
// Make newform 4368.2.h.s in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4368_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4368_h_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_4368_2_h_s();".
// 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_4368_2_h_s();".
// 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 := [16, 0, 116, 0, 212, 0, 124, 0, 21, 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], [0, 1004, 0, 632, 0, -296, 0, -89, 0, -5], [808, 0, 632, 0, -296, 0, -89, 0, -5, 0], [-20, 98, -396, 0, -423, 0, -81, 0, -4, 0], [20, 98, 396, 0, 423, 0, 81, 0, 4, 0], [501, 0, 1727, 0, 1318, 0, 243, 0, 12, 0], [0, 2340, 0, 4878, 0, 3186, 0, 559, 0, 27], [-3020, 0, -11580, 0, -7768, 0, -1353, 0, -65, 0], [0, 2628, 0, 11580, 0, 7768, 0, 1353, 0, 65], [0, -11588, 0, -42380, 0, -32240, 0, -5889, 0, -289]];
    Rf_basisdens := [1, 392, 196, 98, 98, 49, 196, 196, 392, 392];
    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_4368_h();"
function MakeCharacter_4368_h()
    N := 4368;
    order := 2;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    v := [2, 2, 2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4368_h_Hecke();"
function MakeCharacter_4368_h_Hecke(Kf)
    N := 4368;
    order := 2;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 1, 0, -1, 0, 0, 0], [0, 0, 0, 1, 0, -1, 0, 0, 0, 0], [-2, 0, 1, 0, 0, -1, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, -1], [2, 0, 1, -1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, -1, 0], [0, -1, 0, -2, -2, 0, 1, 0, 1, 0], [0, 0, -2, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, 1, 1, 0, -1, 0, 1, 0], [0, 0, -1, 0, 0, -1, 0, 1, 0, 0], [0, -1, 0, -1, -1, 0, 3, 0, 0, 1], [0, 0, 0, 1, -1, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 1, 1], [0, 1, 0, -1, -1, 0, 3, 0, 2, 1], [0, 2, 0, -1, -1, 0, 0, 0, 0, 1], [-4, 0, 3, -1, 1, 1, 0, 1, 0, 0], [0, -1, 0, 1, 1, 0, -5, 0, -2, -1], [0, -1, 0, -2, -2, 0, 5, 0, 1, 0], [0, -4, 0, -1, -1, 0, 0, 0, 2, 0], [2, 0, 3, -2, 2, -1, 0, -1, 0, 