
// Make newform 9036.2.a.i in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_9036_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_9036_2_a_i();".
// 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_9036_2_a_i();".
// 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, -9, -17, 8, 18, -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], [-5, -17, 15, 13, -8, -2, 1], [5, 17, -7, -18, 6, 3, -1], [-7, -18, 8, 18, -6, -3, 1], [-16, -73, 9, 92, -25, -16, 5], [-29, -125, 15, 136, -35, -23, 7]];
    Rf_basisdens := [1, 1, 3, 1, 1, 3, 3];
    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_9036_a();"
function MakeCharacter_9036_a()
    N := 9036;
    order := 1;
    char_gens := [4519, 2009, 1261];
    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_9036_a_Hecke(Kf)
    return MakeCharacter_9036_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], [0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, -1], [1, -1, 0, 1, 1, 2, -1], [-1, 2, 1, 0, 0, -1, 1], [2, -2, 1, 1, 0, 1, -1], [-2, 0, -1, 0, -1, 0, 0], [1, 1, -2, -2, -1, 0, 1], [0, 1, 1, -1, -1, -2, 2], [-3, -1, 1, 1, 2, 1, 0], [0, -1, -3, 0, 1, 1, 1], [0, 3, -1, -3, 0, -4, 2], [-3, -1, 1, 2, 0, 1, 0], [0, 2, 3, 1, 3, 1, -1], [0, -1, 1, 2, 1, 1, -1], [2, -1, -1, 1, -3, 2, 0], [0, 2, -2, -5, 0, -3, 1], [-4, -1, 1, 1, 3, 3, -2], [3, 0, -2, 0, -1, 2, -1], [1, -2, 2, 0, -3, -1, 2], [-4, 1, 1, -1, -1, -1, 2], [-2, 3, -1, -4, -1, -1, 1], [-1, 1, -2, 3, 1, 1, 4], [-3, 5, -2, -2, 1, -3, 2], [0, 0, 0, 0, -2, 3, 1], [1, -4, 0, 2, 0, 2, -1], [1, 1, -4, -4, 1, -3, 4], [1, -2, 3, 3, -2, 1, -2], [-1, -4, 1, 5, 0, 2, -4], [-2, 0, -1, -3, -3, 1, -1], [0, -1, 6, 1, 4, 1, -6], [-1, 2, 3, 1, -2, -3, 2], [-3, 0, 5, 3, 0, 5, -6], [-4, 1, 1, -2, 1, 0, -2], [-4, -1, -4, -5, -4, 0, 4], [3, 1, 3, 1, 0, -2, 1], [0, 5, -1, -5, -1, -2, 2], [1, -9, -4, 1, -3, 4, -3], [-11, -2, 0, 5, 3, 3, 2], [-1, 2, -2, 1, 2, -5, 2], [7, -7, -3, 5, -1, 6, -1], [-2, 1, -1, 5, 7, 2, -4], [3, 0, -4, 0, -7, 0, 3], [2, -4, -3, -1, -5, 0, 1], [-4, 1, -2, 0, -2, -5, -1], [-10, 0, 5, 7, 7, 6, -5], [0, -2, -8, -5, -2, 5, 3], [-4, 4, 2, -9, -1, -5, 3], [9, -6, 3, -1, -2, 1, 2], [-8, 7, 7, 0, 1, -6, 3], [-3, -1, 2, 0, -1, -3, 6], [-9, 0, 2, 5, 8, -3, -2], [-1, 0, 0, 0, 0, 0, 0], [-2, -2, -7, -4, -5, 0, -2], [-2, 1, 0, -2, -5, -3, 1], [1, 