
// Make newform 7168.2.a.be in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7168_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_7168_2_a_be();".
// 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_7168_2_a_be();".
// 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 := [-8, 48, -4, -76, 9, 32, -6, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [4, 6, -5, -2, 1, 0, 0, 0], [-20, 32, -8, -15, 20, -2, -4, 1], [0, -18, 9, 17, -6, -3, 1, 0], [-6, 16, -7, -17, 6, 3, -1, 0], [8, 0, -42, -1, 26, -4, -4, 1], [-8, -16, 38, 5, -26, 4, 4, -1], [-24, 20, 62, -13, -30, 6, 4, -1]];
    Rf_basisdens := [1, 2, 4, 2, 2, 4, 4, 4];
    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_7168_a();"
function MakeCharacter_7168_a()
    N := 7168;
    order := 1;
    char_gens := [1023, 5125, 1025];
    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_7168_a_Hecke(Kf)
    return MakeCharacter_7168_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, 1, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, -1, 1, 0, 1], [2, 0, 1, -1, -1, 1, -1, 0], [0, 0, 1, 0, -1, 0, -2, 1], [0, -1, 1, -1, -1, 1, 0, 1], [-1, -1, 2, -1, -3, 2, -1, 1], [1, 2, 1, 0, -2, -1, 1, -1], [1, 1, 0, 0, -3, 1, 1, 0], [2, 1, 0, 0, 2, 0, 0, 2], [-2, -1, 1, -3, -1, 2, -2, 1], [0, 1, 1, -1, 1, -2, -1, 0], [-3, 1, 0, 0, 1, 1, -1, 0], [6, 1, -2, -2, 0, 0, 2, 0], [0, 1, -1, 1, 1, -3, 0, 1], [-1, -2, 1, -2, -2, 2, -2, 2], [-6, 1, 1, 3, -1, 0, -3, -2], [1, 1, 1, 2, 2, -1, -1, -1], [2, 3, 0, -3, 2, -2, 2, -2], [-2, -1, 0, -1, 2, 0, -2, -2], [0, -2, 0, -2, -4, 1, 0, -2], [2, 1, 0, 1, 2, 0, 0, 0], [-2, 0, 1, 0, -3, 0, 2, 3], [2, -2, -3, -1, 3, 3, 1, 0], [5, -1, -2, -4, -1, 3, -1, 4], [1, 2, -2, 1, 3, -5, 2, -3], [0, 1, -2, 2, 2, -4, -2, -2], [-2, 2, 0, 3, 0, 1, 0, -3], [0, -1, -1, 2, -1, 3, 2, 2], [0, 4, 2, 0, -2, -1, 2, -4], [-3, -1, -1, -4, 4, -1, 1, 1], [-6, 3, 3, 1, -1, -5, 2, -5], [9, -4, -1, 2, 0, 3, -1, 1], [-3, 3, 4, 3, -3, 2, -3, -3], [4, -2, -1, -1, -1, 5, -1, 6], [7, -4, 0, 1, -5, 3, -2, 5], [-1, -5, 0, 0, 3, -1, 1, 0], [4, -2, 1, 5, 1, 1, -1, -2], [-6, -3, 1, 7, -5, 0, -3, 2], [1, 4, -1, 2, 2, -6, -2, -2], [11, 0, 3, -3, 2, -1, 1, 3], [-5, -1, -1, 3, 0, 2, -1, -2], [2, -1, 