
// Make newform 1045.2.a.h in Magma, downloaded from the LMFDB on 19 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_h();".
// 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_h();".
// 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 := [11, -18, -16, 27, 8, -10, -1, 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], [21, -10, -33, 2, 11, 0, -1], [27, -1, -52, -5, 20, 1, -2], [41, -10, -78, -3, 30, 1, -3], [-62, 15, 111, 2, -41, -1, 4]];
    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_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, 1, 0], [-1, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 1, -1, 1, -1], [1, 0, 0, 0, 0, 0, 0], [-1, -1, -1, 0, 1, 2, 0], [-1, -1, -1, 0, -1, 2, -2], [-1, 0, 0, 0, 0, 0, 0], [-1, 1, -1, -1, -1, 0, -1], [1, -1, 1, 1, 1, 0, 0], [0, 2, -1, 1, 2, 0, 1], [-4, 0, -2, 1, -2, 3, -2], [3, -1, 0, 0, 1, 0, 2], [-1, -3, -1, -1, 3, 1, 1], [-1, -1, 0, 2, 3, 3, 0], [-1, 3, -1, 4, 1, 0, 0], [5, -1, 1, 2, 1, 0, 2], [4, 4, -2, 0, 0, -2, 3], [0, -2, -1, 0, -2, 3, -1], [2, -2, -1, -1, -2, 2, 0], [6, 0, 2, -1, 0, -3, 3], [2, -2, 2, 1, 0, -1, 0], [7, -1, -1, 0, -1, 0, 2], [1, -1, -1, 3, -1, 1, -2], [3, 1, 3, -4, -3, -5, -2], [-2, 0, -1, 4, 0, 6, -2], [-6, -4, 1, 0, 2, 0, -4], [5, 1, -2, -2, -3, 0, 1], [-4, -2, 1, -1, -4, 1, -5], [-2, 0, 3, 0, 0, 2, -3], [-1, -3, 0, -1, 3, -5, -2], [8, 0, 1, -2, 2, -3, 3], [-3, 1, 4, 2, 1, 3, -6], [5, 1, 3, -2, -1, -6, 2], [-6, -4, -1, 3, 2, 5, -4], [5, 3, -3, -3, 1, -2, 8], [1, 5, 0, -1, -3, -1, 1], [0, 2, -5, 1, -2, 0, 1], [12, 0, 0, -4, -2, -6, 3], [2, 4, 5, -3, -6, -3, -3], [14, -2, -1, -5, -4, -3, 3], [-6, 2, 1, 3, -2, 6, -4], [7, 5, 0, 1, -1, -4, 8], [-6, 0, 3, -1, 0, 0, -8], [5, 5, 2, -3, -3, -4, 1], [-10, 0, -2, 5, -2, 3, -1], [9, 3, 2, 1, -1, 3, -3], [-1, 3, 3, -1, -3, -3, 1], [-2, -4, 3, 4, 0, 3, -7], [-3, 5, 0, 0, 7, -8, -1], [-4, 8, 0, 4, 0, -1, 1], [-7, -7, -2, 2, 7, 4, -4], [7, 9, -1, -1, -3, -5, 1], [7, -3, 0, -1, 3, 2, -4], [10, 2, -2, -5, -2, 0, 7], [-1, 1, 3, -3, 1, 0, 0], [4, 8, -4, -4, -8, -8, 3], [10, -2, 0, 0, 0, -7, 1], [-5, -5, -1, 3, -3, 7, -8], [11, 3, -4, -5, -1, -1, 10], [-2, 6, 2, 1, -6, 3, 0], [8, -6, -1, -4, 0, -5, 4], [-3, 3, 5, 