
// Make newform 4608.2.k.be in Magma, downloaded from the LMFDB on 15 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4608_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4608_k_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_4608_2_k_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_4608_2_k_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 := [1, 0, 0, 0, 0, 0, 0, 0, 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, 2, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0], [0, 1, 0, 1, 0, 1, 0, 1], [0, 1, 0, -1, 0, -1, 0, -1], [0, -1, 0, -1, 0, 1, 0, -1]];
    Rf_basisdens := [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_4608_k();"
function MakeCharacter_4608_k()
    N := 4608;
    order := 4;
    char_gens := [3583, 2053, 4097];
    v := [4, 3, 4];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4608_k_Hecke();"
function MakeCharacter_4608_k_Hecke(Kf)
    N := 4608;
    order := 4;
    char_gens := [3583, 2053, 4097];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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], [-1, 0, 0, -1, 0, 0, 0, 1], [0, 0, -1, 2, 0, 0, 0, -1], [0, 0, 1, 0, 0, 1, 1, 0], [-1, 0, 0, 1, -1, 0, 2, 0], [0, 1, -1, 0, -1, -1, -2, -1], [2, 0, 1, -2, 0, -1, 1, 0], [0, 0, 0, 4, 0, 0, 0, 0], [-1, 0, -1, 1, 0, 1, -2, 0], [2, 0, -2, 0, 0, 1, -1, 1], [-1, 1, -2, -1, 0, -2, -2, -4], [0, -1, 1, 0, -1, -3, 0, 1], [2, 0, 3, 2, 0, 3, 3, 4], [4, -2, 0, 0, 2, 0, 0, 0], [1, 0, 2, 1, 0, 2, 2, 5], [4, 2, 2, 4, 0, 2, 2, 4], [-1, 0, 2, 1, 3, -2, 2, 0], [-4, 0, 4, 4, 2, -4, 0, 0], [0, -2, -2, -4, -2, 0, 0, -2], [0, 1, 0, 2, 1, 4, 0, 0], [-6, -2, 0, 0, 2, 3, 3, 3], [-4, 0, 1, 4, 0, -1, 1, 0], [0, 2, 0, 2, 2, -4, 0, 0], [-4, -2, 2, 0, 2, 2, 4, 2], [5, 2, 0, 5, 0, 0, 0, 1], [0, -2, -1, 6, -2, -6, 0, -1], [-8, -2, 2, -8, 0, 2, 2, 4], [5, 0, 0, -5, -3, 0, 2, 0], [-2, 2, 0, 0, -2, 0, 0, 0], [10, 0, 2, 0, 0, 5, 7, 5], [0, 0, -2, 0, -2, 2, 6, 0], [0, -1, 3, 0, -1, 3, 0, 3], [8, -2, 2, 8, 0, 2, 2, 0], [-5, -4, 0, -5, 0, 0, 0, 1], [0, 2, -3, -6, 2, -6, 0, -3], [3, 0, 2, -3, 1, -2, -2, 0], [-2, 0, 5, 2, -4, -5, 5, 0], [0, 4, -2, -8, 4, -4, 0, -2], [1, 0, -1, -1, -6, 1, -2, 