, 1, 12, 6], [14, -16, 0, 0, -7, 5, -1], [8, -12, -8, -2, 21, -22, 9], [3, -28, 9, 1, 0, -11, -10], [-24, 4, -6, 2, 0, 1, 10], [28, -6, 0, -4, 3, -2, 3], [8, -8, 6, -10, -9, -5, 1], [-27, 14, 8, -1, 0, 6, -1], [7, 2, -8, 1, -8, 8, 1], [16, 8, 12, -6, 3, -9, 11], [-16, -22, -2, -8, 9, 2, 1], [-1, 0, -4, 5, -2, -16, -3], [-11, 12, 8, -9, -17, 25, -2], [0, -2, 9, 4, 8, 0, -1], [21, 0, -7, 9, -3, 3, 3], [31, -20, 0, 3, -3, 6, 12], [27, -8, 2, 3, 14, -3, 13], [-20, 0, 6, -6, -14, 25, -8], [12, -26, 2, 4, 2, -11, 2], [-19, -4, 6, -5, -1, -14, 0], [41, -6, -5, 1, 12, -6, 6], [-8, -8, -9, -4, 6, 1, -11], [26, 16, -6, -6, 7, -7, -5], [6, 16, 1, -8, -14, 14, 1], [-8, 16, -12, 4, -1, -14, -5], [-28, 24, -6, -8, -2, 10, -6], [1, 4, 10, -9, 2, -4, 9], [5, -18, -4, 9, 9, -5, -12], [-17, -6, 8, -9, -2, 8, -5], [24, -4, 2, 14, 12, -11, 4], [7, 6, 3, -1, -1, 3, -17], [31, -18, 0, 15, 12, -3, -1], [-5, -6, -6, -1, -6, -4, -11], [8, 16, -5, 4, 8, -12, 5], [7, -6, 2, 9, -15, 1, 4], [6, -4, 0, 10, 0, 4, -10], [2, 0, 2, -6, -5, -11, 5], [18, -14, 12, 2, 6, -6, 2], [2, 6, -6, -14, 3, -8, 3], [-7, 16, -8, -1, -1, -9, 4], [-9, 18, -12, -13, -3, -3, 0], [-4, 4, -6, -2, -19, 13, -9], [-4, 8, -6, -6, -8, 0, -12], [-13, -14, 3, -15, 9, -1, -9], [18, -14, 10, 6, 2, 6, -4], [-12, 20, 4, -8, -11, 12, 17], [-12, -6, 8, -2, -8, 21, 4], [35, -8, 1, -5, 18, -3, 12], [8, 18, -10, 0, 3, 16, 11], [-2, -6, 6, 0, 6, 15, 10], [-4, 4, -9, 10, -13, 3, -2], [13, 6, -6, -1, 21, 5, 16], [-18, 4, -1, 6, -19, 0, -22], [-8, 0, 8, -6, -13, 17, -5], [19, -6, 16, -3, -16, 8, 9], [22, -12, -9, 8, 1, 5, -2], [24, 8, -10, 2, -8, 6, 4], [-17, 16, -4, 9, 7, 16, -8], [11, -4, -12, 7, 8, -3, -1], [6, 26, -18, -10, 8, -7, 6], [-19, 14, 8, 9, -3, -12, -12], [-50, 12, 8, -4, 0, -3, 0], [-63, 4, 12, 3, -20, 0, -11], [-49, 8, -6, 1, -5, 10, -10], [5, 2, -6, 1, -3, 1, -14], [-9, -4, 13, -5, -6, 4, 6], [10, 4, -18, -14, 16, -1, 16], [4, 26, 5, 4, -18, 2, 5], [14, -4, 4, -2, -13, 23, 3], [-12, 4, -15, 2, 0, 7, -9], [15, -18, 4, 9, 8, -10, -7], [-26, 4, 0, -10, 3, -3, 21], [-21, 0, 16, 11, -14, 6, -5], [-16, 14, -3, -10, -8, -6, 1], [16, 8, 1, 4, -4, 6, 15], [-49, 18, -5, -9, -8, 10, 2], [17, -10, 20, 1, 9, -9, -8], [-1, 2, -10, 5, 6, -14, -11], [7, -10, -2, -9, 14, -14, -7], [-12, -10, -8, -2, 2, -3, 2], [-12, 12, 12, 2, -13, 23, -9], [-4, 0, -14, 2, 14, -8, -6], [0, 20, -4, 6, 5, -13, 9], [0, -10, 11, 2, -8, 7, -1], [-19, -4, 0, -1, -24, 15, -5], [-9, -8, 9, -11, -10, 4, -4], [17, 8, -2, -1, 6, 9, 1], [23, 2, 0, -9, 4, 4, 11], [21, 0, 8, 3, 5, 4, -8], [10, 2, 