6, -6, -10, 1, 0, 1], [-2, 2, 4, -4, -3, -6, -4], [2, -3, 2, -1, -4, -8, 0], [3, 5, 4, 1, -3, -4, -1], [-3, -5, 1, 4, 4, 7, -10], [-2, -5, -8, -3, -4, 2, 0], [-4, 1, 4, -4, 2, -1, -5], [4, -5, 10, 13, 10, 9, -11], [1, -7, -6, 1, 1, 4, 1], [-9, 3, 1, 8, 5, -1, 0], [-6, -6, -3, -1, 1, -6, 0], [-5, 8, -7, -11, -3, -9, 10], [-3, 1, -4, 1, -1, -2, 3], [0, -3, 3, 0, 1, 5, -7], [-6, 3, -9, -6, -1, -5, 5], [0, 1, 7, 0, -2, 5, -3], [-3, -1, 8, 2, -3, 3, -4], [-1, 3, -7, -11, -12, -5, 8], [-12, 9, -7, -7, -1, -2, 6], [-1, 1, 3, -5, -8, -10, 5], [-8, 9, -1, 5, -3, -4, -2], [0, -1, -4, 1, 2, 0, 0], [1, 3, -11, -6, -4, -1, 6], [-8, 1, -4, 1, 2, 0, -2], [-6, -5, 0, 1, -4, -2, -6], [2, -11, -2, 5, 4, 10, -12], [2, -1, 4, -1, -2, -7, -1], [1, 3, -2, 3, 3, 0, 7], [-13, 14, -3, 2, -2, -4, 5], [-16, 5, -3, -8, 3, -7, 1], [12, -15, 4, 6, -1, 1, 1], [-12, 0, 9, 8, 1, -2, -2], [8, -11, -3, 0, -1, 5, -3], [1, -6, 3, -2, -2, 2, -3], [9, -9, 5, 0, -8, -3, 6], [4, -4, -1, -1, 1, 1, -5], [-7, 7, 9, 3, 2, -2, 9], [-2, -3, 4, 3, 0, 3, -8], [-6, 7, 2, -6, 4, -3, 1], [4, -11, -1, 2, 1, 6, 1], [-3, 0, 0, -5, -3, -10, 4], [18, 3, -5, -5, -11, 0, 4], [-5, 3, -1, -8, -6, -16, 1], [-3, -4, -5, 6, -2, -3, -2], [-19, 11, 3, 0, 0, -8, 7], [10, -13, 1, 6, 3, 11, 3], [5, -1, 2, -5, -3, 0, 5], [5, 9, -5, -3, -4, -6, 9], [3, -9, 7, -4, -8, -5, 2], [-9, 3, 1, 7, 0, -6, -1], [-16, 4, 2, 7, 2, 6, -3], [-1, -4, -3, 3, 2, 6, 5], [18, -9, -5, -10, -1, 7, 1], [-15, 6, -4, -6, -1, -4, 7], [-2, 7, -12, -5, 4, 4, 4], [12, -7, 8, 9, 8, 2, -6], [11, -7, 9, 14, 12, 8, -11], [-2, 8, 8, -6, 2, -5, 2], [-10, -5, -1, 9, 3, 8, 0], [11, 2, 4, -8, -7, 0, 5], [4, 0, 0, -4, 4, 6, 0], [-5, 15, -1, -5, 4, -4, 7], [0, -7, 6, 2, -4, 5, -7], [-1, -3, 1, -7, -4, 2, -1], [-11, 10, 2, 3, 9, -11, 6], [2, -11, 2, 16, 8, 9, -13], [-1, 3, -5, -4, -2, 10, -5], [-9, 9, -2, 4, -1, -8, 6], [-4, 3, 0, -3, -2, 7, 10], [22, -4, -6, -6, -2, -2, 2], [0, 9, 2, 6, 9, -11, 5], [-5, -2, 5, 0, -5, -8, 9], [1, -11, 3, -1, 1, 4, -3], [19, -7, -4, 10, 3, 17, -12], [-5, 6, -5, -2, -7, 2, 7], [-6, 2, 3, 6, 15, -3, -5], [5, 1, 4, 1, -4, 4, 3], [2, 1, -2, -7, -4, 11, -3], [6, -11, 5, -4, 1, 7, -1], [3, 19, -1, -10, -4, -16, 16], [-7, 10, 4, 3, 11, 4, -5], [1, -16, -1, 10, -4, 14, 1], [6, -9, 8, 11, 8, 6, -4], [2, 18, 6, -6, 4, -8, -4], [0, -3, -1, 3, 