0], [-4, 0, 4, 4, -2, -4, 0, 0], [1, -3, 2, 1, 0, 2, 2, 2], [-8, 2, 4, 0, -2, 6, 10, 6], [4, -3, 0, 0, 3, -4, -4, -4], [-9, 4, 2, -9, 0, 2, 2, -1], [0, 6, 1, 2, 6, -2, 0, 1], [0, 0, 2, 0, -2, -2, 2, 0], [2, 2, -4, 0, -2, 7, 3, 7], [4, 0, 3, -4, -4, -3, -5, 0], [-3, 1, 2, -3, 0, 2, 2, 6], [0, -4, -4, -2, -4, 0, 0, -4], [0, -2, -4, 0, 2, 6, 2, 6], [-12, 3, -6, 0, -3, -2, -8, -2], [0, -4, 1, 0, 0, 1, 1, 0], [6, -2, 0, 0, 2, 4, 4, 4], [0, -2, 8, -8, -2, -4, 0, 8], [-5, 0, 5, 5, -6, -5, 8, 0], [6, -6, 0, 0, 6, 1, 1, 1], [-5, -1, -2, -5, 0, -2, -2, 6], [0, 0, -4, -2, 0, 8, 0, -4], [4, -2, 2, 4, 0, 2, 2, 4], [7, 8, -2, 7, 0, -2, -2, -5], [4, 0, 8, -4, 2, -8, 4, 0], [0, 2, -4, 0, 2, 4, 0, -4], [0, 1, 4, 4, 1, 0, 0, 4], [-1, 0, -7, 1, -2, 7, 8, 0], [0, 2, 2, 0, 0, 2, 2, 8], [6, 3, 0, 0, -3, -12, -12, -12], [-4, 0, 3, -4, 0, 3, 3, 8], [-3, 0, -6, 3, 1, 6, 2, 0], [10, -8, 4, 0, 8, 0, 4, 0], [0, 6, 2, 4, 6, -4, 0, 2], [6, 4, 2, 0, -4, -1, 1, -1], [7, -1, -2, 7, 0, -2, -2, 0], [-2, 8, 3, -2, 0, 3, 3, 4], [8, -2, -8, 0, 2, -10, -18, -10], [-3, 10, -6, -3, 0, -6, -6, -5], [-3, 0, 6, 3, -5, -6, -2, 0], [-8, 3, 1, 0, -3, 1, 2, 1], [0, 4, 6, 4, 4, -2, 0, 6], [4, 0, 1, -4, 4, -1, 9, 0], [5, 5, 6, 5, 0, 6, 6, 10], [-4, -10, 0, 0, 10, -4, -4, -4], [4, -6, -4, 0, 6, -4, -8, -4], [0, -2, 7, 14, -2, 2, 0, 7], [-4, 4, -1, -4, 0, -1, -1, -8], [-8, 1, 5, 0, -1, 9, 14, 9], [0, -2, -10, 4, -2, -2, 0, -10], [-13, 0, 1, 13, 2, -1, -6, 0], [-2, -4, -2, 0, 4, 7, 5, 7], [12, 0, 1, -12, 4, -1, 1, 0], [12, -4, 4, 0, 4, 2, 6, 2], [0, -4, 5, -10, -4, 4, 0, 5], [4, -2, -4, 4, 0, -4, -4, 8], [-4, 0, -10, 4, -6, 10, -6, 0], [0, 6, 2, 4, 6, 4, 0, 2], [5, 0, 5, -5, 4, -5, -6, 0], [0, -7, -7, 0, -7, -3, 0, -7], [-10, 8, 3, -10, 0, 3, 3, 0], [-1, 0, 4, 1, 13, -4, 2, 0], [14, 0, 5, -14, -4, -5, -7, 0], [17, 0, -7, -17, -4, 7, 0, 0], [16, 0, -3, -16, -4, 3, 5, 0], [0, 1, -5, -16, 1, -9, 0, -5], [12, -2, -2, 12, 0, -2, -2, 4], [-10, 5, 8, 0, -5, 4, 12, 4], [4, -2, 2, 4, 0, 2, 2, -8], [14, 2, 4, 0, -2, -12, -8, -12], [0, -4, -4, -20, -4, 4, 0, -4], [0, 5, 0, -2, 5, 4, 0, 0], [-14, 0, 10, 0, 0, -3, 7, -3], [9, 1, -2, 9, 0, -2, -2, -8], [0, -2, 8, -6, -2, 0, 0, 8], [0, 