0, 0, -6, 4, -6, 2], [-11, -3, -2, 4, -1, 1, -3, 2], [-6, -3, 1, -7, 3, 2, -5, 2], [-4, 6, 2, -2, 2, -4, 2, -2], [-2, -1, -3, 1, 1, -1, -4, -3], [4, 4, 1, 5, -3, 1, -1, 0], [5, -3, -1, -4, -8, 5, -1, 1], [-4, 3, 1, -2, -3, -1, 2, -4], [4, -6, 1, 4, -5, 10, -2, 3], [8, 4, 2, -4, 2, 1, 0, 0], [-2, 2, -2, 6, -2, -4, -2, -2], [-5, -3, 1, -6, -4, 3, -3, 1], [0, 2, -3, 1, 9, -3, -1, -2], [4, 4, 0, -4, 0, 0, 6, -4], [-5, 4, 1, 6, 4, -3, -3, -5], [-6, 2, 0, -2, 2, 4, -4, -4], [8, -4, -2, 4, 6, -1, 0, -2], [12, -2, 1, -3, -3, -5, 5, 0], [2, 2, -6, 0, -4, 1, 2, 4], [4, 0, -2, 10, -2, 2, -2, 4], [0, -1, 3, -9, -1, 0, 0, -1], [18, 1, 0, -6, -2, 2, 2, 0], [4, -1, 3, 5, -5, 2, -7, 0], [-2, -2, -4, 1, 4, 3, 4, 3], [-4, 5, -7, 5, 5, -4, 1, 0], [7, -8, -3, 6, 2, -4, -4, 2], [-4, 4, 0, 4, -2, 0, 0, -2], [-9, 3, 0, 7, 11, -6, -1, -1], [-4, 0, -4, -2, -4, -2, 0, 2], [1, 0, 5, -2, -2, 1, 3, -3], [-8, 3, 3, -11, -9, 2, -1, -2], [-9, -3, -6, 0, 9, -3, 3, 0], [2, -7, -4, -6, -6, 6, 0, 4], [5, -2, -5, -2, -2, 2, 2, 0], [1, 1, -5, -5, -2, -4, 5, -2], [-2, 3, 7, -1, -7, 4, 0, -3], [-10, 2, 2, 0, 12, -7, 0, -2], [9, 2, 3, 6, -4, 3, -1, -3], [0, -3, -5, 6, 3, -7, 2, -4], [6, -4, 1, 10, 1, -2, -2, 1], [2, -4, -6, 0, 4, -4, -2, 0], [-8, 1, 5, -5, -11, 4, -3, -2], [5, -7, -7, 0, 4, 9, -3, 5], [2, 4, 2, 5, 0, -11, 2, -3], [-7, 4, 3, -2, -2, 0, -4, -6], [3, -2, 5, 7, -4, -1, -7, 5], [-4, -3, 9, -3, -5, 3, -10, 1], [11, 7, 2, -6, 5, -3, 1, 0], [1, 3, 6, -1, -5, -10, 5, -1], [-8, -1, 3, 7, -1, -2, 1, -6], [3, -4, 4, 5, 3, 1, -6, 5], [-4, 6, 4, 0, 0, -10, -4, -6], [11, 2, -3, 6, -6, -2, -2, 0], [6, -3, 1, -3, 3, 8, 2, 7], [14, 5, -3, -1, 9, -3, 2, -1], [-9, -2, 3, -2, -2, 9, -7, 5], [-5, 0, 0, 11, -1, 1, 6, -1], [6, -1, 2, -4, 8, 2, -6, -2], [6, 1, -5, 15, 3, -3, 0, -3], [-12, 8, 6, 11, 2, -5, -4, 1], [-10, -5, -1, -9, 5, 4, 3, 4], [10, 0, 4, -2, 0, 6, 2, 2], [-10, 3, -3, -3, -3, -1, 6, -3], [-10, 4, 2, 0, 6, -8, 2, -2], [0, -6, -6, -4, -14, 10, -4, 4], [16, -3, -2, -9, -6, -2, 2, -2], [-2, 0, 2, 0, -8, -4, 0, 4], [3, -6, -5, -8, 6, 7, -1, 3], [-5, -5, -7, 1, -4, 8, 1, 0], [10, 0, -8, 2, 2, 3, 8, 0], [2, -4, -2, -12, 8, 0, 6, 4], [-5, -5, 5, -1, -6, 10, -1, 8], [-14, -1, -5, -3, 15, -7, -6, 