4, -4, -4], [-3, -11, 0, 7, -1, 11, -7], [-13, 4, 4, 11, 9, 17, -12], [4, -2, -8, -3, 4, 11, 3], [-1, 14, -11, -12, -4, -10, 15], [-11, 2, -9, -9, 4, 2, 4], [8, -7, 7, 12, 5, 6, -15], [7, 5, -5, -4, 2, 11, -2], [3, 3, -3, 8, -1, 7, -14], [-20, 8, 3, -8, 8, 2, -12], [15, -3, -8, -14, -12, -3, 8], [18, -2, -7, -9, 5, -1, 5], [-19, 23, 2, -7, 1, -18, 1], [-14, -6, 3, 15, 1, 1, -1], [11, 2, -1, -9, 2, -9, -8], [-11, -22, 1, 10, 4, 6, -11], [-6, -3, -7, 2, 3, -11, 1], [-8, -6, -10, -6, -10, -6, -6], [-3, 10, -4, -1, 5, 1, -4], [12, -7, 0, -2, -18, -5, 11], [7, -10, 1, -6, -11, 2, 7], [2, -9, -1, 10, 11, 9, -1], [-6, 6, -14, -11, 2, -8, 12], [-11, 7, -2, -6, -7, -10, 12], [-8, 11, 1, -10, -5, -11, -3], [-14, 5, 12, 7, 2, 12, -4], [-19, 10, 3, 12, 10, 8, -1], [11, -3, -4, 2, -1, -5, 8], [-8, -14, 5, 2, -10, 10, -10], [-8, 3, 16, 14, 8, 15, -9], [-9, -10, 8, -5, -13, -12, -2], [-1, 10, -13, -8, -18, -5, 17], [-10, 11, 2, 3, 10, -9, -6], [2, -2, 4, -2, 6, -10, -4], [15, -10, 3, 11, -4, 20, -5], [5, 7, 3, 4, 4, 10, 5], [-17, 8, 2, -11, -7, -23, 2], [7, -6, -3, 6, 0, 8, -13], [18, -7, -10, -10, -12, -7, 9], [11, -4, 7, 17, 12, 20, -13], [14, 2, 10, 2, 0, 1, -4], [-19, 5, 2, 19, 11, 0, -1], [0, -20, -5, 6, -9, 7, -2], [20, 0, 1, 3, 3, 0, -12], [-10, -5, 6, -4, -2, -1, -9], [11, 3, -9, -9, 0, 1, 1], [0, -3, -4, 15, 0, 6, 10], [-1, -14, 6, -1, -9, -8, 1], [6, -4, -2, -10, -6, -5, 15], [-3, 4, 4, -6, 1, -2, 3], [3, 2, -4, -3, 15, 6, -2], [-6, 3, -1, -10, -5, 1, -11], [-11, 10, -14, -13, -5, 5, 10], [-26, 24, 1, -10, 1, -20, 4], [14, 0, 19, 5, -5, 11, -5], [-6, -6, -6, 18, 7, 4, 0], [-14, 11, -5, -7, -3, 3, -9], [-3, 7, 6, -1, 15, -5, -8], [11, -7, -11, -7, -2, -12, -1], [-10, 9, 1, -5, 5, -19, 0], [27, 11, 0, -2, 3, -9, 14], [-2, 11, 6, 2, 10, 4, -11], [11, -4, 1, 0, -3, 6, -7], [-7, -8, 13, -7, -7, 3, -12], [-6, -7, -7, -2, 15, 4, -4], [19, 5, -6, -9, -15, -15, 12], [-16, -4, 8, 7, 12, 7, -1], [1, 5, 12, 15, 11, -1, -13], [-14, 18, 10, 2, 4, -12, -2], [-17, -9, 2, 7, 12, 0, 1], [-7, 10, -9, 12, 8, -2, -5], [-2, 5, -4, -9, -2, -2, 4], [-17, -14, 2, 20, 14, 18, -11], [-11, 1, 4, -1, 1, -13, -4], [-1, 9, 12, 8, 2, -9, -8], [13, -12, 4, 11, -5, -5, 6], [13, 6, 7, 4, 4, -14, 5], [-2, 1, -5, -3, -1, -13, 12], [18, -10, -3, 5, 2, 9, -19], [15, 9, 4, -1, -5, 9, 0], [-4, -17, 11, 