
// Make newform 6045.2.a.t in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6045_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_6045_2_a_t();".
// 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_6045_2_a_t();".
// 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 := [-5, -4, 41, -5, -52, 14, 22, -7, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0], [-2, -1, 1, 0, 0, 0, 0, 0, 0], [-4, 8, 7, -6, -2, 1, 0, 0, 0], [-3, -4, 9, 7, -6, -2, 1, 0, 0], [3, -4, -26, -2, 29, 4, -10, -1, 1], [3, 4, -17, -16, 20, 11, -8, -2, 1], [7, 1, -31, -4, 30, 4, -10, -1, 1], [-7, -11, 54, 28, -56, -17, 19, 3, -2]];
    Rf_basisdens := [1, 1, 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_6045_a();"
function MakeCharacter_6045_a()
    N := 6045;
    order := 1;
    char_gens := [4031, 4837, 1861, 2731];
    v := [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_6045_a_Hecke(Kf)
    return MakeCharacter_6045_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 := [[-1, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 1, 0, 0, 1], [-2, 1, 0, 0, -1, 1, -1, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 2, -1, 0, 0, -1], [1, 0, -2, -1, 0, -1, -1, -1, 0], [-3, 0, 0, 0, 0, 0, 0, 2, -1], [0, -2, 1, 2, -1, -2, 2, 2, -3], [1, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, -1, -1, 2, 0, 1, -3, -1], [0, -1, 2, 1, 1, -2, 4, 1, -1], [-1, -2, 0, 1, -2, -1, 1, 2, 2], [-5, 1, -2, -2, 0, 2, -2, -1, -1], [-5, 1, 1, 0, -2, -1, 3, 0, 2], [-2, 1, 1, 2, 0, -1, 1, 2, 2], [-2, 3, 3, 2, -1, -1, -1, 3, 2], [0, 0, -4, 1, 3, 2, -1, -3, -2], [-5, 0, -3, -2, 0, 0, -4, -2, -1], [-2, 2, 0, -3, 0, 2, -6, -1, 1], [0, -1, -5, -3, 2, 1, -4, -4, -1], [1, -4, 0, 1, 2, 2, 3, 1, 1], [0, -2, 4, 0, 1, -4, 5, 3, -1], [-3, 4, 1, -1, -2, -2, 0, 3, 1], [-1, 2, 2, 0, -1, -2, 3, -3, 2], [0, -6, 1, 2, -1, -2, 0, 2, 2], [-3, 4, 2, -1, 2, 1, -4, -1, -1], [-3, 3, 2, 0, 1, 2, -2, -3, 3], [-11, 2, 2, 2, -3, -2, 1, 3, 1], [0, 3, -5, -4, 2, 2, -7, -6, 2], [-3, 0, 3, -1, 1, 0, 0, 3, 0], [-2, -2, -1, 1, 0, 2, 2, -3, -1], [-7, -4, 2, 2, -1, 3, 4, 2, -3], [-1, 1, -7, -3, 5, -2, -2, -4, -5], [0, -4, 0, 1, 1, -3, 4, -2, 1], [1, -2, 7, 4, 0, -1, 10, 4, -1], [-11, 1, 2, -3, -2, 2, 0, 0, 2], [-5, 1, 1, 0, -2, 4, -3, 5, 3], [3, -2, 2, 2, -3, 1, 4, 5, 1], [-1, -1, -1, 1, 0, 5, -4, -1, 0], [-4, -4, 2, 4, 1, 2, 5, 1, -3], [-2, 2, 4, -1, 0, -2, 1, 8, -2], [-2, -4, -2, -1, 0, -1, -3, 0, -1], [2, 