
// Make newform 1045.2.a.j in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1045_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_1045_2_a_j();".
// 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_1045_2_a_j();".
// 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 := [-2, 10, 13, -36, -20, 34, 11, -11, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0], [13, -126, 26, 112, -110, -35, 47, 6, -4], [177, 78, -400, -50, 307, 41, -94, -12, 8], [9, -288, 76, 488, -94, -196, 37, 22, -5], [-170, 162, 472, -202, -277, 103, 39, -16, 1], [-2, -110, -178, 43, 356, 50, -150, -21, 14], [7, -108, 565, 154, -695, -117, 235, 30, -20], [-3, 96, -180, -327, 418, 220, -196, -46, 21], [35, 388, -249, -738, 324, 372, -130, -53, 16]];
    Rf_basisdens := [1, 29, 29, 29, 29, 29, 29, 29, 29];
    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_1045_a();"
function MakeCharacter_1045_a()
    N := 1045;
    order := 1;
    char_gens := [837, 761, 496];
    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_1045_a_Hecke(Kf)
    return MakeCharacter_1045_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, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, -1, 0, 0, 1, 0, 0], [0, -1, -1, 0, 0, 1, 0, -1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, -2, 0, 0, 1, -1, -2], [-1, 0, 0, -1, 0, 1, 2, -1, 1], [-1, -1, -1, 1, 0, 0, 2, -1, 1], [0, 1, 2, 0, 0, 2, -1, -1, -1], [-2, 0, 2, 1, -1, 1, 0, 1, 0], [-2, -1, 0, -1, 1, 1, 1, -3, 0], [4, 2, 2, -1, 1, 0, -2, 1, 0], [0, 1, 0, -1, -1, 1, 0, 1, -1], [7, 1, 0, -1, 2, 0, 1, 0, 0], [-1, 1, 2, 0, 1, -3, -1, 2, -1], [0, 0, -2, 2, 0, -1, 0, 1, 3], [2, 0, 3, -1, 1, 1, 1, -1, 0], [-5, -2, 1, -1, -2, 1, -2, -1, 0], [-2, -1, 1, 1, 1, 0, -1, -1, 1], [7, -3, -1, 0, 1, -2, 1, 0, -4], [0, -2, -2, -1, 1, 1, 3, -4, 0], [0, 5, 1, -2, 0, 2, 0, 1, -1], [5, -3, 0, 0, 1, -3, -1, -1, -2], [-7, -3, 1, 1, -2, 1, -1, 1, 0], [1, 3, -2, 3, -4, -2, -1, 1, 3], [-1, 1, 2, -2, -3, 3, -2, -3, -1], [-2, 2, -1, -1, 1, 0, 2, -3, 1], [-6, 1, -1, 0, 0, -1, -1, 2, -1], [6, 4, 1, -1, -3, 1, -2, 1, -1], [3, 6, 0, -1, 2, 1, -1, 2, -1], [0, -1, 3, 0, 2, 1, 0, -1, -3], [-1, 2, -1, 1, -2, -3, 0, 2, 1], [4, -2, -3, 2, 0, -2, 1, 0, 2], [0, -2, 0, -1, 3, -1, 1, -1, -3], [-7, -1, -2, 4, -1, 2, 0, -1, 1], [2, -3, -1, 1, -1, 0, 2, -1, -2], [7, -3, -4, -2, 1, -5, 0, -1, 1], [-1, 