6, -12, 0, 0, -12, -12, -16], [0, 0, -1, 10, 0, 4, 0, -1], [0, -3, 1, 0, 3, -7, -6, -7], [10, 0, -1, -10, 4, 1, -1, 0], [0, -4, 12, 4, -4, 4, 0, 12], [-21, 0, -3, 21, -10, 3, -6, 0], [4, 0, 0, -4, -10, 0, -8, 0], [-19, 1, 2, -19, 0, 2, 2, 4], [12, -2, 0, 0, 2, 0, 0, 0], [5, 0, 16, 5, 0, 16, 16, 19], [12, 8, -1, 12, 0, -1, -1, -8], [-14, 0, -5, 14, 8, 5, 3, 0], [-3, 0, 3, 3, 6, -3, -4, 0], [-17, -1, 2, -17, 0, 2, 2, -6], [-28, 4, 4, 0, -4, 2, 6, 2], [0, 6, 7, -2, 6, -2, 0, 7], [-3, 0, 0, 3, -5, 0, 6, 0], [-4, 0, -10, 4, 10, 10, 2, 0], [0, 4, -6, -8, 4, 8, 0, -6], [-2, 6, -4, 0, -6, -11, -15, -11], [9, -3, -6, 9, 0, -6, -6, -2], [0, -3, 1, 8, -3, -11, 0, 1], [4, -5, -4, 0, 5, 8, 4, 8], [5, -2, -8, 5, 0, -8, -8, -11], [-6, 0, -11, 6, 0, 11, -7, 0], [5, 0, -7, -5, 4, 7, -8, 0], [0, 1, -5, 8, 1, -1, 0, -5], [-10, -4, -9, -10, 0, -9, -9, -4], [9, 2, 4, 9, 0, 4, 4, 13], [0, -4, 3, -14, -4, 0, 0, 3], [-8, 10, -4, -8, 0, -4, -4, -4], [-5, 0, -8, 5, 11, 8, 6, 0], [0, -2, 10, 4, -2, -12, 0, 10], [7, 11, -2, 7, 0, -2, -2, -12], [0, -1, -1, 24, -1, -1, 0, -1], [-2, 8, -3, -2, 0, -3, -3, -8], [-16, 2, 12, 0, -2, 2, 14, 2], [1, 0, 10, -1, 5, -10, 10, 0], [-22, -6, -4, 0, 6, -8, -12, -8], [-10, 0, -7, 10, -12, 7, -3, 0], [0, 2, -16, 0, 2, 8, 0, -16], [-14, 4, -1, -14, 0, -1, -1, 0], [8, 6, 8, 0, -6, -2, 6, -2], [0, 4, 7, 2, 4, 8, 0, 7], [-16, 3, -1, 0, -3, -13, -14, -13], [0, 0, -16, 2, 0, 0, 0, -16], [19, 0, -3, -19, 2, 3, 6, 0], [-12, 0, -4, 12, -14, 4, -8, 0], [0, -5, -5, 0, -5, 15, 0, -5], [0, 8, 11, 10, 8, 0, 0, 11], [-4, 12, -3, -4, 0, -3, -3, 8], [0, 1, -19, 0, -1, 9, -10, 9], [0, -2, 12, 8, -2, 12, 0, 12], [-6, 2, -4, 0, -2, -1, -5, -1], [-1, 1, -14, -1, 0, -14, -14, 0], [-4, -4, -14, 0, 4, -6, -20, -6], [19, -2, -6, 19, 0, -6, -6, 1], [4, -6, 8, 4, 0, 8, 8, 0], [1, 0, 10, -1, -1, -10, 6, 0], [0, 0, -6, -32, 0, 8, 0, -6], [0, 0, 8, -2, 0, 0, 0, 8], [22, 2, 8, 0, -2, -7, 1, -7], [0, -4, -8, -2, -4, -4, 0, -8], [4, -6, -10, 4, 0, -10, -10, -12], [13, -4, -12, 13, 0, -12, -12, -17], [0, -2, -3, 26, -2, 2, 0, -3], [11, 0, -10, -11, -1, 10, 10, 0], [14, -2, 12, 0, 2, -3, 9, -3], [12, 0, -5, -12, -12, 5, -5, 0], [-15, -17, 