2, -4, -4, 5, -5, -2, -2, -5], [-1, -2, 6, -3, 0, -4, 3, 3, 0], [0, -1, 3, 6, -5, 0, 5, 1, 1], [-1, 2, 3, 0, -4, 0, -4, 4, 5], [-4, 5, -5, -2, 0, -4, -5, -4, 1], [4, 0, 2, -1, -2, 0, 3, 0, 6], [5, -4, 8, 5, 1, 4, 3, 0, -2], [-6, 6, -3, -7, 3, 2, -3, -3, -3], [-5, 0, -2, -4, -1, 1, -2, -4, -2], [-3, -1, 2, 6, -4, 1, 3, 3, 4], [1, -8, -1, -2, 1, 3, 0, 2, -1], [-7, 6, 2, 0, -7, -4, 0, 5, -4], [5, -1, -1, 6, -1, 0, 2, 0, 4], [2, 2, -1, -1, 0, -4, 1, -1, 0], [4, 0, 3, 6, -2, 4, 1, 7, 2], [-5, -2, 7, 5, -2, 2, 2, 7, -5], [-2, -7, -3, -3, 5, 3, 2, 1, -6], [0, 0, 1, 2, 3, 3, -4, -1, 5], [1, 0, 10, 1, -10, -5, 3, 8, 3], [3, 6, -1, -5, 1, 4, -6, -3, 8], [-1, -2, -5, 1, -2, 2, -3, 1, 7], [-8, 2, -1, 5, -2, -4, 3, 3, -1], [11, -3, -9, -3, 5, -2, -7, -3, -6], [-9, 3, -12, -8, 6, 5, -8, -6, -4], [4, 2, 3, 2, -3, 2, -1, -2, 6], [-3, 0, 3, 1, -3, 5, 1, -2, 3], [2, -5, -5, 0, 5, 4, 3, -7, -1], [4, -6, 2, 7, -8, -3, 10, 2, -5], [-3, 4, 5, 4, 0, 5, 0, -2, 9], [-2, 6, -2, -4, -7, 4, -7, -2, 2], [0, -1, 4, 3, -3, 2, 3, -4, 5], [-14, 3, 0, -1, 3, 1, -4, -5, -1], [8, -1, 4, 6, 1, -7, 1, 4, -4], [-1, 3, -3, 0, 3, -5, -4, 1, 1], [-12, 5, 8, 1, -4, 0, 7, 3, 6], [2, -1, 0, -1, -5, 5, 2, 2, 2], [-4, 8, 6, 1, -7, 4, 0, -4, 0], [1, 9, -13, -3, 0, 5, -13, -4, -4], [0, 4, 1, -3, 7, -2, 0, -2, -7], [-6, 8, -5, 0, -4, 1, -6, 2, 1], [4, -5, 2, 3, -5, -1, 5, 4, -4], [-2, 2, 14, 7, -11, -2, 9, 3, 8], [-7, -3, -7, 0, 2, 8, -8, -8, 6], [4, 9, 4, 9, -6, 0, 1, 5, 0], [-9, 7, 7, 4, -5, -2, -4, 3, 1], [-2, -1, 0, -2, 0, -8, 4, -7, -4], [0, 1, -3, 4, 0, 2, -6, -2, 0], [-1, 4, 7, -3, -3, -3, 6, -2, 4], [-11, 16, 0, -1, -2, -5, -2, 0, 1], [-4, 2, -6, -5, 1, 1, 5, -4, -5], [-10, -7, 4, 1, -5, 4, 2, -1, 3], [-3, -3, -9, 0, 7, 6, -10, -8, -9], [7, -9, -4, -6, 1, 1, 5, -3, -1], [2, 5, 1, -2, -4, 4, 0, 2, -1], [6, 5, 2, 2, 1, -7, 8, 5, -3], [-9, -3, -1, -6, 0, 5, 6, -6, 2], [-4, 1, 2, 6, -5, -2, 3, 8, -7], [4, -6, 3, 0, -3, 0, 8, 3, 5], [-13, -1, 3, -2, -3, -1, 5, 0, -2], [2, 4, 2, -9, 8, 1, -5, 0, 8], [0, 3, 11, 7, -9, -2, 4, 7, 14], [-7, 1, -6, -4, 8, 2, -2, 3, -5], [-3, -2, -10, -11, 12, 4, -12, -9, 5], [-6, 4, 2, 3, 2, -5, 2, 1, -5], [-11, -11, 3, 1, -3, 3, 4, 3, -5], [11, 2, -1, -4, 1, 6, -8, -9, 0], [9, 0, 7, -4, -1, -4, 8, 4, 1], [10, -2, -7, 7, -4, -1, 2, -2, -2], [8, -9, -7, 1, 17, 3, 0, -9, -2], [2, -4, -4, -6, -5, 3, -5, -3, -6], [0, -5, 3, 8, 0, 2, 12, 5, 2], [-2, 14, -7, -6, 0, 6, -11, -8, 