1], [17, 3, -4, 6, 9, -7, 5, 0], [15, 4, 1, 12, 0, -3, 3, -1], [-5, -4, 0, -5, 3, 5, 6, 3], [-14, -4, 1, -1, -13, 9, -5, -2], [-4, 4, 10, 2, -8, 6, -8, -8], [3, 2, -5, 8, 16, -12, 4, 0], [-9, 2, 6, -1, 1, -3, 0, 1], [-6, 6, -8, 4, 2, -1, 6, -2], [7, 8, 1, 4, -4, -1, 3, 5], [-12, -1, -2, 6, 2, -2, -2, -4], [-11, 5, -6, -2, 3, -5, 1, 0], [5, 13, 4, 4, 1, -7, -1, 0], [-21, 0, 3, -4, 0, -6, -2, -8], [-3, 6, 2, 1, -5, 9, 4, 3], [7, 1, 8, -7, -5, 10, -1, -1], [-2, 3, 1, -6, 15, -5, 0, -2], [-13, -2, -3, 8, -8, 5, 3, -1], [10, 3, 5, 9, -1, -2, 0, 1], [-12, -4, 3, 2, -3, -10, -2, -7], [-1, -2, -1, 4, 4, -8, -4, 2], [6, -5, -3, -7, 1, -1, 0, -7], [10, -6, -1, -7, -15, 5, -5, 2], [0, -2, 0, 5, 12, -1, 4, -5], [-8, 4, 10, 4, -6, 5, -4, -2], [9, 2, -1, 0, 4, 1, 11, -1], [-3, 5, -1, -6, 12, -9, 7, 1], [-13, 6, -2, 3, 17, -7, 12, -7], [-11, 2, -1, 2, -2, 0, 0, -6], [-7, -5, 0, -6, 13, 1, 1, 0], [-16, 6, 5, 1, -7, 3, 1, 2], [-16, -1, -1, 21, -1, -6, -8, -7], [-6, -6, -10, 20, 4, -3, -4, 4], [5, 0, -1, -3, -8, 7, 3, -3], [10, 3, 4, 12, 2, -2, 4, -8], [2, -2, 0, -4, 12, -10, -6, 2], [-11, -6, 2, -1, 7, -3, -6, -1], [-3, 9, 6, 4, -9, -3, 11, 2], [0, 5, 9, -7, -3, -2, 7, 6], [-2, -1, -1, -6, -3, 7, -12, 8], [-1, 1, -1, 6, -2, -5, 9, -5], [20, -4, 3, -8, -11, 0, 8, 3], [12, 9, 4, 1, 0, -4, 2, 2], [8, 6, -3, -11, 5, 3, 1, 4], [5, -10, 2, 11, -5, 3, -4, 5], [-4, -8, -12, 4, 12, 4, -4, 8], [-17, 1, -6, -7, 5, -6, -1, 1], [6, 1, -3, -1, -11, -1, 4, 3], [-11, 1, -1, -8, -6, 1, -3, 3], [9, 1, -10, -4, 7, -3, 5, 6], [-8, -17, 2, -5, -10, 16, -4, 6], [-1, 2, 3, -8, 0, 2, 6, -2], [1, 9, 5, -7, -8, 2, 1, 2], [10, 8, 3, -3, 1, -5, 9, -8], [-6, 13, 5, -1, -1, -8, -1, -8], [-29, 2, 5, 6, 2, -8, 4, -8], [-3, -5, -9, -2, 4, 1, -9, 11], [0, 8, -6, 8, 14, -4, 6, -8], [-13, 3, 4, -16, -1, 7, 5, -4], [4, 3, -14, 17, 14, -2, 4, 2], [-1, 0, 8, 13, -9, 1, -4, 3], [-5, 4, 11, -10, -10, -9, -5, -1], [10, -4, 2, -8, -16, 20, -4, 8], [-20, -4, -1, -1, -1, 11, -3, 2], [-12, -1, 1, 2, 5, -9, -6, -12], [-14, 3, 7, -3, -9, 5, 0, -3], [4, -1, 0, -10, 8, 10, -2, -2], [-2, -7, -8, 7, 6, 6, 4, 6], [-4, -1, -1, 18, 3, -1, 2, -6], [16, 0, 1, 9, 1, -1, 1, 4], [-8, 1, -10, 4, 18, -16, 0, 2], [2, -9, -9, 9, 3, -3, -2, 