1, 2, 0, 1, -1, 0, 3, 3], [1, -4, 0, 2, -1, -2, 2, -2, 1], [1, -1, -1, 2, -1, 1, 0, -1, 4], [-4, 0, 1, -1, 1, 3, 1, 1, 0], [0, -8, -1, 2, 0, 2, 3, -1, 1], [5, 4, 1, -1, -2, 1, 2, 3, 0], [-4, 2, 4, -3, -1, 5, -1, -3, 1], [1, -1, 1, 0, -1, 2, 4, 2, 1], [7, 2, 0, 0, -1, 0, -1, 2, 0], [2, -3, -1, -2, 2, -1, -2, 0, -2], [-1, -3, 3, -1, -2, 2, -1, 0, -1], [-1, -4, 3, 0, -1, -2, 3, 2, -1], [-8, -3, -1, 4, 0, 3, 3, -3, 0], [7, -3, -1, 2, 1, -1, 0, -1, -4], [3, 6, 1, 3, -4, -4, -6, 3, 2], [3, 2, -1, 0, 1, -5, -2, -1, 1], [-1, -1, 0, -2, -1, 3, -1, 0, 3], [1, -5, -1, 3, -2, 4, 3, 0, 3], [-8, -4, 0, -1, 5, -3, 3, 0, -2], [-8, -1, -1, 4, -2, 5, 1, 0, 5], [1, 7, 3, -3, 0, -5, -2, 2, 0], [-8, -7, 0, -2, 2, 1, 2, -2, 1], [13, 0, -1, 2, -1, 0, 0, -1, 1], [9, 4, 3, 0, -1, -2, -7, 1, 0], [7, 3, -2, 4, -3, -5, 1, 0, 1], [-2, 4, 3, 1, -3, -2, -4, 4, -4], [1, -3, -6, 4, -3, -5, -2, 2, 4], [0, 2, -5, -1, -1, -5, 1, 6, -1], [-5, 1, 0, -2, 1, 0, 6, -2, -2], [1, -4, -1, -2, 1, 0, 3, -2, 1], [15, 0, -1, -4, 1, -2, 1, 0, -7], [-1, -7, 0, -1, 0, -2, -2, -1, 0], [-4, -2, -1, 3, -3, -2, -1, -2, 1], [-11, -2, 3, 0, 1, 0, 4, 1, 1], [14, 4, -2, 1, -1, 1, -2, 0, 5], [2, -1, -6, 1, -1, -3, 4, -3, 1], [-2, -3, 3, 3, -1, 2, 1, -2, 4], [-5, 1, 5, 4, -1, 1, -2, 4, 3], [0, 5, -1, 0, 0, 1, 2, -1, 3], [-5, 0, 2, 0, -1, -2, 3, 2, 2], [4, 1, -2, 3, -3, -2, -6, 2, 4], [3, -5, 4, -3, -2, 2, -4, -2, -3], [-2, 2, 0, 4, 0, 6, 0, -1, 1], [7, 7, -3, 0, 1, 2, -4, 2, -3], [-1, 2, -5, 6, -3, -4, 5, -2, 5], [0, 2, 1, 0, -2, 9, 2, -4, 5], [-13, 1, 3, 0, -1, 5, -2, 2, -1], [4, 3, -3, 3, -1, 4, -1, 1, -5], [-5, 11, 2, -2, 1, 3, -1, -3, 0], [1, 0, 2, -4, -1, -5, -1, -4, 2], [6, -3, 0, 1, 3, 1, -1, 1, 4], [-5, -3, 6, -1, 2, 6, 0, -2, -3], [6, 5, -5, 1, -1, 5, 1, 0, -2], [7, -2, -4, 2, -5, 2, 2, 2, 3], [6, -6, -2, -1, -1, -3, -1, 1, -1], [4, 0, 6, 3, -1, 4, 0, -2, 1], [2, 0, 1, 6, 0, 2, -3, 0, 8], [4, 5, -3, -3, 1, 4, -1, -1, -1], [-3, -1, 2, -2, 3, 9, 2, -1, -1], [1, 2, 5, -1, 2, -1, -2, 2, 1], [-12, 2, 7, 2, -2, 2, 0, 4, 1], [8, 5, 1, 1, 1, 2, -5, 5, -1], [7, 8, -7, 1, -2, 3, 1, -2, 2], [-3, 9, 8, 0, 1, -3, -6, 7, -1], [0, -13, 0, -4, 0, 1, 5, -3, -7], [-8, 3, 3, -7, 1, 0, 1, -3, -1], [14, -3, 1, -5, -1, 5, 6, -7, 2], [9, -5, -3, 0, -1, -1, 2, -7, 0], [2, -9, -2, 3, -1, 6, 5, 0, -1], [-8, -4, -3, -2, -4, -2, 4, 1, -6], [-8, -8, -1, -2, 