5], [1, 9, 1, 1, 3, -5, 1, -4, -6], [-13, -3, -3, 5, 3, 9, -4, -3, 7], [-2, -4, -1, -5, 8, -3, -1, -9, -2], [-2, -2, 8, 3, -11, -6, 7, 6, 6], [5, -5, -8, 3, 9, -6, -2, 1, 1], [3, -1, 14, 1, -3, -4, 11, 11, 2], [5, 4, -4, 8, 2, 6, -8, -1, -4], [-15, 2, -1, 0, -2, -1, -6, 0, -1], [-17, 15, 6, -1, -3, 9, -11, 3, 15], [18, -7, -10, -5, 2, 2, -1, -5, 2], [1, 7, 3, -1, -2, 7, 3, 4, 4], [-1, 7, -2, 2, 2, 4, 4, -5, 8], [-1, 1, 4, -5, 2, 1, -4, 4, 4], [-10, 3, -4, 0, 5, 3, -12, -9, -1], [-5, -2, -8, -3, 6, 4, -13, -5, -4], [-12, 4, -1, -1, -4, 10, -10, -2, 1], [-1, 8, 7, 6, -3, 7, 2, 8, 4], [0, 0, 2, 7, -2, -3, 0, 6, -13], [1, 1, -12, -2, 11, -3, -8, -11, 5], [8, 2, -19, -5, 11, 0, -8, -8, -6], [-8, 11, 5, -5, -6, -6, 0, 1, -1], [13, -11, -2, -2, 1, -8, 9, -1, 1], [8, -6, -10, -5, -1, -5, -3, -1, -3], [-2, 3, -9, 1, 10, 3, -3, -10, 0], [-10, 1, -6, 0, 6, 6, -9, 0, 5], [-3, 4, 11, 9, -10, -2, 5, 1, 4], [-9, -1, -3, -1, 14, -7, -3, -2, -3], [20, -15, -6, 0, 4, 4, -1, 1, 9], [12, 1, -6, 5, 4, -1, -8, -8, -1], [6, -8, -4, 0, 6, 0, -8, -2, 1], [16, -9, -5, 4, 6, -8, 9, -5, 1], [-3, -7, 2, 2, -1, 9, -7, 9, -9], [-6, -7, 4, 9, -9, 0, 8, 16, 1], [-14, 0, 4, 7, -9, -4, -6, 3, 4], [-10, 4, -2, 7, 1, 2, 11, -2, -6], [-6, 16, 0, -6, 4, -7, -5, 1, -2], [-7, -8, -6, 0, 9, -1, -1, -1, 7], [1, 2, -10, -4, 13, 1, -8, -19, 5], [-7, 1, -5, -13, 8, 2, -7, -1, -11], [-3, 4, -1, -10, 4, 4, -11, -7, -5], [6, 4, 2, -1, -5, -3, 6, -2, 13], [-5, 5, -9, -2, -2, -1, -12, 7, 7], [16, -13, -5, 5, 3, 4, 6, 1, -2], [8, -9, -2, -5, 6, -2, 3, -14, -1], [-6, 2, 2, -3, 5, -3, 4, 3, 7], [21, -7, -12, -6, 6, -2, -3, -4, -6], [-11, -14, 4, 2, 3, 1, 0, -2, 7], [5, -7, 18, 5, -9, -9, 30, 18, -2], [-9, -3, 1, 6, -6, -5, 6, -3, 5], [-12, 5, 9, -3, -6, -1, -4, 1, 1], [-8, 15, 2, -4, 7, -3, -2, -8, 1], [14, 5, 1, -12, 6, 9, -15, -11, 12], [10, 4, -7, -7, 7, 11, -11, -14, -1], [-8, -5, 5, 5, -5, 5, 9, 4, -7], [7, -1, -7, -6, -2, 5, 7, -5, -5], [-3, 1, -2, 4, -7, -3, 1, 8, -9], [15, -10, -12, -6, 3, 7, -10, -5, -8], [-4, 6, -7, -7, 2, 0, -16, -6, 0], [6, -6, -9, 0, 3, 7, -4, 2, -5], [8, -14, 0, -1, 4, -5, 1, 9, -4], [-7, 0, -3, -3, 4, -7, 6, 1, -6], [-27, 1, 4, 4, -15, -3, -5, 8, -1], [-7, 16, 5, -5, 2, -8, -1, 3, 2], [-20, 1, -7, -5, -2, 2, -1, -3, 8], [-8, -15, -7, -8, 5, 5, 11, -5, -3], [9, -12, 1, -1, 1, -2, 5, -1, -13], [1, 8, -2, -2, -2, -6, 2, 1, -10], [-16, -2, 4, -8, 0, 1, 9, -3, 9], [4, -3, 7, 2, -1, -10, 