0], [0, 0, -2, -1, 1, 0, 0, -1, 0, 0], [0, 0, 0, -1, 1, 2, 0, 0, 0, 0], [0, 0, 0, 2, 2, 0, -2, 0, -1, 0], [2, 0, -2, 2, -2, -2, 0, -2, 0, 0], [8, 0, 1, -1, 1, -1, 0, -1, 0, 0], [-4, 0, -2, -2, 2, 0, 0, 1, 0, 0], [0, -1, 0, 1, 1, 0, -3, 0, 2, 0], [2, 0, 3, 3, -3, -1, 0, 1, 0, 0], [0, -3, 0, -1, -1, 0, 1, 0, 2, 1], [0, 8, 0, 0, 0, 0, -2, 0, 2, -1], [0, 0, -3, 0, 0, 3, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 0, -1, -1], [0, -1, 0, -1, -1, 0, 5, 0, 1, -1], [-6, 0, -5, 2, -2, -1, 0, -1, 0, 0], [8, 0, -2, -1, 1, 0, 0, -2, 0, 0], [0, 0, 1, -4, 4, 1, 0, -1, 0, 0], [2, 0, 1, -3, 3, 1, 0, -2, 0, 0], [0, -2, 0, 0, 0, 0, 6, 0, 2, 0], [0, 3, 0, -1, -1, 0, -5, 0, 0, 1], [-2, 0, -1, 0, 0, -3, 0, 0, 0, 0], [-8, 0, -1, 1, -1, -1, 0, 0, 0, 0], [0, 2, 0, -2, -2, 0, 2, 0, -2, 0], [0, 5, 0, -1, -1, 0, 1, 0, 2, -1], [0, 6, 0, 3, 3, 0, -2, 0, 0, 0], [6, 0, 0, -2, 2, 0, 0, 0, 0, 0], [0, -11, 0, -1, -1, 0, -1, 0, 4, 1], [0, 2, 0, -1, -1, 0, 2, 0, 2, 2], [0, 0, -4, 2, -2, -2, 0, 1, 0, 0], [-2, 0, 7, -2, 2, -1, 0, -1, 0, 0], [-10, 0, 1, -3, 3, 3, 0, 1, 0, 0], [2, 0, 1, -2, 2, 1, 0, 3, 0, 0], [0, 2, 0, -2, -2, 0, 2, 0, 6, 0], [2, 0, 0, 0, 0, 4, 0, -2, 0, 0], [0, 11, 0, 1, 1, 0, -5, 0, 0, 1], [8, 0, 0, 2, -2, 0, 0, 2, 0, 0], [0, -9, 0, -2, -2, 0, 1, 0, 4, 1], [0, 4, 0, -2, -2, 0, 4, 0, -4, 0], [0, 0, -4, -4, 4, 0, 0, 0, 0, 0], [2, 0, -5, 3, -3, -3, 0, 1, 0, 0], [0, -1, 0, -1, -1, 0, 11, 0, 2, 1], [0, -2, 0, -2, -2, 0, 6, 0, 1, 3], [0, 0, -3, -1, 1, 3, 0, 0, 0, 0], [4, 0, 8, 1, -1, 2, 0, 2, 0, 0], [0, -2, 0, -3, -3, 0, 10, 0, 6, 0], [0, -1, 0, 0, 0, 0, -3, 0, 1, 0], [0, 7, 0, -3, -3, 0, 9, 0, 8, 1], [-4, 0, 2, 1, -1, -2, 0, -2, 0, 0], [-8, 0, 1, 5, -5, 1, 0, -1, 0, 0], [0, -2, 0, -2, -2, 0, -2, 0, -3, -1], [0, -17, 0, -1, -1, 0, 5, 0, 1, 1], [-8, 0, -5, -4, 4, 3, 0, 1, 0, 0], [0, -8, 0, 3, 3, 0, -8, 0, -8, 0], [0, -3, 0, 5, 5, 0, -7, 0, 0, -1], [0, -16, 0, 3, 3, 0, -6, 0, 0, -1], [-8, 0, 0, 2, -2, 2, 0, 1, 0, 0], [0, 14, 0, 0, 0, 0, 2, 0, -1, -1], [0, -1, 0, 5, 5, 0, -11, 0, -2, -1], [-2, 0, -3, -3, 3, -1, 0, 