7], [1, -1, 3, 0, -4, -5, 9, -7], [-9, 7, 6, -6, 1, -3, -3, -12], [-8, 4, 6, -4, -4, 4, 2, 0], [-7, 8, 8, 9, 5, -11, 4, -9], [15, 0, 4, -13, -9, 11, -2, 3], [8, 6, 9, -15, -11, 19, -3, 6], [13, 2, 10, -1, -9, 9, -8, 1], [-8, 1, -10, 19, 2, 2, 0, -2], [0, 8, -10, 13, 16, -7, 6, -3], [-1, 4, 1, -12, 4, 3, 3, 5], [-4, 0, 4, -4, -10, -4, 0, -6], [-8, -2, 2, 14, -6, 8, 10, -2], [-2, 5, -4, 6, 10, 0, 2, 2], [9, -1, 0, 0, -7, -5, -9, -6], [13, 4, 9, -8, -12, 2, 6, 4], [-4, -2, 2, 2, -12, 8, -12, 4], [2, -13, -7, -15, 1, 1, 4, -1], [10, 0, 3, -3, -3, 1, 3, 2], [5, 7, 12, 0, 5, -1, -1, -2], [4, 1, -3, -7, -3, 4, 3, 2], [-7, -1, 3, 19, -8, 2, -1, -6], [10, 2, 2, -4, 4, -1, -6, -4], [5, -3, -5, -1, 2, 4, 5, -2], [-11, 8, 17, 4, -12, 2, -2, -4], [-9, 5, -2, -1, 17, 0, 7, -5], [-2, -2, 6, 8, 8, 1, 0, -2], [-10, -12, 8, 8, 2, 12, -18, 6], [-6, 7, -11, 9, 11, -2, 8, -3], [17, 4, 1, 13, 6, -9, -1, -5], [0, -10, 1, -8, -9, 10, -2, -5], [2, -14, -14, 8, 12, -2, -6, 6], [-1, 6, 5, -6, 4, -11, -5, -9], [-15, -4, 5, -12, -6, 15, -3, 3], [-14, -4, -12, -8, -2, 0, 2, -6], [-12, 4, 4, -6, -6, -2, -4, 2], [4, 7, 5, 6, -7, -1, -6, -8], [-4, 1, 5, -11, -1, 13, -6, -3], [-19, -2, -1, 4, 12, -1, -11, -3], [22, -7, -10, 5, 20, 0, 6, 6], [-4, -11, -1, 6, -5, 17, -18, 4], [-12, 0, 8, -4, -4, -8, -6, -4], [5, -6, -6, -3, 11, 9, 8, 11], [-3, -4, 9, 2, -22, 7, -3, -1], [26, 2, -10, 12, 12, -5, 4, -2], [9, 8, -7, 17, -2, -5, 7, -9], [0, -6, -6, 6, -10, 8, -2, 4], [-18, 4, -6, 10, 8, 5, 0, 2], [16, -4, 0, -20, 4, 0, 10, 0], [-14, -2, -5, -8, -17, 10, -6, 3], [-3, 2, -3, 12, -2, -5, -5, 3], [-26, 9, 1, 11, 3, -4, -7, 2], [-16, 8, 6, 0, -14, -8, 4, -14], [-26, 3, -5, 9, 5, -10, -3, 4], [-8, -5, -11, -13, -1, 3, 6, -5], [10, 0, -10, 26, 8, -14, 0, -2], [-8, 3, 6, -8, -10, 6, 4, 8], [-12, -8, 0, -12, -14, 12, -2, 2], [-7, -3, 2, -10, -9, 11, -5, 0], [8, 10, -12, 12, 24, -26, 10, -2], [32, -12, -6, 0, 6, -3, -6, 4], [3, -10, 10, -9, -15, 7, -4, 13], [36, -3, -6, 1, -6, 12, 0, 14], [-15, -2, -11, 8, 14, -7, -1, -3], [8, 8, 10, 0, -8, 4, -2, -12], [0, 4, 2, -28, -14, 12, 0, 6], [22, -11, 2, 9, -12, 16, 2, 10], [7, 2, -1, 8, 4, -4, 16, -6], [-20, 7, 7, 3, -9, -18, -3, -14], [-25, 0, -5, -2, 18, -17, 