1, -3, -3, -4], [-11, -9, -1, 4, 5, 8, -5], [-16, 0, 0, 2, 8, 0, 3], [12, 0, 0, 2, 4, -4, 6], [6, -8, -2, -1, 2, 3, -2], [-4, -4, 0, 5, 0, 2, 7], [8, 6, 2, -1, 0, -10, -2], [9, -9, 0, -10, 5, -8, 10], [3, 1, -4, 0, -1, 2, 8], [5, 1, -6, 1, 9, -3, 10], [6, -6, 4, -5, 8, -9, 7], [-5, 7, -3, 1, -5, -1, 0], [-7, 5, -3, -2, 7, -10, 9], [7, -7, -3, 6, 1, 4, -4], [-6, -2, 3, 3, -4, 1, -3], [5, -1, 5, -5, -11, 0, -3], [-7, 7, 1, 0, -7, 4, -6], [16, -4, 5, 0, -2, 7, -1], [5, 1, -2, -4, 1, -5, -1], [-7, -7, 3, 3, 9, 3, 3], [4, 6, -5, 4, -2, 10, -5], [-9, 1, -9, 5, 1, 1, 1], [1, 9, -2, 6, -5, 8, -10], [9, -3, 1, -6, -1, -10, -1], [3, 1, -5, -7, 3, -4, -3], [10, -2, -2, -11, 6, -7, 10], [-14, 4, -3, 6, -4, 4, -4], [-4, -4, 1, -2, 4, -2, 9], [11, 7, -1, 3, -5, -7, -9], [3, 5, -5, 1, -3, 3, -3], [-18, 4, -1, 9, -6, 4, -5], [2, 4, 1, 3, 0, -2, 0], [1, -13, 0, 1, -1, 5, -5], [-15, -7, 1, -4, -5, 7, -11], [11, 3, 9, -9, 1, -16, 4], [-6, -4, -2, 3, -2, -5, -4], [16, 8, 1, -3, 0, -9, -3], [6, 8, 9, -1, -6, -3, -2], [-5, -3, 1, 4, 7, 5, -1], [-8, 4, -2, -11, 0, -11, 4], [22, -6, 5, -6, 4, -4, 9], [-8, -2, -3, 1, 16, 3, 13], [4, -2, 4, 1, -10, 4, -8], [-4, -6, -2, -3, 0, -1, 4], [-4, -2, -2, 2, 8, 5, 7], [1, -1, 0, 1, 7, 4, -6], [-7, 1, 2, -1, -5, 4, -10], [13, 3, -4, -6, 3, -5, 18], [-17, 3, -2, -4, 5, -5, 0], [1, 1, 0, 2, 7, 1, 13], [-10, -2, 5, -2, -2, 8, -1], [5, -3, 6, -2, -3, 3, -15], [-17, 5, 0, -3, -9, 4, -8], [-2, -4, 5, 0, -6, -5, 3], [-11, -1, 3, -4, 5, 4, 4], [9, 3, -4, -5, 7, 1, 6], [0, 0, 2, 1, 0, 0, -6], [11, 7, 1, -1, -7, 2, -7], [-5, 1, 2, -4, -3, -5, -3], [20, 6, 3, 4, 0, -15, 8], [-12, 8, -5, 10, 4, -2, -6], [9, 3, 8, -9, 3, -10, 9], [8, 4, 1, -7, -8, 2, 9], [3, 11, -3, 3, 1, -10, 12], [-6, -20, -2, -1, 0, 14, -11], [6, 2, 4, 0, -12, 5, -1], [-7, -11, 7, 6, 1, -1, -4], [5, 3, 6, 3, 1, -5, -5], [7, -3, 5, -4, 5, 5, -4], [-16, 0, -5, 9, 6, 10, -6], [5, -3, 6, 5, -3, -3, 7], [-31, -5, -3, 6, 1, 14, -8], [-1, -3, 1, 2, -3, -3, -5], [4, 12, 7, -1, -14, -9, 1], [20, 0, 5, -1, -6, 5, -10], [6, -4, -9, -2, -8, 5, 9], [14, 6, 3, -16, 2, -18, 16], [6, 14, 3, 1, -12, 3, 1], [8, 0, -7, 2, 8, 8, 0], [1, 1, 1, -1, -9, -14, -2], [-10, 4, 1, 15, 0, 10, 3], [14, 