3, -4, 11, -10, 4], [23, -24, -2, -7, -8, 7, 9], [10, 8, 2, 4, -13, 0, -5], [-6, -20, 3, 0, 1, 6, -4], [-13, 16, 6, -19, 2, 19, 11], [-49, 8, -10, 1, 6, 3, 3], [2, 8, 2, -8, 12, 4, -10], [14, 4, 6, 10, -15, 4, -17], [-20, 6, 4, 0, 0, -13, -6], [-1, 20, -20, 3, 19, -4, 10], [0, 18, -6, -6, -6, 3, -8], [10, -10, 20, -8, -3, 12, -7], [29, -12, -2, 9, 8, -27, -7], [3, 2, -11, -11, -3, 15, 15], [5, -12, 0, -3, -16, 8, -1], [5, 2, -10, -5, 12, -16, 13], [40, -14, 4, 8, -7, -14, -1], [21, -26, 0, -3, 8, -4, 1], [-9, 12, -5, 7, 10, 0, -14], [-4, -12, 12, -10, 18, -6, 22], [-6, -34, 15, 12, -7, 4, -10], [6, 12, -3, -8, 5, 2, 12], [-14, 4, -16, 2, 16, -22, -2], [-13, 2, -20, 9, 10, -12, 11], [1, -6, 6, -7, 6, 7, 17], [-23, 6, -12, 1, 2, -2, -17], [31, 0, -1, 5, -8, 11, 2], [-10, -8, -4, -14, 4, 9, 6], [7, -2, 10, -3, 7, 7, 20], [4, -16, 0, -4, -3, 2, 1], [4, -24, 22, 0, 3, 8, 3], [-3, 14, -12, 1, -3, -8, -16], [12, -2, 10, 4, -9, 5, 1], [-5, 30, 2, -11, -27, 25, 8], [-6, -2, 10, -16, 3, -4, 1], [14, -28, 9, 6, 1, -10, -2], [-11, 24, -6, 1, -3, -6, 12], [8, 24, -12, -10, 4, 0, 2], [5, 2, -6, -5, -9, 3, 8], [13, 6, 3, 7, 10, -5, -12], [-1, -6, -3, -13, 9, 7, 5], [22, -10, 10, -8, -6, 16, 8], [-3, -6, 14, 17, -12, -8, -23], [35, -30, 2, 11, 5, 9, -4], [-27, 20, 10, 1, -12, 11, -3], [-19, 2, 8, 13, -16, 30, 5], [-9, 12, 2, 5, 4, 4, -3], [15, 22, -4, 5, -4, 10, 5], [-47, 22, 0, -15, -8, 7, 3], [24, -20, 10, -4, -7, -7, -5], [10, -12, 4, -12, 17, -3, 19], [-11, -26, 16, -11, -9, -8, -2], [-7, 16, 9, -3, -25, 23, -11], [-1, -6, -3, -21, 18, 11, 30], [-4, -12, 4, 12, -7, 0, -9], [12, 0, -16, -10, 1, 5, 9], [17, 22, -18, 1, 7, -3, 4], [27, -14, 2, 11, -5, 2, -14], [6, 0, 0, -4, 0, 4, 0], [0, 6, -14, 6, 1, 4, 1], [-2, 28, -9, -6, -6, -8, -3], [-41, -2, -13, 3, -12, 0, 16], [1, 10, 0, 9, 8, -5, 15], [8, -36, 20, 10, -4, -1, -4], [-29, 6, -10, -5, 5, -15, -4], [-18, 10, 16, -12, -10, 11, -4], [9, 24, 0, -5, 4, -9, 9], [-20, -14, -1, -4, 13, -10, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1015_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1015_2_a_k();".
// 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_1015_2_a_k(:prec:=7)
    chi := MakeCharacter_1015_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_1015_2_a_k();".
// 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_1015_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1015_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 14, -31, 3, 21, -6, -3, 1]>,<3,R![-16, -64, -32, 68, 13, -16, -1, 1]>],Snew);
    return Vf;
end function;