11, -5], [6, 7, 3, 13, -5, -7, 2, -5], [19, 2, 1, 10, 2, 4, -4, -6], [-10, 9, -1, -3, -11, -4, -8, -3], [2, 13, 15, 2, 1, -11, 8, 2], [-16, -9, 3, -11, -9, 14, -5, 2], [-9, 2, -7, 6, 6, 6, 6, 6], [-7, 2, 5, -12, -8, 4, -12, -2], [-14, 0, 5, -12, 13, 0, 0, -13], [2, -3, -5, -13, 7, -7, -4, -7], [-9, -8, 5, -6, -4, 9, -15, -5], [10, 5, 0, 5, 6, -4, -2, -4], [11, 4, 12, -1, -9, -1, -2, -9], [0, 3, 6, 0, -10, 2, -10, 0], [-6, 14, -3, -1, 7, -3, 11, -18], [-16, 1, 13, 3, 5, 2, -7, -2], [2, 3, -6, 7, 8, -4, -4, -10], [8, 1, -3, -10, 5, 5, 2, 2], [-10, 8, 0, -8, 0, 4, 14, 8], [-6, 2, 4, 12, 18, -2, -2, 4], [31, 4, -8, -5, 3, -7, 6, -1], [-2, 13, 2, -10, -20, 0, -2, 0], [1, -5, -13, -9, 12, 2, -5, 2], [16, 12, -4, -4, 4, -19, 8, -6], [4, 2, 14, -4, -2, -2, 8, 0], [5, 5, 15, 6, 12, -5, -1, -3], [11, -9, 12, -18, -13, 13, -1, 14], [-14, 6, -3, -9, -5, 5, 3, 4], [-20, -8, 2, -8, 2, 11, -6, 6], [18, -9, -1, -2, 1, 3, 8, -8], [-6, -14, -2, 2, -10, 12, -2, -4], [20, 13, -6, -4, 6, -14, 8, -4], [21, 9, 0, 30, 9, -13, 5, -6], [18, 0, -4, -10, 0, -2, 2, 2], [10, 5, 2, 6, -20, 2, 6, 2], [24, -4, -6, 0, -10, 9, -2, 16], [-1, -5, -11, 11, -4, -6, 5, -2], [28, 0, -4, 0, 12, -5, -2, 12], [-27, 10, -1, -10, -8, -7, -9, -9], [-3, -14, -1, 6, 2, 8, -8, 0], [-4, 0, 20, -12, -24, 16, -14, 8], [0, -18, -9, -21, -5, 9, 3, 2], [-4, -4, 4, -4, 8, -3, -20, -2], [-8, -6, 4, -18, -16, 1, 0, 0], [-16, 5, -1, -5, 3, 2, 0, 1], [26, 5, 16, -6, -6, 8, -4, 10], [23, 14, 13, -5, 0, -13, 1, -11], [16, 11, 3, -35, 3, -2, 17, -4], [1, -16, -4, 5, -3, 11, -12, 7], [-5, 11, 13, 0, -14, 3, -1, -3], [4, -1, 15, -9, -13, 8, -19, 2], [-5, 0, 3, 16, 16, -20, -12, -4], [-23, -5, -10, 15, 3, 10, 5, 11], [-3, -10, -7, 6, 6, -3, -1, 5], [0, 10, -4, 4, 20, -10, 4, 6], [22, -1, 2, 14, -8, 2, 12, 2], [14, 2, 4, -18, 8, 0, -10, 10], [-14, 11, -9, 27, 17, -8, 7, -4], [12, -6, -4, 0, 0, -14, -4, 2], [-10, -19, -14, -7, 8, 8, 10, 10], [-2, 9, -7, 15, 33, -17, 8, -9], [17, 6, -11, 7, 10, -13, 3, 11], [0, 0, 2, -8, 8, 0, -8, 12], [5, -5, -13, -10, 22, -7, -5, 1], [6, -12, -12, -10, 14, 9, -6, 20], [-12, -6, -22, 26, 26, -24, 4, -10], [10, 18, 1, -9, 15, -19, 11, -4], [-22, 7, -8, 11, 6, -8, -2, 2], [2, 4, 2, 17, -4, 