1, 0, 0], [12, 0, 0, 1, -1, -6, 0, 1, 0, 0], [0, 0, -2, -1, 1, 0, 0, -2, 0, 0], [0, -7, 0, 1, 1, 0, -5, 0, 0, 0], [0, -2, 0, 6, 6, 0, -2, 0, -6, 0], [0, -9, 0, 2, 2, 0, -1, 0, 4, -2], [0, 14, 0, -2, -2, 0, 0, 0, 2, 1], [-4, 0, 2, 0, 0, -4, 0, 3, 0, 0], [0, 1, 0, -5, -5, 0, 7, 0, 1, -1], [0, -16, 0, 0, 0, 0, 0, 0, -2, -2], [12, 0, 2, -1, 1, 0, 0, 0, 0, 0], [0, 6, 0, 2, 2, 0, -6, 0, 1, 1], [-4, 0, -6, 0, 0, -2, 0, 0, 0, 0], [0, 7, 0, -4, -4, 0, 3, 0, 6, 0], [2, 0, -13, 0, 0, 1, 0, -2, 0, 0], [-6, 0, 7, -1, 1, -5, 0, 1, 0, 0], [0, -2, 0, 4, 4, 0, -8, 0, -3, 2], [-16, 0, 0, 4, -4, -4, 0, 0, 0, 0], [0, -5, 0, 5, 5, 0, -1, 0, 4, 1], [8, 0, 0, -2, 2, 2, 0, -3, 0, 0], [-18, 0, -12, 4, -4, 4, 0, 0, 0, 0], [-12, 0, -1, -1, 1, 1, 0, 1, 0, 0], [0, 8, 0, 3, 3, 0, -4, 0, 0, 2], [0, 7, 0, 1, 1, 0, -5, 0, -4, 3], [0, 23, 0, 0, 0, 0, 5, 0, 1, 0], [-10, 0, -5, -1, 1, 3, 0, 2, 0, 0], [10, 0, -3, -1, 1, -1, 0, 1, 0, 0], [12, 0, -8, 3, -3, -2, 0, -1, 0, 0], [0, 6, 0, 8, 8, 0, -12, 0, -5, 2], [0, -11, 0, 1, 1, 0, 7, 0, -6, 2], [0, 22, 0, 0, 0, 0, 0, 0, -6, -1], [0, 10, 0, -2, -2, 0, 4, 0, 4, -1], [-10, 0, 2, -6, 6, 6, 0, 0, 0, 0], [0, -4, 0, 6, 6, 0, -10, 0, -6, -3], [16, 0, 8, 0, 0, 0, 0, 2, 0, 0], [16, 0, 5, -2, 2, -1, 0, 2, 0, 0], [8, 0, 12, -3, 3, -6, 0, 0, 0, 0], [0, 6, 0, 1, 1, 0, 6, 0, 4, 0], [-26, 0, 0, 4, -4, -2, 0, 1, 0, 0], [10, 0, 1, 0, 0, 5, 0, -1, 0, 0], [0, -5, 0, 1, 1, 0, 3, 0, -6, 0], [0, 14, 0, -4, -4, 0, -2, 0, 0, -2], [-8, 0, 3, -2, 2, -5, 0, 1, 0, 0], [0, 16, 0, 6, 6, 0, -12, 0, -7, -1], [16, 0, -12, 2, -2, -4, 0, 0, 0, 0], [-26, 0, -3, 10, -10, -5, 0, 0, 0, 0], [0, 6, 0, -5, -5, 0, -6, 0, -2, 4], [0, -12, 0, 8, 8, 0, -8, 0, -5, -1], [0, 1, 0, -5, -5, 0, 11, 0, 8, 1], [20, 0, 4, -8, 8, 4, 0, 0, 0, 0], [16, 0, 1, 5, -5, -3, 0, -1, 0, 0], [0, -19, 0, -2, -2, 0, -1, 0, -5, -2], [0, -14, 0, -5, -5, 0, 0, 0, 6, 3], [0, -11, 0, -2, -2, 0, 1, 0, -2, -2], [0, -6, 0, 4, 4, 0, -4, 0, -6, 3], [-18, 0, -9, -4, 4, -1, 0, -1, 0, 0], [10, 0, 6, 0, 0, 2, 0, 0, 0, 0], [0, 20, 0, 2, 2, 0, 2, 0, -8, -3], [0, 