9, -3, -4, -6], [6, -10, -5, -14, -13, 1, -1], [1, -7, -15, -10, 4, 6, 0], [14, 8, 8, -5, -6, -7, 15], [10, 13, 2, 0, 10, -1, -1], [-6, -7, 0, 6, -5, -7, -5], [-15, -12, 0, 12, 19, 18, -19], [1, 2, -8, 1, 5, -3, 18], [14, 3, -7, -14, 9, -8, 5], [6, 6, 9, 6, 5, 3, -11], [25, 10, 0, 3, 1, -12, 6], [-18, 5, 6, 7, 8, -13, 5], [-23, 15, 2, 7, 1, 0, 3], [8, -2, -7, -9, -1, 11, -7], [4, -5, -21, -18, -5, -3, 19], [-6, -11, -15, 2, 1, 8, -1], [-1, 5, -10, -18, -15, -15, 18], [13, -10, 4, 3, -5, 5, -14], [0, 20, 3, -4, -5, -28, 16], [20, 10, -2, 7, 11, 9, -11], [22, -1, -12, -3, -17, -2, 10], [-1, -4, -18, -7, 6, -3, 0], [9, -10, 15, 13, 14, 19, -16], [9, 0, 20, 11, 3, 1, -2], [-23, 18, -13, -4, -6, -6, 25], [24, -20, -2, 1, 2, 3, -1], [9, -2, -7, 7, 0, 9, -8], [9, -15, 13, 3, -4, 0, -3], [0, -18, -7, -2, -7, -5, -2], [11, 8, 3, 9, 4, -19, 4], [9, 4, 2, 10, 3, -7, 3], [12, 1, 2, -4, -4, -15, 9], [24, 11, 15, 0, -3, 0, -7], [17, -5, -1, -4, -14, 2, 0], [0, -13, 11, 14, 9, -13, -9], [26, -27, 11, 13, -3, 3, -1], [-6, 28, 11, -11, -1, -13, 9], [8, 17, -4, -2, 0, 2, 14], [9, 22, -2, -7, -4, -5, 2], [25, -9, 7, 15, 0, 4, -3], [-9, -9, 6, 4, 13, -1, 2], [25, -2, -17, -8, 1, 0, 7], [28, 16, -4, -16, -12, -12, 4], [9, 7, -2, 3, 1, 6, 5], [26, -1, 12, 1, 4, -4, 10], [-5, 16, 6, 1, 1, 5, 14], [1, 22, 6, -7, -5, -15, 2], [0, -1, 3, 0, -5, -5, -9], [-5, -8, -4, 8, 4, 12, -13], [11, -6, 5, -16, 6, 0, -1], [28, -19, -18, -4, 0, 17, -3], [9, -15, -7, -4, -2, 12, -17], [30, -13, 12, -4, -4, 5, -3], [-24, -5, 18, 21, 16, 10, -10], [-8, -2, -2, -14, -4, -13, -3], [12, -26, 13, 18, 5, 4, -12], [6, -7, -5, 4, 13, 3, -15], [9, 4, 10, 10, 13, -6, -9], [-8, 4, 13, 17, 9, 23, -5], [-1, 10, -15, -11, 8, 7, 2], [8, 17, -4, 0, -6, 0, -9], [-16, 15, 2, -8, -10, -26, 6], [-3, 2, 0, 3, -11, -17, 6], [16, 9, -6, -26, -12, -23, 11], [14, 8, -9, -13, 3, -26, 16], [-22, 7, 9, 15, 3, 13, -15], [31, -5, 4, 13, -13, 24, -3], [4, 4, 7, 1, -1, -11, 25], [-26, 6, -10, -5, 4, 4, -7], [-10, -4, 11, 13, -11, 1, -9], [-2, -5, -17, -7, -5, -3, 10], [12, -1, 6, 11, 12, -8, -6], [2, -4, -13, -14, -7, -13, 8], [13, -9, 13, 25, 4, 21, -21], [32, -14, -9, 8, 1, 18, 0], [-4, -1, -15, -15, -15, -8, 6], [-19, 7, 4, 3, -9, -17, 5], [-10, -12, -9, -7, 5, 15, 5], [5, -23, 2, 12, 11, 10, -25], [23, -9, 12, 11, 1, 14, -1], [47, -11, -1, 12, 6, 