9, -6, -3], [12, -2, -3, 10, -1, -4, -2, 9], [9, 14, 14, 5, 7, -15, 2, 1], [20, 2, -10, -10, -6, 7, 6, 14], [4, 19, -1, -14, -5, -13, 14, -10], [27, -8, -7, 4, 16, -1, -5, 5], [-24, 3, 2, -10, 2, 6, 4, 8], [5, -6, 13, -26, -6, -3, -7, 5], [27, -11, -10, 16, 13, 17, 3, 4], [26, 1, -3, 7, 5, -15, 6, -5], [13, -2, -14, -21, 3, -3, 6, -1], [18, -7, 9, -9, -3, 7, 8, 7], [30, 1, 8, -18, -26, 20, 4, 6], [2, -14, 2, -24, -26, 18, -6, 12], [-18, 11, -12, -15, -10, -10, 4, -6], [-7, -3, 14, -14, -1, -5, 7, -4], [4, 3, 14, 0, 14, -16, 8, -6], [52, -8, 6, 3, -6, -5, 8, 9], [1, 8, -5, 2, 22, -20, 0, -10], [8, -8, -10, -8, -10, 8, 8, -2], [19, 0, -3, 18, 28, -9, 7, -3], [-14, 21, 5, -11, -1, 0, 15, 2], [-21, 8, -5, -8, 20, -24, 4, -10], [8, -11, -11, 3, 13, -2, -7, 6], [16, 14, -2, 18, 14, -28, 2, -18], [27, 8, -9, -4, 4, -6, 6, 8], [24, 10, 4, -10, -4, 17, -2, 16], [-26, -6, 6, 0, -16, 2, -10, -10], [12, -9, -6, -20, -6, 4, 8, 14], [-15, 21, 0, -10, -7, -13, -5, -6], [10, -7, 8, 6, -30, 12, -6, 6], [-12, 1, 6, 11, 18, -18, 10, -10], [-23, -11, 10, 0, -21, 13, -11, -6], [-2, 7, -7, 1, 17, 9, -6, -5], [25, 1, -11, -12, 0, -3, -7, 1], [-29, 1, 2, -3, 1, 18, -9, 5], [-11, -6, 5, -16, -10, 7, -1, 15], [-22, 11, 0, 5, 6, 2, -4, -14], [2, -5, -9, 14, 25, -13, -4, 0], [12, -21, -9, -1, -13, 14, -17, 4], [6, -17, -3, 5, 1, 17, -12, 1], [-37, -1, 9, 15, -8, 2, -23, -2], [42, 2, 3, 21, -3, -7, 7, 2], [-21, 3, 12, 1, -5, 4, -5, 1], [-16, 2, 0, -11, -8, -13, 4, -17], [7, -4, -8, 3, 3, -1, 14, 3], [18, 5, 1, -5, 13, -3, 8, -11], [-4, -5, -6, -13, 10, -4, -4, -2], [9, 12, -15, 6, 18, -27, 15, -11], [-42, -16, -14, 4, -12, 12, 4, 12], [35, 8, 3, 2, -6, 1, 13, 1], [12, -17, -1, -17, -7, 3, 4, 19], [6, 13, 3, 18, 9, -15, 4, -26], [0, 6, -14, -6, -12, -12, 10, -4], [-2, 4, 5, 0, 29, -4, -2, -5], [-10, 23, -9, 11, 13, 0, -2, -7], [-3, -12, -1, -18, -10, 12, 0, -6], [-6, 20, -10, 11, 20, -21, 2, -1], [13, 7, 1, -2, -10, -11, 11, -7], [23, 1, 9, -22, -2, 19, 1, 7], [0, 21, 9, 5, -9, -5, 6, -9], [16, -8, -6, 14, -10, 10, -8, 4], [-2, -9, 8, -1, -6, 24, -6, 22], [23, 2, -21, -16, 8, -12, 24, 2], [8, -3, -3, 15, 29, -20, -1, 0], [1, -11, 0, 16, -15, -11, 9, 0], [0, -4, -2, -1, 4, -9, 6, 11], [47, -2, 