0, 4, -8, 6, -14, -2], [4, -6, -5, 1, 8, -7, 4], [7, 3, 6, -2, -15, -4, -5], [4, 4, 0, 7, 4, 1, -4], [-19, -1, -4, -2, 5, 10, 7], [2, 6, 0, -2, -6, 4, 8], [5, -5, -3, -9, 5, -7, -2], [-21, -3, 5, -7, -9, 6, -10], [-25, -5, -10, 7, 5, 10, -6], [-7, 11, -8, 0, -9, -9, 1], [21, -5, -4, -4, -7, 11, -3], [3, 3, -1, -1, 15, -16, 24], [-10, -6, 1, -4, 6, -2, 11], [-6, -8, 0, 6, 4, 8, 7], [-7, -1, -8, -3, 1, 4, 7], [-17, -1, -3, 8, 5, 7, 5], [-5, -9, 2, 1, -11, 12, -15], [18, -8, 3, -1, -6, -9, 2], [10, -4, 2, -12, -10, -7, -4], [-32, 8, 9, 3, 2, 0, 1], [-5, -5, 3, 7, 13, 4, 6], [-10, -16, 1, -2, 8, -2, 11], [14, 18, 4, -3, -6, -11, -1], [-10, -8, 12, -9, -18, 1, -12], [-1, -3, 8, -1, 9, -12, 10], [23, -13, -7, -14, 1, 4, -6], [-16, -12, -3, 12, -10, 15, -26], [16, -6, 5, 0, 16, -11, 8], [3, 5, 10, -4, 5, -29, 13], [-14, -16, 2, 1, 0, 2, -2], [-14, -6, 4, 13, -8, 8, -13], [15, -11, 4, 0, 1, -13, 5], [-11, 1, 4, -3, -5, 4, 4], [-2, 6, 4, 8, -2, 4, -2], [-13, -3, -9, -8, 7, 6, 2], [-15, -3, 3, 7, 5, 6, -4], [2, -14, -7, -2, 2, 12, -7], [-1, 1, 3, -6, -15, -3, -9], [1, 3, 0, -1, -11, -2, -16], [19, 1, -19, 4, 3, 6, 7], [5, -1, -1, -1, 5, -2, -6], [8, -16, 0, 7, 4, 8, 2], [-1, -15, 5, 4, 9, 3, 11], [2, 0, 1, -3, 4, -12, 20], [-33, 9, 0, 10, 3, 15, -17], [-4, -8, -18, 5, 12, 2, 11], [35, -5, 11, -9, -13, -5, 8], [6, -12, -5, -5, -4, 2, -10], [19, -9, 0, -9, 13, -19, 24], [-23, -7, -6, 5, -1, 12, -14], [-7, 9, 3, -6, 11, -18, 19], [7, 1, 0, -1, -3, 12, 12], [-3, -5, -4, -8, 5, -6, -9], [11, -17, 2, 2, -7, 2, 3], [34, -6, -1, 4, 12, -9, 16], [-11, 9, 10, -5, 7, -19, 8], [-24, -2, 4, 1, 10, 5, 1], [-12, 6, -1, 9, 6, 2, 2], [29, 5, 7, -9, -11, -19, 6], [22, -4, 5, -10, 10, -11, 20], [2, -4, 12, 5, 4, 8, 2], [9, 9, 7, -1, -13, -6, -1], [14, 18, -7, -1, 4, -7, 5], [3, 1, -3, -1, 3, -16, -8], [2, 20, -11, 16, 4, -11, 5], [-9, -3, 4, -7, -3, 9, -8], [-7, 9, -3, 10, -11, 13, -16], [8, -16, -4, 5, 2, 0, 7], [1, -5, 12, -17, -1, -13, 2], [-4, -8, -5, -7, -6, 11, -7], [20, 6, 10, -1, -2, -13, -6], [-3, 5, -4, 6, -7, 17, -13], [1, 13, -4, 8, -3, 3, 4], [6, -8, 13, -16, -4, -17, 0], [9, -3, 6, -2, 1, -21, -9], [16, 6, 6, -9, 2, -11, 2], [31, 7, 6, -12, -3, -24, 13], [-9, 