1, 3, 0], [10, -9, 4, 3, 1, 7, -3, 3, 5], [2, 13, -3, 0, 1, 12, -7, -4, 1], [-10, -1, 13, -4, 4, 3, 10, -4, 8], [0, 4, -2, 5, -7, 7, -3, 7, -1], [17, 5, -6, 1, 13, 2, -7, -4, -5], [0, -8, -9, -1, 0, 4, 9, 2, -1], [1, 15, -14, -2, 1, 0, -10, -1, -9], [-1, 2, -8, -9, 14, 4, -9, -2, 1], [26, -7, 12, 5, -3, 2, 14, 9, 4], [-2, -13, 20, 8, -17, 7, 21, 18, 2], [26, -11, 3, -1, 0, 2, 3, 2, 15], [-5, 2, -3, 1, 3, 8, -6, -8, -5], [-13, 3, -19, -15, 0, 4, -8, -13, 5], [2, 2, 14, 4, -11, -2, 2, 7, 11], [-7, 5, -14, -2, 15, 4, -10, -7, -9], [16, -12, -15, -10, 10, 1, -6, -15, -14], [9, -1, 4, 6, 5, -8, 14, 3, 7], [-6, 16, -20, -3, 6, -8, -7, -11, -1], [-3, -2, 22, 12, -7, -9, 21, 11, 2], [6, -1, 4, -8, 1, 5, -13, -2, 1], [-10, 8, -6, -7, -4, 2, -11, -8, -4], [7, -7, -2, -4, 8, -4, -7, 11, 3], [12, -5, 11, 6, -3, -9, 8, -1, 4], [-8, -7, -7, -3, 8, -14, -4, 5, -12], [-2, 7, 8, 4, -4, -14, 6, 1, 0], [0, 14, -1, -5, 7, 8, -5, -7, 10], [-7, -3, 5, 1, -14, 2, 7, 1, 6], [10, -9, 1, 12, -7, 0, 10, 3, -12], [21, -6, -7, -4, 3, 15, -8, -14, -9], [10, -6, -9, 8, 10, -5, 3, 10, -10], [-6, 7, 9, 7, -5, -11, 15, 8, -11], [2, 6, -12, 1, 8, -2, -12, -8, 4], [17, -1, 5, 12, 7, -2, 16, 7, 3], [-12, -2, 5, -1, -9, -9, -2, 16, -5], [-32, 4, 0, -5, -2, 9, -15, -3, 5], [4, 5, 0, -5, 2, -12, -8, 0, -15], [1, 0, 8, 6, -5, 3, -2, 8, 1], [9, -2, -8, 4, 9, 7, 10, -14, -2], [5, -6, 8, 16, 2, 15, 6, 3, 2], [3, 2, 2, -1, -8, 4, -3, 11, -8], [0, 6, 9, -3, -10, 0, 11, 2, 8], [-9, -4, -7, -2, 0, 5, 1, 7, -6], [12, -2, -20, -18, 17, 3, -13, -16, -3], [24, 1, -15, -12, 1, -4, -11, -6, -2], [2, 12, 2, -10, 11, -2, -10, -6, 9], [27, -15, -7, 5, 7, 12, -7, -7, -5], [12, -9, 11, 0, -4, -6, -1, 7, -5], [-8, 5, 10, 7, -11, 4, 12, 3, 8], [23, -4, 3, 3, 7, -1, 9, -4, -6], [7, 2, 2, -7, 3, -13, 4, 8, -2], [5, -16, 7, -6, 8, 3, 15, 0, 5], [5, -5, 10, 14, -12, -6, 28, 16, 6], [12, -10, -10, 4, -3, -2, -11, 6, 2], [-1, -17, -8, -3, 6, -1, 3, 9, 6], [15, 1, 4, 5, -1, 2, -1, 2, 11], [-14, -6, 18, 11, -3, -6, 23, 7, 15], [-5, 18, -1, -2, 9, 0, -6, 11, 6], [8, 6, 11, 8, -7, -2, -4, 22, -5], [-1, -3, 21, 5, -4, -14, 16, 14, 1], [3, -12, -4, 1, 1, -2, 3, 5, 9], [9, 8, -26, -4, 8, -2, -17, -14, -5], [-8, 11, -2, -6, -4, 11, -16, -7, 8], [26, -14, -4, 2, 7, 6, -7, 4, -10], [11, -11, 11, 11, -12, -1, 9, 12, -3], [-7, -8, -14, -14, 1, -2, -9, -12, -7], [-9, 12, -2, -1, -14, -2, -16, 6, 3], [0, 3, 3, 0, -1, 1, 2, -3, 