3, -12, 8, 6, 6, -6], [-3, 8, 8, 19, 5, 7, -2, -5], [10, 18, 2, 4, -4, -11, 6, -6], [-23, -1, 6, 37, -5, 4, -15, -3], [3, 1, 5, 12, 0, 1, 7, -9], [11, 11, 8, -24, 3, -13, 3, -12], [24, 3, -1, -7, -9, 14, 6, 17], [-33, 10, -2, -3, 1, 3, 10, -15], [-11, 8, -3, 8, -2, 7, 1, -5], [10, -12, 5, 11, -25, 3, 1, 6], [-2, 26, 10, -10, -2, -16, 18, -8], [15, 6, 13, 15, -8, 11, -3, 13], [16, -11, -15, 11, 5, 18, -12, -1], [14, 14, 3, 5, 17, -21, 1, -10], [9, 11, 2, -12, 11, -21, 11, -4], [-3, -18, -15, -2, 2, 11, 7, 11], [25, 9, 17, 7, 16, -8, -3, 8], [32, -8, -10, -12, 26, 1, 10, 6], [-12, 11, -5, 3, 13, -17, -4, -3], [-12, -10, -8, -2, -14, 20, -6, -2], [1, -14, -17, 16, -4, 13, -15, 3], [-10, 7, -1, -19, 5, -14, 11, 2], [-19, -10, -14, 11, 7, 5, -16, 5], [-9, 19, -11, -12, 16, -21, 17, -9], [3, -18, -3, -14, -30, 16, -4, 4], [12, 1, -16, -11, 0, 4, -4, 12], [-27, 5, 2, 4, 11, -7, -1, 2], [-24, -8, 3, -10, -11, 18, -16, 5], [-14, -6, -6, -14, -4, 16, -14, 4], [20, -3, -6, 14, 6, -10, 6, 0], [31, -10, 1, 10, -10, 20, -12, 2], [-10, 7, 12, 29, 6, 2, -10, -4], [6, -6, 14, -4, -24, 19, 0, 16], [8, 3, -9, -21, -1, -12, 14, -1], [31, -12, -3, 6, -18, 10, -18, 8], [2, -7, 9, -37, -29, 30, -5, 20], [8, 0, -6, 30, 10, -14, -6, -6], [34, -10, 2, 0, -20, 13, -2, 0], [41, -11, -6, -2, -5, 19, 5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7168_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7168_2_a_be();".
// 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_7168_2_a_be(:prec:=8)
    chi := MakeCharacter_7168_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_7168_2_a_be();".
// 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_7168_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7168_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -32, -152, 24, 94, -4, -18, 0, 1]>,<5,R![-128, -128, 512, -176, -162, 84, 6, -8, 1]>,<11,R![64, 128, -192, -640, -492, -88, 32, 12, 1]>,<13,R![4096, 0, -6784, 4960, -882, -284, 138, -20, 1]>,<17,R![116608, 15872, -46272, -4992, 3528, 272, -100, -4, 1]>,<23,R![-70592, -56064, 12800, 15680, 1428, -688, -84, 8, 1]>,<31,R![-251648, 366848, -59584, -25792, 5304, 576, -132, -4, 1]>],Snew);
    return Vf;
end function;