23, -6], [-21, 24, -4, 11, 1, -2, -5], [-9, 1, -14, -8, -5, -7, 12], [-2, 12, 9, 17, 11, 11, -21], [-14, 7, -1, 2, 12, 13, -1], [-2, -20, 14, 20, 13, -4, -16], [-9, 20, 11, -1, 8, -5, -14], [-8, -7, 23, 21, 14, -10, -12], [23, -14, -16, -16, -13, -7, 11], [2, -11, -2, 6, 11, 1, -9], [-7, 11, -1, -9, 14, -4, 1], [-20, -11, -8, 8, 12, 19, 5], [-28, -6, 19, 30, 7, 1, -4], [13, -1, -10, -4, 8, 9, -8], [30, -8, -17, -15, 3, -4, -10], [15, 7, -1, -10, 6, -2, 11], [-7, 7, 12, 0, 10, 5, 4], [-6, 2, 20, 8, 8, 7, -1], [1, -3, 19, 14, 8, 14, -7], [6, -18, -8, -10, -7, -20, 4], [-15, -20, 3, 20, 12, 9, -21], [26, -6, 5, -1, 11, 1, 5], [-6, -12, 10, -5, -4, -7, 3], [1, 13, -13, -21, 10, -13, 7], [-1, 9, -6, -3, -3, 0, 19], [11, -28, 2, -8, -19, -13, 7], [3, -4, -2, 5, 5, 13, -8], [-9, -11, -9, -6, -16, 16, -4], [-3, 9, -8, -17, -3, 14, 7], [16, -19, -7, -14, -10, 11, -3], [13, 13, -20, -20, -27, -3, 14], [0, 9, 9, 9, 5, -4, -16], [7, -15, -18, 1, -1, 14, -17], [10, 12, -7, 4, 3, 2, -12], [-4, 26, 9, -8, 1, -13, 12], [-34, -8, 0, 21, 8, 13, -3], [13, 15, 12, -9, -5, 0, 3], [-22, -17, 2, 20, 4, 17, -21], [2, 6, -15, -8, 15, 7, -8], [17, -7, -15, -8, -8, 1, -2], [20, 7, 14, 0, 12, 13, 1], [16, 3, -11, -22, -17, -20, 17], [9, -4, 19, 22, 22, 20, 3], [-10, 22, -13, -16, -9, 1, -1], [-13, -6, -10, -11, -6, -5, 8], [35, -17, 8, 24, 13, 32, -16], [-2, 22, 4, -7, 6, -11, -7], [-6, -1, -6, -14, -16, -12, 6], [-2, 17, 17, 2, 2, 3, 9], [31, 1, 6, 13, 9, -6, -11], [-10, 0, 13, 9, 15, -13, -5], [22, -16, -3, -6, -6, 4, -2], [17, -11, -20, 15, -7, 2, -1], [23, -20, 1, 8, -8, 19, -7], [-4, 12, -16, -10, 18, 8, -12], [27, -16, 1, -9, -4, -5, 6], [-10, -9, 17, 32, 28, 11, -11], [-27, 15, -26, -20, -3, -10, 22], [-28, 1, 12, 6, 22, -17, 3], [34, -11, -20, 3, -10, 1, 15], [0, 18, 0, -17, -12, -29, 25], [-41, 8, 6, -6, -11, -18, 5], [14, -24, -2, 3, -6, -1, 5], [-14, -10, 13, 9, 1, -17, -11], [25, 29, 4, -9, 3, 1, 7], [-2, -19, 8, -2, 9, 21, -13], [3, 8, -2, -3, 3, -10, 3], [-16, 8, 8, -13, -8, -29, -5], [12, -44, -4, 8, -16, 14, 0], [-19, 23, -18, -18, -9, -22, 23], [-2, 1, 25, 6, 1, 9, -19], [-29, -10, -13, -7, -2, 1, 10], [22, -19, 3, -2, -23, 6, 2], [-42, 13, 5, 5, -2, -2, 12], [6, -12, 4, 7, -6, 3, 9], [-11, -8, -4, 23, 11, 20, -13], [4, -2, 24, 9, -5, -1, -9], [21, -12, 9, 19, 5, 1, 2], [-5, 6, -9, -7, 