2, -15, 0, 2, 2, 14], [0, 0, 12, -14, 0, 0, 0, 12], [8, 2, -16, 0, -2, -2, -18, -2], [-13, -2, -2, -13, 0, -2, -2, -15], [-9, 0, 12, 9, 9, -12, -10, 0], [12, 0, -10, -12, 2, 10, -6, 0], [0, -3, 6, 20, -3, 10, 0, 6], [8, -4, -4, 0, 4, -16, -20, -16], [6, 10, 0, 0, -10, 0, 0, 0], [-12, 2, 10, -12, 0, 10, 10, 12], [4, 0, -6, -4, 14, 6, 6, 0], [-1, 0, 9, 1, -6, -9, 6, 0], [0, 0, 2, 0, 18, -2, -6, 0], [0, -2, -12, -22, -2, 4, 0, -12], [10, 11, 0, 0, -11, -4, -4, -4], [-1, 0, -16, 1, -7, 16, -6, 0], [-26, 0, -12, 0, 0, 8, -4, 8], [0, 2, -8, 8, 2, 4, 0, -8], [-23, 0, 9, 23, 0, -9, 4, 0], [-22, 0, -2, 0, 0, 1, -1, 1], [-17, 13, -6, -17, 0, -6, -6, -2], [-6, 9, -8, 0, -9, -4, -12, -4], [-16, -8, -3, -16, 0, -3, -3, -8], [-1, 0, -5, 1, -16, 5, 14, 0], [-6, -6, 4, 0, 6, -17, -13, -17], [8, 0, -17, -8, -4, 17, 7, 0], [0, 10, -8, -10, 10, 8, 0, -8], [-2, 16, -3, -2, 0, -3, -3, 4], [-20, -7, 4, 0, 7, 0, 4, 0], [-21, 8, -8, -21, 0, -8, -8, -3], [0, -2, -1, -18, -2, 6, 0, -1], [12, 6, 2, 12, 0, 2, 2, -20], [0, 2, -10, 28, 2, 8, 0, -10], [0, 0, -4, -4, 0, 4, 0, -4], [-2, -8, -2, 0, 8, -5, -7, -5], [-8, 1, -1, 0, -1, 19, 18, 19], [0, -4, 10, 8, -4, -4, 0, 10], [-1, 0, 7, 1, -4, -7, 12, 0], [1, -9, 6, 1, 0, 6, 6, 10], [0, 4, 7, 2, 4, 4, 0, 7], [8, 7, -1, 0, -7, -1, -2, -1], [-46, 6, -4, 0, -6, -1, -5, -1], [-4, 0, 1, 4, -12, -1, 17, 0], [-1, -5, 18, -1, 0, 18, 18, 28], [0, 0, -8, -22, 0, -8, 0, -8], [28, -4, 12, 0, 4, -6, 6, -6], [0, 8, 15, 10, 8, -8, 0, 15], [-12, 22, -4, -12, 0, -4, -4, -8], [-3, 0, -20, 3, 5, 20, -10, 0], [18, 0, -5, -18, 16, 5, -9, 0], [34, 4, -6, 0, -4, 1, -5, 1], [0, 8, 0, 2, 8, 12, 0, 0], [-10, -12, 5, -10, 0, 5, 5, 12], [4, -8, 16, 0, 8, 14, 30, 14], [-12, -6, -14, 0, 6, 2, -12, 2], [-13, 8, -16, -13, 0, -16, -16, -15], [28, 0, -7, 28, 0, -7, -7, 0], [0, -8, 0, -36, -8, -4, 0, 0], [8, 0, 5, -8, 8, -5, -19, 0], [12, 2, -12, 12, 0, -12, -12, -12], [0, -2, 5, 18, -2, 18, 0, 5], [-9, 0, 2, 9, -15, -2, -2, 0], [-30, -10, 0, 0, 10, 0, 0, 0], [0, 2, 14, -4, 2, 4, 0, 14], [2, 2, -4, 0, -2, 15, 11, 15], [0, 0, -10, 0, 14, 10, 10, 0], [4, -2, 0, 4, 0, 0, 0, 12], [40, 8, 4, 0, -8, 4, 8, 4], [-15, 0, -8, 15, -19, 8, -14, 