20], [-22, -1, -18, -13, 10, 7, -20, -16, -2], [20, -5, 10, 15, -1, -10, 18, 16, 0], [9, -4, 8, -5, -11, -2, 5, 1, 3], [18, -2, -9, -4, 10, 9, 6, -5, -3], [-14, 14, -1, -9, 1, 14, -9, -4, -5], [-6, -11, 0, -7, -4, 0, -4, 11, -8], [17, -16, -6, 9, 6, -6, 6, -9, -8], [-4, 24, 6, 5, -2, 13, -18, -7, 4], [12, -16, 7, -3, -8, 4, 2, 4, 0], [16, 4, 9, 10, -11, -1, 9, 2, 2], [3, 7, 9, 1, -7, -2, -1, 3, -12], [14, 7, -10, 1, 3, -6, -4, -8, -3], [4, -7, -14, -2, 2, -10, 0, 5, -6], [14, 11, -12, 1, 4, 9, -7, -7, 0], [-14, 14, 9, 10, 4, -1, 5, 10, -5], [21, -12, -13, 1, 10, 8, -11, -5, -13], [1, -13, 4, 10, -2, -1, 10, 10, -4], [-10, -12, 23, 6, -11, 0, 14, 5, 12], [7, 19, -1, -10, -9, -9, -15, 1, 7], [-17, -5, 1, -2, -5, 0, -11, 2, 0], [18, 1, -16, -12, -2, 5, -18, -8, 3], [23, 1, 8, -8, 12, 3, -5, 9, 3], [-3, -26, 1, -1, -1, 3, 7, -9, 2], [-19, -5, 13, 15, -2, -18, 32, 15, -16], [-19, -9, 13, 13, -6, -14, 31, 10, -4], [20, -14, 24, 11, -9, -5, 24, 17, -3], [-15, 10, -26, -3, 10, 15, -13, -23, 5], [-20, 7, -4, -4, -11, -12, -6, 0, -4], [-13, 11, -10, -12, 11, -8, -12, -6, -2], [11, 2, 6, 14, -18, 1, 5, 13, 6], [-2, -13, -12, 0, 2, 2, -7, 9, -10], [11, -10, 0, 13, -4, -8, -1, 11, -6], [-10, -6, 5, 7, -8, -2, 10, -2, 13], [19, -4, -13, 2, 11, -9, -6, -7, -23], [20, -1, 4, 8, 2, -6, -1, -6, 5], [8, 11, -2, -1, -10, -7, -6, 11, -9], [2, 0, 1, -3, -2, 7, -8, -13, 15], [19, -8, -9, 8, 5, 5, 3, -11, -3], [-1, -16, 6, -11, 7, 3, 8, 1, -6], [-21, -4, 3, 1, -8, 4, 4, -11, 6], [-18, -12, 5, -13, -11, 8, 10, -2, 12], [8, 4, -11, 3, 11, -8, -10, -7, -4], [19, 3, 16, 14, -17, 1, 21, 25, 16], [-14, 18, 8, 3, -15, 2, 5, 7, -1], [14, -9, -8, -12, 10, 3, -3, -16, 5], [20, 6, -7, -6, 18, 13, 1, -12, 3], [0, 34, -12, -11, -4, 8, -30, -9, 13], [17, -5, 18, 6, 0, -3, 20, 1, -3], [-14, -6, -13, -9, 13, -13, -1, -17, -7], [-41, 0, 0, -10, -7, 9, -10, 7, 1], [-21, -10, 5, 16, 3, -4, 21, 9, -12], [0, -3, -21, -16, 5, 3, -22, -15, -4], [0, 0, 2, 1, 12, -20, 19, 1, -29], [-10, -11, -4, -7, 2, -16, 4, 1, -7], [-7, -12, -3, 15, -10, 4, 2, 4, -3], [-4, 8, -11, -11, 14, -1, -5, -6, -11], [-1, -9, 17, 10, -10, 7, 9, 5, -1], [9, 8, 10, -7, -4, 11, -13, 3, 0], [-12, 3, -4, 9, -4, 6, 2, 5, 4], [-15, 11, -20, -3, -6, 3, -23, -13, -9], [1, 7, 13, 8, 5, -2, -7, 9, -17], [-19, 16, 9, -16, -7, 3, -6, -3, 6], [-29, 11, 8, -12, 1, -7, -3, 3, -3], [-16, 25, -21, -14, 1, 3, -16, -11, 18], [12, -7, 15, 3, 2, -3, 18, 