17, 0, 3, 3, 0, 1, 0, 0, -3], [16, 0, 2, -8, 8, 6, 0, 2, 0, 0], [0, 11, 0, 1, 1, 0, -9, 0, -8, 0], [-8, 0, -2, -5, 5, 2, 0, -3, 0, 0], [0, 5, 0, -3, -3, 0, 9, 0, 5, 6], [0, 16, 0, -7, -7, 0, 0, 0, 2, -2], [10, 0, 1, 6, -6, 1, 0, 1, 0, 0], [-8, 0, -4, 6, -6, -4, 0, -2, 0, 0], [0, 7, 0, 1, 1, 0, -3, 0, 0, 3], [0, -14, 0, 0, 0, 0, -2, 0, -7, 1], [-14, 0, -3, 6, -6, -1, 0, 0, 0, 0], [-24, 0, 5, -1, 1, 1, 0, -2, 0, 0], [-12, 0, 10, -4, 4, 2, 0, 0, 0, 0], [4, 0, -17, 5, -5, 5, 0, 1, 0, 0], [-2, 0, 5, -1, 1, -11, 0, 0, 0, 0], [8, 0, 7, -5, 5, 1, 0, 3, 0, 0], [0, 19, 0, -2, -2, 0, 13, 0, 3, 4], [22, 0, 11, 5, -5, -7, 0, -1, 0, 0], [0, -3, 0, 0, 0, 0, 3, 0, 8, -1], [0, -7, 0, 3, 3, 0, 1, 0, 2, 0], [-14, 0, 4, 6, -6, -4, 0, 0, 0, 0], [0, -10, 0, 6, 6, 0, -8, 0, -4, -1], [-4, 0, 8, 4, -4, 0, 0, 0, 0, 0], [0, 11, 0, -1, -1, 0, -3, 0, 2, -4], [0, 29, 0, -3, -3, 0, 3, 0, 5, -1], [-24, 0, -2, -4, 4, 6, 0, 2, 0, 0], [0, 0, 1, 0, 0, 1, 0, 3, 0, 0], [0, -4, 0, -6, -6, 0, 12, 0, -2, -2], [2, 0, 5, 0, 0, -7, 0, -1, 0, 0], [0, 15, 0, 1, 1, 0, -5, 0, -6, -4], [0, -4, 0, -7, -7, 0, 20, 0, 10, 2], [2, 0, -3, -5, 5, 3, 0, 1, 0, 0], [0, -4, 0, -4, -4, 0, 0, 0, -4, -2], [-24, 0, -14, -1, 1, 8, 0, 1, 0, 0], [-6, 0, -3, 4, -4, -7, 0, 1, 0, 0], [0, -16, 0, 0, 0, 0, 8, 0, -1, -3], [0, -1, 0, 1, 1, 0, 7, 0, 6, 5], [8, 0, -8, 5, -5, 4, 0, 0, 0, 0], [4, 0, 8, 3, -3, -12, 0, -1, 0, 0], [0, -6, 0, 6, 6, 0, -8, 0, 2, 3], [4, 0, 4, -4, 4, 4, 0, 4, 0, 0], [-4, 0, -1, 9, -9, -13, 0, -1, 0, 0], [0, 1, 0, 4, 4, 0, -13, 0, -9, -4], [0, 3, 0, 9, 9, 0, 5, 0, 0, 1], [-42, 0, 3, 0, 0, -1, 0, 1, 0, 0], [10, 0, -10, 4, -4, 2, 0, 2, 0, 0], [0, 4, 0, 10, 10, 0, -6, 0, -6, -1], [0, -24, 0, 0, 0, 0, 0, 0, -6, 2], [0, -5, 0, 7, 7, 0, -11, 0, -5, -7], [-10, 0, 6, 4, -4, -10, 0, 2, 0, 0], [0, 13, 0, -1, -1, 0, 1, 0, 6, -2], [0, 0, -8, 2, -2, 8, 0, 4, 0, 0], [0, -1, 0, -2, -2, 0, -9, 0, 4, -2], [16, 0, -2, 9, -9, -4, 0, 2, 0, 0], [6, 0, 7, 6, -6, 1, 0, 2, 0, 0], [0, -10, 0, -2, -2, 0, -10, 0, -4, -2], [-28, 0, 9, 3, -3, -11, 0, 5, 0, 0], [0, 35, 0, 