5, 5, 0, -1, 8, -23], [2, 12, 3, 3, -12, -10, 10], [16, 4, 2, 6, 4, -2, 9], [3, 7, -16, -8, 7, -3, 2], [7, -5, 6, -4, -23, 1, -18], [11, 17, 0, 6, 3, -20, 2], [30, -6, -1, -5, 14, -12, 13], [-5, 21, 1, -7, -13, -2, -7], [-14, 12, 0, 4, 16, -9, 9], [-11, -13, 8, -6, 3, 5, -10], [-3, -3, 3, -13, 3, -16, 16], [20, 2, -3, -2, -8, -3, -2], [15, 5, -8, 5, -3, 23, -7], [-9, -15, 4, 7, 3, 12, 8], [-10, 4, -10, -2, -6, 1, -5], [3, 1, -3, -10, -1, -15, -4], [24, -2, 5, -6, -10, -18, -4], [-17, 3, -12, 2, 17, -7, 19], [-3, -3, -13, 16, 9, 13, 5], [15, -1, -6, -4, 9, 2, 3], [-19, 19, -11, 15, 1, 3, -8], [26, 0, -15, -5, -6, 3, 1], [14, 8, 5, 1, -16, -8, -3], [0, 6, -4, -12, 10, -16, 14], [29, 11, 8, 3, -9, -21, -9], [12, -2, 3, 10, -8, 19, -2], [19, -1, 4, -15, -5, -33, 15], [6, -8, -3, 0, 8, -4, -10], [16, -2, 9, -5, -24, -1, -7], [-22, 2, 5, 13, 2, 9, 0], [-26, -6, 5, 3, -2, -9, -10], [25, 3, 15, -24, -3, -20, 12], [8, 2, 1, 2, 20, -5, 7], [-10, 4, -12, 8, 0, 36, -6], [8, -10, -6, -9, 10, -13, 15], [0, 2, 6, 5, 8, 9, -9], [38, -4, -1, -7, -2, -17, 18], [-10, -6, -18, 6, 8, 0, 9], [-30, -16, 1, 2, 4, 10, -11], [7, 5, -11, 4, -5, 21, -10], [-34, 4, -8, 7, -10, 4, -7], [-25, 13, -13, 14, -9, 19, -17], [18, 14, -5, -2, -12, -15, -12], [49, -1, 6, -9, -3, -6, -6], [6, -2, -4, 16, 2, 8, -7], [8, -2, 8, 7, 12, -13, -4], [-10, -6, 10, 4, 4, 19, -18], [13, 5, 7, -3, 5, -19, 5], [-21, -3, -6, 7, 1, 26, -17], [-16, 0, 2, 1, -8, 5, -2], [6, -20, 11, 5, 2, 13, 4], [-13, 15, -1, 11, 11, -10, 14], [22, 12, 4, -5, -14, -15, -10], [23, -7, -5, -15, 9, -1, 9], [-6, 16, 3, 2, 6, 1, -4], [-6, -4, 1, 4, -6, 11, 5], [29, 7, 5, -10, -21, -11, -11], [-12, -10, 6, 3, 6, 6, -20], [19, -5, -1, 0, 5, -12, -6], [-9, -11, 4, 13, -1, 20, 8], [-1, 17, 3, 0, -19, -8, 2], [-4, 16, -4, 1, 0, -8, 9], [-3, 5, -15, -1, -7, 8, 1], [-28, -8, -8, 16, 20, 20, 4], [19, 5, -1, -4, 5, -2, 7], [24, 2, 7, -8, -4, -21, 8], [-17, -7, -9, 1, -3, 13, -22], [12, 8, 3, 3, 8, -20, 19], [3, 7, -3, -3, -17, 9, -6], [-2, 4, 4, 2, -14, 11, -26], [-38, -28, 4, 1, 6, 11, -6], [24, 16, -3, 10, 10, -15, 2], [4, 10, -2, 2, -12, 1, 12], [29, -3, 2, -11, -19, -19, -12], [37, 3, 5, -21, 3, -15, 15], [13, -3, 1, -11, 