0], [-14, 6, -16, 0, -6, 0, -16, 0], [0, 12, 8, -4, 12, 12, 0, 8], [0, -6, 8, 10, -6, 8, 0, 8], [-21, 0, 7, 21, 12, -7, 6, 0], [-32, 0, 0, 32, -2, 0, -12, 0], [5, -5, 2, 5, 0, 2, 2, 2], [16, 18, 0, 16, 0, 0, 0, -8], [29, -12, 14, 29, 0, 14, 14, 9], [0, -10, -12, 12, -10, 4, 0, -12], [22, 0, 6, 0, 0, 7, 13, 7], [4, 0, 14, -4, -6, -14, -6, 0], [3, 5, 18, 3, 0, 18, 18, 22], [9, 0, -2, -9, 3, 2, 18, 0], [-16, -3, -1, 0, 3, 11, 10, 11], [-4, 0, 16, 4, -14, -16, -4, 0], [3, 0, 11, -3, 4, -11, 12, 0], [0, 2, 4, -6, 2, -12, 0, 4], [-6, -12, 13, -6, 0, 13, 13, 12], [-11, 4, -4, -11, 0, -4, -4, -1], [-21, 0, -14, 21, 7, 14, 10, 0], [-8, 0, 4, 8, -18, -4, -12, 0], [0, -6, 16, 10, -6, 0, 0, 16], [14, 4, 2, 0, -4, 15, 17, 15], [-10, 8, 24, 0, -8, -8, 16, -8], [0, 0, -1, -6, 0, -8, 0, -1], [8, -4, -13, 8, 0, -13, -13, 8], [-5, 0, -6, 5, -3, 6, 2, 0], [0, 15, 0, 2, 15, 4, 0, 0], [0, 0, 4, 0, -6, -4, 0, 0], [-4, 0, -4, 0, 0, -6, -10, -6], [0, 6, -19, 18, 6, 2, 0, -19], [0, 4, 16, 12, 4, -8, 0, 16], [-15, 5, -10, -15, 0, -10, -10, 0], [-10, 8, 19, -10, 0, 19, 19, 12], [24, 0, 8, 0, 0, 8, 16, 8], [4, -5, -18, 0, 5, 10, -8, 10], [-7, 4, 12, -7, 0, 12, 12, -13], [0, -8, 5, 6, -8, -8, 0, 5], [26, 8, 8, 0, -8, -20, -12, -20], [11, 0, -3, -11, -6, 3, 10, 0], [4, 0, -4, -4, 10, 4, -8, 0], [0, -8, 4, 10, -8, -12, 0, 4], [4, -6, -16, 4, 0, -16, -16, 4], [-23, 0, 2, 23, 21, -2, -2, 0], [-27, 0, 7, 27, -6, -7, 6, 0], [14, -6, -16, 0, 6, -3, -19, -3], [19, -10, -4, 19, 0, -4, -4, 7], [0, 16, 1, 0, 0, 1, 1, 8], [-34, 0, 3, 34, -8, -3, 7, 0], [0, -11, -8, -4, -11, 12, 0, -8], [-23, 0, -21, 23, 14, 21, 6, 0], [14, 10, 0, 0, -10, 13, 13, 13], [-8, 0, -12, 8, 6, 12, 28, 0], [-26, -8, -3, -26, 0, -3, -3, 12], [28, 13, 6, 0, -13, -14, -8, -14], [0, -12, 19, 0, 0, 19, 19, 8], [3, 0, -4, -3, 17, 4, 22, 0], [0, -2, -10, -28, -2, -4, 0, -10], [11, 15, -6, 11, 0, -6, -6, -24], [16, -14, 12, 0, 14, -2, 10, -2], [-13, -14, -4, -13, 0, -4, -4, -1], [0, 13, 1, 0, -13, -7, -6, -7], [18, 0, 9, -18, -12, -9, -3, 0], [0, -2, 0, -8, -2, -24, 0, 0], [0, 10, -8, -14, 10, 0, 0, -8], [-28, 0, 2, 28, -14, -2, -14, 0], [-12, -4, -24, 0, 4, 2, -22, 2], [-22, 1, 0, 0, -1, 20, 