2, 2, 0, -7, 0, -5, 0], [20, 0, 10, -6, 6, 6, 0, 2, 0, 0], [0, 9, 0, -7, -7, 0, 5, 0, 6, 3], [8, 0, -2, -1, 1, -4, 0, 1, 0, 0], [0, -10, 0, -1, -1, 0, -10, 0, -10, 2], [-10, 0, -6, 0, 0, 2, 0, 6, 0, 0], [0, -1, 0, 3, 3, 0, 11, 0, -6, 1], [2, 0, -15, -4, 4, 5, 0, -1, 0, 0], [0, 2, 0, -2, -2, 0, -10, 0, -2, 2], [-12, 0, -6, -9, 9, 8, 0, 0, 0, 0], [0, 19, 0, 1, 1, 0, 9, 0, -8, -2], [-4, 0, 6, -2, 2, 2, 0, 6, 0, 0], [-24, 0, -17, 11, -11, 1, 0, 0, 0, 0], [6, 0, 13, -8, 8, 5, 0, 1, 0, 0], [-4, 0, -17, -3, 3, 5, 0, -1, 0, 0], [0, 21, 0, -3, -3, 0, -7, 0, -4, 1], [0, -17, 0, -1, -1, 0, -7, 0, 3, -1], [0, 28, 0, -5, -5, 0, -6, 0, -2, 3], [0, 0, 3, -5, 5, 1, 0, -3, 0, 0], [6, 0, 1, -4, 4, 3, 0, -6, 0, 0], [0, 13, 0, -9, -9, 0, 15, 0, 0, -3], [0, 11, 0, -7, -7, 0, 3, 0, 6, -3], [26, 0, -12, 4, -4, 4, 0, 2, 0, 0], [0, 2, 0, -2, -2, 0, 2, 0, -6, 0], [0, -17, 0, 1, 1, 0, -7, 0, -4, 0], [0, -4, 0, 4, 4, 0, -4, 0, 2, -2], [4, 0, 6, -6, 6, 0, 0, -3, 0, 0], [6, 0, -8, 0, 0, -4, 0, -4, 0, 0], [14, 0, -11, -2, 2, 3, 0, 2, 0, 0], [-6, 0, -11, 7, -7, -5, 0, -3, 0, 0], [10, 0, 13, -8, 8, -1, 0, 4, 0, 0], [0, 21, 0, 3, 3, 0, -23, 0, -4, 0], [-18, 0, -8, 4, -4, -4, 0, 0, 0, 0], [-26, 0, 9, 3, -3, -3, 0, 7, 0, 0], [0, -20, 0, -4, -4, 0, 10, 0, 4, -3], [-22, 0, -6, 6, -6, -2, 0, -4, 0, 0], [2, 0, 3, -3, 3, 7, 0, 9, 0, 0], [0, -7, 0, -5, -5, 0, 3, 0, -1, 1], [0, 22, 0, -1, -1, 0, 8, 0, 12, -1], [0, -21, 0, 1, 1, 0, -1, 0, 0, 1], [28, 0, 6, -7, 7, 4, 0, 6, 0, 0], [-16, 0, -4, -4, 4, -4, 0, -4, 0, 0], [0, -7, 0, -5, -5, 0, -5, 0, -4, -3], [16, 0, -2, 0, 0, -6, 0, 0, 0, 0], [2, 0, -17, -2, 2, 9, 0, 0, 0, 0], [0, 16, 0, -6, -6, 0, 20, 0, -1, 1], [0, 13, 0, -10, -10, 0, -1, 0, 5, -4], [8, 0, 4, 0, 0, 0, 0, -8, 0, 0], [0, 2, 0, -2, -2, 0, -6, 0, -6, 4], [0, -5, 0, -3, -3, 0, 11, 0, 4, 5], [0, 12, 0, -8, -8, 0, 8, 0, 9, -3], [-10, 0, 3, 11, -11, -9, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 6, 0, 11, 0], [0, -37, 0, 8, 8, 0, 1, 0, -1, 0], [0, 21, 0, 7, 7, 0, -5, 0, -6, 3], [-14, 0, -11, 1, -1, -5, 0, -7, 0, 0], [-26, 