21, -7, 23], [-34, 16, -1, 12, 0, 26, -20], [4, -20, 7, 2, -2, 19, -16], [-11, 7, -4, -13, 1, -14, -14], [30, -2, -4, 0, 16, 0, 12], [23, 13, 15, -3, 3, -27, 13], [-3, -11, 16, 9, 3, -4, 5], [-4, 12, 4, -2, 6, -19, 13], [-35, 17, 7, 3, -5, 7, 3], [-7, 27, -4, 6, -5, -9, 2], [-6, -2, 19, 7, 14, -18, 11], [47, 11, 1, -1, 3, -8, 6], [-35, 1, 4, 1, -9, 14, -4], [3, -7, -21, -13, 5, 15, -1], [-3, -3, -14, 7, 29, 4, 11], [-53, 1, 1, 13, 5, 8, -22], [-15, -25, 1, 11, 9, 20, -25], [2, -10, -7, 0, -6, 10, 8], [-25, 1, -15, 20, -7, 32, -14], [12, 10, 7, 12, 6, -17, 22], [7, 5, -2, 0, 3, 4, -13], [17, -1, 9, -4, -9, -13, 0], [7, 3, 3, 0, -3, 14, 2], [-23, -9, 11, -9, -9, -5, -6], [21, -1, 14, -19, -7, -42, 3], [2, 18, -15, 7, -14, 14, -11], [-25, 9, 4, -1, -3, 25, -14], [-10, 0, -1, 2, -4, 4, 1], [-2, -10, 10, -4, 20, -18, 21], [15, -9, 11, 3, 23, -16, 23], [-5, 13, -20, 8, 9, -17, 7], [9, 7, 14, -6, 19, -37, 19], [25, 15, 12, -11, 1, -11, 8], [-8, 4, 6, -3, 8, -17, 7], [3, 25, -4, -12, -9, -5, -6], [20, 10, -9, 3, 0, 1, 6], [-4, -2, -3, -13, -4, 3, -17], [1, -7, -6, 3, 29, -9, 28], [50, 10, 5, -2, 0, -16, 18], [-3, -5, -20, 14, 1, 32, -1], [3, -1, -13, 2, -9, -13, -4], [31, -9, -1, -26, 7, -11, 2], [-19, -11, 0, 6, 11, 20, 14], [-3, 13, -2, -1, -1, -16, 8], [-13, 7, -12, -14, 11, -6, 14], [-7, 13, -6, -10, -17, 8, 8], [-23, -7, -5, -10, 17, -11, 28], [35, 11, 18, -17, 5, -40, 19], [-2, 14, -5, 8, -16, 5, -18], [4, -26, 1, 1, 22, -2, 12], [22, 24, -1, 5, -8, -24, 10], [15, -7, -1, -4, -1, 4, 10], [10, 2, 4, 12, -16, 18, -33], [-14, -34, -5, 11, 2, 16, -15], [-3, -3, -7, 6, -7, 39, -2], [-22, 4, -3, 14, -4, 10, -17], [-11, 19, 10, -9, 1, -8, 0], [-35, -5, -9, 23, 13, 36, -15], [-7, -11, -13, 12, -9, 23, 2], [15, -1, 3, 5, -7, -2, 1], [-28, 0, -8, 23, 20, 26, -5], [-32, -8, 3, -5, -10, -3, 2], [9, 21, -3, 5, -15, -1, -2], [-3, 7, -15, 17, 9, 7, 3], [-22, -10, -5, 8, 10, 15, -10], [-14, -28, 4, -1, 8, 7, -15], [-32, -16, 10, 13, 16, 20, -22], [12, -10, -6, 5, 10, -1, -8], [2, -12, 2, 11, 10, 19, -5], [30, 4, 9, -16, -2, -15, -15], [21, 9, -5, -19, 9, -9, 18], [-3, -19, -10, 0, 5, 4, 12], [-23, -17, 3, -21, -1, -2, 3], [23, 19, -3, -5, 7, -1, 0], [0, -8, 1, 