0, 2, -2, -5, -6], [-2, 7, -2, -1, -7, 6, -2, -6, 4], [-2, 0, 0, 0, 6, 0, 2, -5, 3], [4, 13, 0, -3, 3, 0, 0, -2, 2], [-3, -1, -1, 6, -1, 9, 0, -1, 4], [0, 4, -1, 1, 3, -1, -6, 5, -1], [1, 5, -1, -2, -3, 4, -5, -4, 4], [0, 3, 4, -1, -3, 3, 1, -2, 3], [2, 7, -4, 10, -2, -3, 3, 1, 9], [5, -1, 5, -12, -1, 1, 2, -4, -11], [-2, -7, 2, -1, 1, -2, 0, -6, 2], [-12, -3, 0, 2, -2, -1, -2, -2, 1], [-4, 3, -3, 7, 1, 2, 0, 0, 5], [28, 8, 2, 5, -5, 2, -7, 5, -1], [10, -1, -8, 4, -4, -3, -3, 2, 6], [-10, -10, -1, 0, 6, -2, 1, 2, -4], [-3, 6, 2, -10, 1, 4, 6, -1, -4], [-1, -9, 1, -1, -4, -10, -3, 4, 1], [-11, 5, 6, -7, -2, 7, -7, -2, -2], [0, -10, -2, -1, -1, 3, 1, -5, -3], [8, 7, 4, 1, -3, 0, -2, 9, -5], [6, -2, -1, -1, 3, -5, 5, 5, -4], [-11, -4, 1, -4, 9, 2, 4, -2, 2], [-8, -5, 1, 4, 4, 3, -2, -5, 3], [5, -4, 4, -1, 2, -1, -3, 7, -4], [4, 11, -9, 3, -3, 2, 3, 2, 2], [-11, -8, 2, 0, -5, -1, 0, 2, 3], [-7, 0, 4, 0, 1, 6, -1, -2, -12], [-7, 5, -3, 5, -2, -1, 3, 1, 2], [-8, -3, -7, 3, -3, 6, 5, -6, 8], [-7, 5, 2, -2, 1, 3, -1, -1, -4], [0, -8, -3, -4, 0, -6, -1, 0, -2], [-15, 1, -2, -2, -3, 4, -2, -3, -3], [3, -2, -3, 1, 0, -3, -3, 3, -9], [-7, 2, 3, -3, 0, 3, 8, 0, 1], [5, -11, -2, -1, -2, 4, 3, -1, -5], [4, 17, -1, -1, 3, 4, 1, 3, 3], [-11, -3, 4, 9, -6, 8, -1, -1, 5], [0, 2, 6, -4, 0, 0, 2, 0, -2], [8, -2, -11, 10, -4, -5, 5, 2, 4], [5, 10, -4, 5, 0, -5, -8, 5, 3], [1, -7, 3, 0, -1, -15, -4, 3, -2], [-3, -5, -4, -5, 2, -5, 3, -1, 1], [-21, -3, -5, 4, 1, 9, -1, -5, 7], [-8, 8, -6, 7, -5, 2, 5, 4, 10], [-5, 0, 7, 5, 0, 1, -5, 6, 2], [11, 9, -2, -4, -1, 3, -9, 2, 5], [-4, 0, -1, -6, 0, -4, -5, -1, -5], [-17, -6, -2, -3, 6, 9, 5, 1, 0], [-17, 2, 0, 5, -4, 1, 4, 0, 2], [4, -10, -5, -5, -1, -8, 7, -5, 0], [-5, -22, 0, -4, 1, 0, 9, -5, -13], [1, 7, -4, 2, 1, -4, 2, -3, -1], [3, -10, 8, -1, 2, 5, 4, -1, -5], [-4, -8, -1, -9, -1, 4, 6, -5, 3], [-7, 10, 4, 4, 1, 7, -2, -3, -4], [11, 1, -2, 0, 3, -5, 0, 2, -4], [-3, 1, 5, -6, -7, 7, 0, -3, -2], [-9, 9, -4, 3, 0, -8, -1, 8, 4], [-15, 5, 11, -3, -4, 6, 1, -3, -2], [-9, 5, -11, 1, -4, 8, 3, -3, 6], [2, -7, 0, -3, 7, -6, 4, 6, 2], [16, -1, -2, -4, 4, 1, 1, 5, -5], [5, -8, 2, 6, 1, -8, -6, 3, 0], [5, 13, -3, 0, 1, 7, -3, -3, -1], [1, 7, 1, 4, 7, 3, -2, 1, -6], [2, -18, -2, 0, -2, 4, 2, -10, -2], [6, 9, -13, 4, -2, -3, -2, -1, 9], [-9, 6, 0, -1, 0, -3, 4, 1, 1], [-8, 