20, 20], [-26, -10, -8, 0, 10, 12, 4, 12], [-20, 0, -6, 20, -6, 6, 14, 0], [0, 3, -16, 20, 3, 12, 0, -16], [-9, 0, -17, 9, -4, 17, -10, 0], [10, -4, -2, 0, 4, 5, 3, 5], [0, 5, -7, 24, 5, -7, 0, -7], [12, -12, -2, 0, 12, 14, 12, 14], [-30, 0, 13, 30, 0, -13, 1, 0], [22, 0, -11, 22, 0, -11, -11, -12], [-16, 4, 8, 0, -4, 12, 20, 12], [-1, 2, -6, -1, 0, -6, -6, -3], [7, 0, -8, -7, -1, 8, 22, 0], [-1, 0, -5, 1, 4, 5, -16, 0], [-14, 2, 4, 0, -2, 11, 15, 11], [-32, 0, 10, 32, 14, -10, -10, 0], [8, -10, 6, 8, 0, 6, 6, 24], [-8, 12, 9, -8, 0, 9, 9, 16], [1, 0, -18, -1, 11, 18, -6, 0], [2, 8, -16, 0, -8, -24, -40, -24], [0, -9, 0, -2, -9, 20, 0, 0], [-14, -10, 0, 0, 10, 27, 27, 27], [-11, 17, 22, -11, 0, 22, 22, 20], [0, -3, 27, -8, -3, 19, 0, 27], [-11, 12, 6, -11, 0, 6, 6, 21], [0, 0, -5, -14, 0, -24, 0, -5], [-11, 0, 1, 11, 4, -1, -18, 0], [16, 0, -5, -16, 12, 5, -13, 0], [9, 7, -6, 9, 0, -6, -6, -18], [12, -14, -4, 12, 0, -4, -4, -20], [20, 8, 8, 0, -8, 18, 26, 18], [11, 16, 2, 11, 0, 2, 2, 23], [38, 0, -13, -38, -4, 13, 7, 0], [0, 8, 6, 12, 8, 6, 0, 6], [1, 0, 5, -1, -4, -5, 18, 0], [14, 6, -12, 0, -6, -11, -23, -11], [-11, 13, 14, -11, 0, 14, 14, -4], [0, 6, 20, 14, 6, -32, 0, 20], [56, -2, -8, 0, 2, -2, -10, -2], [20, 2, 26, 20, 0, 26, 26, 40], [-48, -5, 11, 0, 5, -13, -2, -13], [0, -14, 0, 8, -14, -24, 0, 0], [-1, -15, -6, -1, 0, -6, -6, 4], [0, 2, -4, 14, 2, -8, 0, -4], [-8, -14, 4, 0, 14, 10, 14, 10], [16, -4, 5, 16, 0, 5, 5, -8], [24, 0, 0, -24, -6, 0, -16, 0], [0, 13, 8, -2, 13, 4, 0, 8], [-7, 0, -33, 7, -2, 33, -4, 0], [0, -8, -9, 10, -8, -12, 0, -9], [0, -8, 16, 4, -8, 0, 0, 16], [8, 0, 15, -8, -24, -15, -1, 0], [-6, 0, 25, -6, 0, 25, 25, 12], [32, 12, 0, 0, -12, 4, 4, 4], [25, 8, -10, 25, 0, -10, -10, -11], [0, 18, -9, 14, 18, 14, 0, -9], [5, 0, 16, -5, 15, -16, -2, 0], [24, 0, -3, -24, 8, 3, -3, 0], [-32, -2, -4, 0, 2, 6, 2, 6], [-46, 10, -16, 0, -10, 0, -16, 0], [0, -9, -7, 0, 9, 13, 6, 13], [0, 11, -2, -20, 11, 10, 0, -2], [23, 0, 9, -23, -6, -9, -14, 0], [0, 3, -21, 24, 3, 7, 0, -21], [0, -6, -3, 26, -6, -6, 0, -3], [30, -16, -28, 0, 16, -4, -32, -4], [-14, 0, 7, 14, -8, -7, -1, 0], [0, 0, 10, 48, 0, 4, 0, 