7, -4, -22, -1], [33, -3, 12, -24, 13, -23, 16], [3, -7, 15, 1, -19, 10, -2], [43, -9, 1, -11, 1, -16, 7], [29, 5, -13, -20, -3, -15, 21], [-7, -3, 8, 5, 7, 4, -24], [-13, -9, -21, 5, 15, 9, 13], [34, -12, 6, -22, 6, -21, 18], [0, -2, -13, 11, 2, 25, -5], [-11, 31, -3, -7, 3, 1, -10], [2, 8, -18, 2, -10, -5, -18], [23, -5, 3, -8, -29, 3, -10], [18, -4, 9, -11, -10, -41, 10], [-21, 9, 5, 2, -19, 0, -1], [2, 8, -14, 7, -2, -8, 6], [-16, -8, 0, 5, 12, -4, -19], [18, -32, 7, -16, -18, 8, 0], [-14, -16, 1, 9, -14, 29, -29], [-4, 14, 13, -13, 4, -29, 14], [-9, -15, -6, -7, -1, 11, -7], [2, -2, 13, 13, -8, 26, -14], [25, -25, -6, -16, -1, 3, -15], [25, -1, 3, 6, 3, 11, -16], [-40, 6, 8, 11, 2, 14, -36], [34, 2, 10, -2, -16, 1, -12], [49, 11, 10, -5, -19, -12, 3], [7, 1, 21, -5, -17, -28, -4], [-19, -15, 0, 25, -1, 26, -29], [3, 25, 18, -3, -27, 1, -6], [-1, -27, -2, 16, 7, 16, -4], [4, 16, -15, 13, 10, 1, 7], [-28, -14, -17, 9, 4, 31, -7], [3, -7, -4, -9, 9, -9, -4], [-14, 8, -6, -5, 4, -17, 6], [27, -21, -3, -7, 11, 1, 12], [26, 2, 12, -7, 6, -28, 11], [-8, -16, 10, -19, -10, 7, -32], [-9, -3, 8, 7, 21, -3, -6], [-4, -14, 13, -1, 20, -7, 25], [-11, 9, 10, -3, 5, -4, 4], [14, 8, 14, -8, -10, -21, 10], [33, -11, 8, -14, -15, -5, 5], [13, 25, 7, 6, -7, -20, 18], [7, 29, 12, -8, 1, -24, 6], [52, 14, 11, 2, -10, -28, 19], [32, -2, -17, 11, 12, 0, 10], [-12, -4, 7, 13, 6, 21, -15], [-15, -19, 9, -3, 13, -7, 8], [-34, -22, -17, 0, 18, 13, -4], [-5, 17, -5, 0, -27, 9, 3], [-2, -4, -1, -8, -30, 20, -14], [16, -28, 2, -3, 10, 17, -21], [31, 31, 8, -2, -21, -9, 5], [-46, 6, -4, 7, 4, 9, -9], [15, 7, -3, 5, -1, -15, 6], [-3, 1, 18, -6, -31, -9, -9], [-41, 13, 1, 0, 5, -1, 16], [23, -3, 8, 2, -25, -9, -19], [1, 7, 11, 2, -17, -23, 1], [-24, 8, -13, 17, 16, 12, -3], [14, -26, -1, -8, 28, -19, 27]];
    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_h();".
// 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_h(:prec:=7)
    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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1045_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![11, -18, -16, 27, 8, -10, -1, 1]>],Snew);
    return Vf;
end function;