4, 2, 2, -2, -4, 2, -5, 5], [-6, 6, 0, -3, -1, -2, 8, 4, -3], [-7, 5, -7, 6, -1, -7, 0, 3, 0], [7, 6, -2, -3, -4, -2, 1, 1, -10], [5, 6, 0, -1, 6, 7, 2, 5, -1], [-11, -3, -2, 4, 1, 14, -2, -4, 4], [23, 4, -4, 5, -2, 3, -7, 5, -6], [6, 0, -3, -5, 11, -1, 1, -3, 2], [26, -3, -3, -2, 0, -14, 0, 0, -4], [-6, 0, -1, -5, -3, 13, 3, 0, -5], [-6, 6, 0, 5, -1, -5, 7, 6, 6], [7, 4, 11, 1, 0, -6, -2, 8, -9], [20, -1, 12, -9, 1, -1, -3, -4, -13], [14, 17, 0, 1, -3, 0, -5, 10, 5], [1, -6, -1, 4, 5, -6, -1, -3, -2], [18, 9, 9, -2, 4, -4, -5, 6, -1], [-4, -5, 8, 1, -5, 13, -4, 0, 0], [-10, -5, 1, 2, 10, 9, 2, -6, 0], [-3, 1, 3, 3, -2, -12, -3, 1, 4], [-13, 7, 6, -1, -6, 0, -2, -1, 4], [-4, -10, 11, -1, 1, 6, -1, -8, 1], [-12, -4, -7, 1, 7, -7, 3, -9, 6], [-10, -6, 8, -7, 7, 1, -3, 0, -6], [-5, -17, -2, 0, -3, 0, 3, -1, 4], [-23, 11, 7, -7, -2, 10, 4, -2, 8], [25, 5, 1, 2, 1, 9, 2, 0, 1], [-16, -13, -5, 3, 1, -4, 11, -4, -4], [16, 2, 5, 8, 2, -10, -5, 6, 0], [-7, 4, -4, 8, -3, -4, -3, 8, 8], [1, 5, 2, 1, -6, -2, -6, 5, -6], [15, 5, 7, 4, -3, -1, -11, 4, -12], [-9, -16, -8, 2, -1, -4, 8, -9, -2], [-5, 1, -8, 7, -6, 0, 6, -1, 6], [-17, 5, 5, -1, 2, 5, -2, 1, 7], [20, -11, -6, 8, 0, -8, -3, 1, 1], [-4, 1, 0, -1, 5, -4, 8, -8, 2], [5, 6, -6, 5, 2, -5, 5, 4, 13], [-10, -9, -3, 7, 1, -9, -10, 4, -9], [-3, -11, -5, 8, 3, -3, -1, 1, 1], [-2, -3, -4, 6, 8, -4, 3, -1, 15], [12, -7, -9, -8, 2, -9, 4, -2, 6], [6, 12, 4, 3, -1, -9, -6, 6, -7], [-7, -8, 0, 0, 3, 8, 0, 8, -3], [5, 1, -6, 0, -5, -11, -5, 2, -11], [-22, -6, 2, 3, 3, 2, 10, -1, 4], [36, -2, -3, 4, -6, -12, 2, 5, 2], [-3, 0, 3, 0, -1, 6, 7, -8, 9], [2, 11, 3, -7, 3, -5, -13, 0, -4], [0, -7, 0, 3, -1, -6, -3, -1, 6], [4, 9, 6, 2, 0, 13, -4, 3, 10], [23, 10, -2, -3, -2, -9, -1, -4, -9], [22, -4, -9, 2, 4, 2, 7, -2, 6], [11, 5, 3, 11, -2, -6, -11, 9, 12], [-4, 4, -2, -3, -1, 3, -1, -1, -13], [-12, -1, 4, 5, 1, -2, 5, -4, 11], [-2, -7, 3, -1, 3, 7, -2, 3, 2], [-5, 1, 6, 1, 0, 9, -5, 0, 6], [17, -5, 6, 0, 1, -3, -1, 5, 4], [10, 5, 2, 1, -1, 0, -2, 2, 12], [-7, -20, 4, -3, 2, -3, 1, -1, -12], [13, 11, 4, -3, 4, 8, 5, -3, 9], [-18, 0, 15, -8, 4, -1, 3, -2, -6], [-2, 5, -3, -3, -9, -2, -1, 6, 8], [-9, -3, -4, -4, -3, -6, 6, 3, 5], [4, -6, 2, -4, 6, -8, 2, -6, -2], [-7, 22, 5, -4, -1, 6, 0, -2, -2], [-1, -3, 3, 1, -4, 12, -3, 5, -6], [-13, 0, 10, -5, 4, 