24, -5, -20], [-21, 29, 18, 11, 19, 10, -19], [5, -25, 6, 13, 3, 34, -25], [1, 10, 5, 8, 12, 21, -6], [-24, 14, 0, -4, 4, -10, -4], [-25, 17, 22, 7, 12, -18, 9], [-23, 3, 9, 20, 12, 23, -4], [-6, 13, 0, -22, 1, -3, -11], [-11, 20, -18, -25, -3, -16, 15], [5, -10, 26, 18, 27, 12, -29], [-12, 28, 7, -3, -7, -12, -4], [-32, 8, 0, -6, -10, 8, 8], [-18, -7, 14, -10, 6, 2, -11], [-6, 25, 15, -6, 5, -23, 15], [-1, -4, -2, -9, 1, -20, -4], [15, 8, -6, -28, -16, -16, 7], [-15, 6, 16, 9, 6, 7, 4], [30, 16, -4, -16, -10, -20, 20], [-16, 12, 5, 16, 1, 1, -2], [-35, 7, -12, -12, -19, -6, 7], [-4, 1, -13, 0, 3, -14, 4], [-8, -17, -7, 1, 17, 12, -2], [13, 17, -3, -2, -18, -19, 20], [-15, 26, 8, -16, -2, -6, 13], [-4, 21, -1, 14, 3, 9, -13], [-10, 25, -23, -30, -21, -14, 30], [34, -14, -2, -6, -1, -12, -12], [-37, 32, 7, -2, 8, -20, -3], [13, -2, 3, 1, -2, -16, 5], [13, 38, 10, -10, 7, -8, 5], [-40, 30, -7, -12, -1, -26, 2], [-5, -8, 10, -5, -23, 7, 6], [-4, 1, -7, -11, 19, -16, -12], [12, -23, 17, 28, 9, 3, -1], [18, -17, -10, 12, 4, 17, -5], [-35, 11, 6, 3, -9, -4, 7], [-17, 7, -20, -29, -27, -18, 11], [14, 2, -12, 5, 9, 7, -5], [25, 27, 4, -11, -13, -7, 11], [-28, -18, 12, 6, 14, -4, -4], [-7, 1, -6, 24, 17, 9, -16], [25, 11, 4, -6, -7, -18, -1], [17, -15, 7, 9, 8, 28, -19], [0, -8, 11, -1, 17, 9, -11], [-32, 16, -9, -1, -5, -5, 15], [-6, 1, 13, -5, 7, -10, 14], [24, 16, -13, -18, -21, 7, 6], [28, 23, -8, -11, 2, -15, 24], [0, 4, -18, -12, -10, -2, 8], [-17, -13, -2, 3, 15, 11, -7], [-9, -11, -9, -6, -4, 25, -2], [0, -19, 4, 10, 2, -1, -23], [31, -25, -5, 2, -3, 1, -2], [40, -11, -25, -4, -1, 19, -19], [-20, 11, -2, 4, -14, 2, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_9036_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_9036_2_a_i();".
// 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_9036_2_a_i(:prec:=7)
    chi := MakeCharacter_9036_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(3019) - 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_9036_2_a_i();".
// 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_9036_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_9036_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![27, 20, -81, 26, 25, -11, -2, 1]>,<17,R![-21, -344, 865, -702, 217, -7, -8, 1]>],Snew);
    return Vf;
end function;