0, 7], [-12, 7, -5, -8, -3, 4, -10, 0, -3], [2, -21, 17, 7, -14, -1, 16, 24, -7], [4, 11, -4, -9, -5, 21, -16, -12, 23], [0, -14, -9, 10, 2, 13, 15, 4, 0], [-22, -4, 37, 11, -26, -6, 13, 18, 10], [20, -16, -16, 1, 13, 7, 5, -18, -8], [5, 12, 0, -1, -4, 10, -18, 6, -4], [-4, 4, -5, 2, -7, 13, -17, -9, 1], [5, 1, -14, -4, 9, 15, -31, -5, -3], [-8, 16, -10, -3, 0, 5, -17, -4, 8], [0, 15, -17, -14, 11, -15, -18, -5, -3], [9, 1, 6, 0, 0, -4, 1, 7, -15], [35, -16, 1, 12, 4, -5, 1, 16, -5], [-16, 20, -13, 2, -13, 2, -18, -7, -7], [16, -3, 3, 1, -7, 5, -10, 8, 24], [-1, -19, 25, 3, -4, 8, 16, 12, -4], [-9, -8, -15, -11, 12, -6, 2, -14, 2], [15, -2, -10, -9, 5, -2, 6, 1, -12], [3, -8, -8, 0, 12, -20, -4, 12, -13], [0, -16, -1, -2, 0, 9, 3, 0, -5], [-8, -1, -16, -8, 14, -2, -25, -28, -17], [-8, 8, -8, 10, -5, -9, 6, 3, -5], [17, 1, -12, 13, 5, 2, -7, -5, -12], [8, -14, 4, 7, -3, -2, 5, -10, -6], [-5, 2, 17, 10, -5, 12, 23, 4, 11], [38, -19, 4, 5, 1, -3, 13, 4, -5], [8, 22, 0, -13, -1, 7, -5, -18, 24], [-7, -13, -14, -9, 12, -7, -4, 7, 9], [23, -29, 12, 12, -2, -7, 14, 12, 6], [-37, 23, 11, 4, -2, 0, 5, 9, 7], [-7, 6, 8, 7, 2, 6, 29, -6, -7], [-5, 19, -3, 5, -5, 1, -1, -1, -7], [17, 23, -9, 1, 11, -2, -6, -13, -4], [-18, 11, -18, -4, 7, 3, -6, -18, -14], [-11, -4, 4, -4, 9, 4, -6, 2, -15], [-5, -21, -1, 2, -9, 13, -7, 4, 21], [-7, -4, 3, -10, 7, -5, -12, 2, -7], [-4, -23, 7, 11, -10, -10, 14, 18, 3], [-4, 9, -2, -17, 3, 13, 3, -15, 16], [39, -11, 14, 10, -3, -8, 9, 14, 6], [-15, 9, 8, -12, 10, -9, 0, -6, -6], [-14, 9, -5, -11, 2, 3, -13, -24, -3], [32, -24, -20, -6, 16, 24, -7, -15, 3], [21, 23, -10, -7, 0, -5, -16, -2, -11], [-6, -8, 7, 21, -5, -9, 29, 23, -7], [-6, -4, 22, 7, 0, 4, 9, 5, -9], [-4, -7, -18, -7, 3, -12, -6, -17, 1], [-31, 28, 10, 3, -14, -6, 8, 7, -6], [-2, 7, -15, -11, -1, -4, -11, -14, 8], [-9, -11, 29, 15, -8, 3, 30, 5, -11], [-16, 13, 11, -11, 7, -2, 5, -4, -13], [-11, 4, 0, 2, -16, -5, -13, 18, -8], [15, -3, -31, -2, 20, -10, -1, -11, -19], [-14, -1, 1, 2, 19, -3, -3, 2, -3], [-9, -14, 3, 17, -6, -4, 29, 1, -12], [22, -4, -1, 9, 7, -11, 10, 4, -8], [10, -8, 6, 4, 4, -7, 16, 20, 1], [18, -7, -13, 1, 12, -13, -7, -4, 0], [-5, -10, -19, -3, 19, 9, -17, -14, -19], [24, -10, -22, -3, 11, 22, -7, -14, 0], [20, -3, -2, -6, 1, -2, -4, -8, -2], [13, -4, 1, -3, -14, -6, 14, 10, 17], [6, -5, -10, -3, 8, -19, -3, -14, -16], [-26, 2, -6, -8, 0, 21, -18, -1, 