2, -4, -6, 2], [-6, 3, -1, 5, -1, 6, 0, -10, -1], [-8, -3, 11, -11, 9, 16, 9, -8, -12], [12, 2, -13, 5, -7, -12, 1, -1, 2], [11, 0, -9, 9, -6, -1, -8, -4, -1], [-10, -9, 5, -8, -2, -1, 2, -7, 3], [15, -12, -7, -9, 6, 0, -3, 0, -8], [26, 6, -2, 2, 2, 0, -12, 6, 4], [9, 14, -1, -5, 4, -1, 4, 11, -2], [11, -6, 1, -1, -4, 13, -6, -5, -8], [21, 18, 5, 5, 0, 7, -4, 8, 3], [-4, -5, -3, 1, 1, 4, 6, -1, 12], [10, 10, 1, -11, 3, 2, 8, -1, 1], [2, -1, -2, 1, -3, -10, -2, 9, 7], [2, -8, -5, 5, -1, -9, 5, 2, -5], [-6, 10, -3, 0, 2, 0, -8, -4, 7], [9, 11, 5, 7, -2, -6, -3, 3, -2], [6, -5, -6, -3, -1, 0, -5, -10, -1], [-13, -13, 4, 8, 5, -5, -1, -6, 3], [17, -7, -6, 2, -1, -7, -2, 2, -2], [10, -10, -1, -7, -1, -3, -4, -4, 2], [-7, -1, 6, 13, 0, 8, -6, -4, 9], [-13, 4, -4, 5, -8, -7, -10, 0, 4], [10, -5, 3, 3, -3, -8, -3, 5, -11], [11, -18, -7, 9, 2, 3, 10, -4, -1], [18, 2, 4, -11, 7, -10, -9, 4, -16], [2, -11, -7, -19, 7, -8, 1, -7, -11], [-7, -7, 4, 1, 2, 0, 10, -4, -5], [-14, 3, 2, 2, -4, 5, -2, -8, 1], [19, 0, 1, 1, -8, -8, -9, 6, -6], [12, -2, 4, -1, -1, 6, -6, -10, -11], [25, 5, 1, -7, 2, 8, 2, 4, -12], [-13, -11, 0, 7, -4, 6, 9, 2, 8], [29, -5, 3, 6, 3, 1, 4, -3, 2], [3, -13, -6, 5, -6, 2, 1, -2, 12], [5, 14, 4, -8, 5, -10, 6, 11, -2], [10, 7, -9, 2, -2, -6, 1, -5, 0], [19, -2, -3, 3, -2, -3, -7, -1, -5], [-14, -10, -5, 0, 0, 3, 10, -16, -5], [5, 9, 7, -9, -8, 10, -5, -1, -14], [12, 2, -8, 13, -5, -1, -5, 2, -6], [-1, 1, 3, -5, -12, 6, -2, -5, -5], [24, 8, -4, 15, -1, -1, -7, 6, 2], [-15, -16, -8, 4, -9, 6, 5, 5, 1], [2, -1, 3, -13, 5, -4, 3, 3, -1], [12, -11, -3, 7, -9, 1, -4, 0, 7], [11, 12, 4, -10, 7, -4, 3, -1, -3], [33, 23, 2, 3, 2, 10, -10, 7, 6], [-10, 13, 1, -4, 2, 13, 13, 3, 0], [5, 9, -8, 1, 0, -12, 0, 0, 3], [-2, 15, -1, -7, 7, -8, 4, 4, -3], [1, 7, -3, 3, -2, 4, -9, 9, -2], [-23, -5, 9, -2, -1, 5, -8, 5, -8], [1, 0, 6, 2, 11, 7, 1, -11, -9], [-5, -9, -8, 3, -4, 0, -6, -10, 3], [-1, -4, 2, 5, -8, 3, 4, 2, 8], [-3, -7, 1, 3, -6, -4, 1, -12, 1], [-7, 25, 1, 2, -3, 14, -4, 2, 17], [-5, 2, 12, -3, -2, 17, 9, 6, -3], [16, 5, -11, 3, -9, -5, 1, -4, 6], [-26, -12, -4, -7, 3, 4, 3, 0, 2], [15, -5, -7, 4, -3, 1, -6, -7, 0], [3, 3, -8, -9, 14, -8, 4, -8, 7], [10, -4, 3, 14, -2, -6, -9, 5, -3], [11, 6, -7, -10, 3, -1, -6, 5, -1], [-5, 3, -1, 6, 5, -7, 6, 10, 7], [-22, 1, 1, 5, 1, 4, 