10], [22, 6, -24, 0, -6, 13, -11, 13], [35, 15, 2, 35, 0, 2, 2, 14], [32, 22, -2, 32, 0, -2, -2, -16], [28, -6, 4, 0, 6, -12, -8, -12], [6, 10, -16, 0, -10, 0, -16, 0], [13, -10, 6, 13, 0, 6, 6, -1], [24, 6, -6, 24, 0, -6, -6, -24], [2, 0, 9, -2, 12, -9, -7, 0], [0, 4, -10, 24, 4, 12, 0, -10], [0, 2, -14, -20, 2, -6, 0, -14], [-50, 0, 14, 0, 0, -5, 9, -5], [0, -1, 5, -88, -1, -3, 0, 5], [-18, -12, 11, -18, 0, 11, 11, 40], [17, -6, -10, 17, 0, -10, -10, -21], [-15, 0, 6, 15, 15, -6, 6, 0], [-64, 1, 3, 0, -1, -9, -6, -9], [-30, 18, 12, 0, -18, 11, 23, 11], [0, 9, -1, 0, 9, -5, 0, -1], [-17, 14, 16, -17, 0, 16, 16, 15], [0, 14, -9, 6, 14, 18, 0, -9], [7, 0, -20, -7, -5, 20, -14, 0], [8, -21, 9, 0, 21, -7, 2, -7], [-14, 0, 17, 14, 20, -17, -15, 0], [8, 0, -5, -8, 8, 5, -13, 0], [-26, 5, 0, 0, -5, 12, 12, 12], [1, 24, 4, 1, 0, 4, 4, 11], [-24, 12, 25, -24, 0, 25, 25, 24], [10, 0, -17, -10, -20, 17, -5, 0], [0, -13, 16, -18, -13, -20, 0, 16], [1, 0, -1, -1, 2, 1, -4, 0], [16, -6, 8, 0, 6, -14, -6, -14], [0, -4, 15, -30, -4, -8, 0, 15], [32, 21, 9, 0, -21, -11, -2, -11], [0, -6, -10, 12, -6, 24, 0, -10], [-11, 0, 25, 11, 16, -25, -4, 0], [11, 15, -22, 11, 0, -22, -22, -28], [0, 16, 8, 0, -16, -8, 0, -8], [-4, -28, 9, -4, 0, 9, 9, 24], [0, 3, -16, 12, 3, 12, 0, -16], [-7, 0, -5, 7, 34, 5, 0, 0], [-20, 0, -5, 20, -8, 5, -5, 0], [0, -14, -20, -30, -14, 8, 0, -20], [10, 12, 3, 10, 0, 3, 3, 12], [0, -8, 0, -52, -8, 4, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4608_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4608_2_k_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_4608_2_k_be(:prec:=8)
    chi := MakeCharacter_4608_k();
    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_4608_2_k_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_4608_2_k_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4608_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 64, 128, -32, 24, 48, 32, 8, 1]>,<7,R![64, 0, 256, 0, 240, 0, 32, 0, 1]>,<11,R![1024, 0, 0, 0, 192, 0, 0, 0, 1]>,<13,R![35344, 51136, 36992, 9120, 1160, 48, 32, 8, 1]>,<19,R![1024, 0, 0, -512, 1088, -512, 128, -16, 1]>],Snew);
    return Vf;
end function;