25], [-40, 12, 2, -12, 2, -4, 0, -7, 2], [16, -5, -17, -21, 2, 0, -26, -13, 4], [-6, -12, -16, -4, -8, 10, -9, -7, 10], [-12, -7, 2, -3, 4, -21, 0, 1, -1], [-9, -13, 8, 0, 16, -16, 16, -1, -5], [0, -20, 17, -1, -19, 7, 12, 14, 16], [25, -3, -15, -1, 20, 12, 3, -34, -3], [-13, -3, 34, 24, -15, -13, 47, 27, 3], [-10, 1, 26, 3, -12, 2, 13, 20, 29], [29, -3, 14, -4, -12, -3, 10, 3, 0], [6, -8, 6, -7, -6, 13, -1, -6, 19], [0, -7, 3, 2, -10, 4, -7, -13, 7], [-15, 20, -11, -8, 10, -5, -7, -17, 6], [12, -8, -15, -9, 11, 5, -24, -4, -26], [0, 12, 5, -2, -15, 14, -15, 3, 13], [-8, -12, 25, 9, -5, -2, 19, 3, 8], [20, 8, -23, -15, 17, 0, -37, -7, 3], [-7, -15, 17, 3, -16, -14, 16, 23, -3], [-39, 31, 10, -9, 2, 6, -1, -15, 15], [19, 13, -7, -5, -3, -1, -1, -5, 10], [-10, -28, -5, -6, 2, 5, 9, 3, 16], [-5, -2, -18, 5, 0, -12, -6, 4, -19], [-16, -6, 17, 7, -11, 9, 26, 6, 6], [-18, 12, 15, -3, -10, -8, 16, 3, 7], [2, 15, -25, -10, 13, 19, -19, -8, -9], [6, 4, -12, 3, 5, 6, -12, -21, 10], [-36, 10, 13, 0, -11, 13, -3, 16, 0], [4, -22, -4, -8, 3, -4, -12, 9, -23], [19, -26, 6, 0, 5, 19, -9, 1, 10], [28, -23, 15, -1, 2, -3, 24, 8, -3], [-23, -5, -12, -8, -5, 3, -6, 3, 17], [2, -5, 10, -4, 5, -2, -8, 12, 21], [-10, -16, 18, 1, -6, 2, 18, 11, 7], [-35, 0, -18, -3, 17, -2, 2, -20, -23], [13, -1, -9, 9, -8, 3, -5, -12, 2], [3, 3, 2, 6, 14, 3, 1, -8, -20], [40, -17, -25, -4, 12, -1, -23, -22, -5], [-15, -5, -4, -8, 0, 12, -11, -14, -20], [-21, -3, 8, 2, 2, -7, -5, 13, -6], [-16, 24, -6, -19, -6, -3, 5, -3, 1], [12, -3, 0, -17, 5, 0, -19, -2, 5], [-12, 12, 9, 7, 1, 0, 17, 0, 0], [2, -9, -5, 0, 14, 0, -9, -7, -2], [16, 17, -2, -6, 29, -5, -5, 5, -9], [-10, 21, 8, 0, -20, 0, -3, -1, 13], [-28, 0, 7, -20, 9, 0, -20, 3, -6], [11, 14, 8, 5, -5, 26, -14, 1, 14], [8, -1, -15, -14, 11, -10, -3, -17, -2], [-9, 0, 3, -5, 10, 2, -2, 12, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6045_2_a_t();".
// 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_6045_2_a_t(:prec:=9)
    chi := MakeCharacter_6045_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_6045_2_a_t();".
// 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_6045_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![2, -7, -11, 38, 19, -43, -27, 5, 6, 1]>,<7,R![-805, -2598, 1127, 2162, -36, -545, -103, 33, 12, 1]>,<11,R![-6854, 4401, 25164, 19768, 4032, -1034, -458, -17, 10, 1]>],Snew);
    return Vf;
end function;