4, -4, -1], [-2, -13, -6, -3, 3, -5, 10, -3, -9], [-32, -25, 5, -7, 1, -5, 8, 2, -5], [-5, 1, 4, -13, 10, 0, 3, 1, -9], [-4, -1, -2, -1, 7, 4, 18, -3, 7], [8, -17, -1, 8, -2, -9, -4, -1, 5], [20, 12, -4, -1, -3, -7, -3, 2, 8], [12, -16, 7, -6, 0, 4, -1, -11, -15], [-8, 7, -2, 2, -4, -3, -3, 0, 14], [-11, 1, 0, -7, -2, -6, 4, 3, 0], [-9, 0, 5, 1, -2, -5, -3, 18, 8], [-13, -11, 7, -9, 0, 5, 13, -6, -9], [5, -7, -1, 8, 9, 3, 6, 8, 5], [0, 17, 0, 1, -7, 4, -6, 6, 10], [-1, -1, 9, -2, 5, 9, -3, 1, 9], [-17, -5, 3, -4, 7, 16, 16, -3, -2], [-2, 4, 10, -10, 14, 6, -1, 10, 1], [-11, -14, 0, 3, 0, 7, 5, -2, 13], [17, 4, -1, 3, -2, 9, -2, 8, -5], [-12, -3, -12, 18, -8, 0, -3, 0, 8], [9, 13, 2, 9, -8, -4, -6, -3, 8], [16, 1, -6, 6, 8, -8, -10, 6, 5], [-2, -3, -1, 0, -2, 17, 4, -16, -4], [37, 17, 0, 1, 2, 1, -11, 0, 2], [2, -9, -6, 11, 7, 2, 10, 3, 9], [36, -2, -4, -7, 5, -9, 12, 7, -12], [-13, -9, -3, 0, -7, -1, 2, -3, -4], [6, -12, -3, 6, 0, -19, -10, 11, -18], [7, 4, -2, 7, 0, 0, 7, -4, 5], [8, -5, 4, 3, 1, 0, -2, -13, 1], [-17, -2, -12, 13, -8, 3, 2, -4, 12], [10, 11, -10, 10, -6, 1, 0, 0, 17], [-23, 10, 5, 5, -6, 12, -2, -12, 9], [-5, -10, -2, 2, 3, -14, 4, 0, 1], [34, 1, -3, 14, -12, -1, -10, 11, 9], [15, -3, 3, 11, 0, 5, 5, 4, 5], [-16, 13, 6, -2, -8, 7, 1, 4, -12], [-2, 3, -2, 1, 1, 12, 4, -5, 17], [-6, 14, 4, -14, -6, 10, -10, -1, 9], [-21, -9, 10, 1, -10, 7, -10, -3, -6], [-33, 3, 3, 2, 5, 9, 2, 4, 1], [-27, 9, 7, -8, 5, 5, 3, -8, 0], [-2, -7, 4, 5, 1, -18, -5, 3, 2], [-14, 8, 3, 7, -3, 3, 1, -6, 3], [27, 8, -7, 5, -8, 11, -2, -10, 1], [26, 28, -8, 15, 1, 7, -7, 8, 14], [15, -5, -4, 9, 2, -8, 6, 10, -3], [-19, 15, -10, 8, -5, 6, 3, 1, 2], [3, -2, -13, 1, 0, -13, -2, 17, 2], [28, 27, -12, 10, 0, 1, -5, 3, 5], [-12, 1, -4, -18, 2, 2, 11, 1, -5], [21, -14, 1, 9, -4, -9, -10, 1, -14], [17, 0, -3, -3, -2, 0, -7, -4, -4], [-14, 8, 11, -4, -8, 13, -7, 2, 2], [-33, 0, -9, 10, -1, -8, -9, 8, 3], [2, 11, -8, -1, 3, 14, 1, -7, -4], [5, 8, -11, 2, 11, -10, 10, -5, 3], [2, -25, -6, -2, 2, 15, 3, -10, 0], [6, -8, -3, -10, -8, -8, 12, 5, 2], [-27, 0, 3, -20, 11, -1, 4, -9, -13], [-40, -11, -10, 3, -3, 2, -3, -1, 6], [-19, -10, 2, -7, -6, 3, 9, -3, 2], [-7, -1, -5, -6, 1, 15, 14, -11, 2], [14, 17, 9, 2, -10, -11, -13, 11, -6], [22, -12, -9, 7, 7, -7, 8, 8, 0], [-14, -1, 13, -3, 1, 2, 9, -6, -4], [2, 4, -10, -1, -5, -7, -5, 16, -4], [-1, 2, 6, -6, 11, -4, -5, 7, -11], [-40, 5, 6, -9, -3, 4, -2, -1, 5], [9, 16, 7, -1, -2, 11, 0, 4, 1], [-42, -2, 3, 0, 0, 3, 8, 10, 7], [8, -5, 8, 4, 2, -7, -3, 3, 7], [1, 18, 2, 0, 9, 4, -8, -3, 4], [5, -13, -9, -5, 4, 5, 3, -3, -6], [12, -1, -1, 1, 9, -2, -1, -4, 12], [19, -14, -1, 5, -10, 13, 12, -10, 9], [-29, -8, 2, -7, -6, -2, 0, -6, 6], [1, 2, 15, 1, 2, 14, -2, -12, -7], [20, -7, 14, -7, 5, -10, -14, 10, -12], [-8, -3, -11, 8, -6, 14, 6, -17, 1], [-40, 5, -2, -6, 2, 3, 5, -1, 1], [11, -4, -9, 7, 8, -10, 13, -3, 1], [15, 8, 4, 1, 2, -1, -8, 4, -18], [-1, -34, -8, -3, 6, -7, 8, -9, -5], [14, 12, 1, 6, 0, -2, -3, -1, 7], [9, 13, 4, 3, 4, 2, -12, 5, 0], [-14, 4, -2, -13, 3, 11, 4, 5, 0], [-10, 21, 5, 1, -7, 0, -9, -4, 4], [-11, 23, -2, -2, 3, 17, 6, -1, 1], [19, -13, -8, 9, 0, -10, 2, -5, -6], [1, -1, 8, 10, 1, -13, -6, 6, 18], [30, -12, 1, -4, 8, -15, 1, 10, -20], [2, 10, 11, -11, -1, 1, -9, -10, -3], [34, 3, -11, -5, 7, -2, -1, -1, 3], [-3, 13, -8, 9, -2, -10, 4, -5, 8], [1, 31, 0, 0, -7, 1, -4, 13, 13], [-23, -21, -6, 1, 8, 11, 2, -7, 6], [8, -6, -3, -5, 9, 11, 12, -12, 10], [-41, 13, 5, -2, -5, 3, -3, 5, -3], [14, 17, -15, 12, -2, 9, -4, -7, 5], [1, -8, 4, 3, 2, -6, -1, -4, -13], [-8, 16, -2, 10, -4, 22, 0, -5, 5], [-5, -8, -4, -6, 1, 2, -12, 6, -13], [-6, -9, -1, -15, 1, 14, -1, 2, -14], [36, -4, -8, -3, 13, -11, 5, 6, -4], [-1, 1, -2, 12, 1, 2, -3, 21, 0], [3, 18, 10, -1, 0, -7, -11, 5, -8], [-28, 3, -5, 3, -3, 0, -6, -4, 3], [3, 14, -15, 19, 2, -3, -10, 11, 14], [3, 9, 15, 6, 3, 1, -11, 11, 1], [-24, 6, 13, -3, -1, 12, 15, 6, 1], [10, -10, 6, 9, -1, -3, 7, 12, 2], [16, -7, -7, 4, -12, -14, 2, -4, 4], [-10, -11, 9, 2, -6, 0, 10, 1, -5], [-14, -13, 9, 6, 4, -2, 4, -2, 4], [5, -4, -4, 9, 0, 13, -3, -9, 10], [20, -7, 2, 2, -4, 1, 7, -4, 6], [-8, -16, 9, -15, 1, 3, -2, -5, -15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1045_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1045_2_a_j();".
// 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_1045_2_a_j(:prec:=9)
    chi := MakeCharacter_1045_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_1045_2_a_j();".
// 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_1045_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -8, 51, -21, -70, 25, 27, -9, -3, 1]>],Snew);
    return Vf;
end function;