0, 1, -2, 2, 5, 0, -3, 0, 0], [0, -13, 0, 9, 9, 0, -13, 0, -2, 0], [-28, 0, 1, 3, -3, 1, 0, 3, 0, 0], [0, -6, 0, 0, 0, 0, -14, 0, -2, -4], [-18, 0, -11, 2, -2, 9, 0, 5, 0, 0], [0, 6, 0, 5, 5, 0, -16, 0, -10, 1], [-28, 0, 11, -9, 9, 5, 0, -1, 0, 0], [-12, 0, -8, 9, -9, -10, 0, -4, 0, 0], [0, 24, 0, 1, 1, 0, 4, 0, -2, -4], [28, 0, 9, -8, 8, -1, 0, 4, 0, 0], [0, 7, 0, 8, 8, 0, -19, 0, -11, -2], [16, 0, 0, 2, -2, -12, 0, 0, 0, 0], [0, -41, 0, -5, -5, 0, 3, 0, 2, -3], [0, -25, 0, 12, 12, 0, -3, 0, -9, 4], [0, 28, 0, 4, 4, 0, 0, 0, 3, -1], [-28, 0, 7, -2, 2, -1, 0, 5, 0, 0], [8, 0, -10, 5, -5, -8, 0, 2, 0, 0], [0, 2, 0, -6, -6, 0, -6, 0, 1, 3], [0, 2, 0, -5, -5, 0, -8, 0, 0, -3], [-4, 0, -4, 2, -2, -8, 0, -6, 0, 0], [-34, 0, 5, -3, 3, -5, 0, -1, 0, 0], [0, 14, 0, 2, 2, 0, -14, 0, -6, 6], [0, -1, 0, 1, 1, 0, 11, 0, -4, 4], [0, -4, 0, 10, 10, 0, -14, 0, -1, 0], [0, 30, 0, 0, 0, 0, 6, 0, 8, -2], [40, 0, 10, 0, 0, -10, 0, -2, 0, 0], [0, 0, 12, -6, 6, 8, 0, 6, 0, 0], [0, 10, 0, -14, -14, 0, 24, 0, 16, 1], [4, 0, -10, 12, -12, -6, 0, 0, 0, 0], [0, -16, 0, 5, 5, 0, -16, 0, 0, -4], [0, -10, 0, -12, -12, 0, 8, 0, 10, -1], [34, 0, 1, 1, -1, 3, 0, -3, 0, 0], [-8, 0, 5, 5, -5, -13, 0, 2, 0, 0], [0, 29, 0, -1, -1, 0, -15, 0, -2, 1], [0, 24, 0, 0, 0, 0, -16, 0, 4, -4], [-18, 0, 19, 0, 0, -3, 0, 0, 0, 0], [20, 0, -1, -2, 2, 9, 0, -4, 0, 0], [-4, 0, -6, 2, -2, 0, 0, 7, 0, 0], [0, -5, 0, -1, -1, 0, 9, 0, 2, 2], [0, 15, 0, 5, 5, 0, 7, 0, 6, -5], [-2, 0, -16, 0, 0, -4, 0, -4, 0, 0], [14, 0, 7, 4, -4, -9, 0, -5, 0, 0], [16, 0, 20, -3, 3, -8, 0, -4, 0, 0], [-14, 0, -7, 2, -2, 13, 0, -3, 0, 0], [8, 0, -8, 10, -10, -6, 0, -3, 0, 0], [0, -6, 0, -8, -8, 0, 16, 0, -2, -3], [22, 0, 14, 0, 0, -6, 0, -8, 0, 0], [0, 33, 0, 3, 3, 0, 1, 0, 0, -3], [0, 0, 6, -6, 6, -6, 0, -2, 0, 0], [12, 0, -6, -7, 7, 0, 0, -2, 0, 0], [12, 0, 4, 4, -4, -6, 0, 3, 0, 0], [0, 30, 0, -8, -8, 0, 14, 0, 6, 0], [-20, 0, 8, 7, -7, -10, 0, 2, 0, 0], [16, 0, -19, -4, 4, 7, 0, -2, 0, 0], [0, 21, 0, 7, 7, 0, -25, 0, -6, -3], [36, 0, 7, 1, -1, -9, 0, -1, 0, 0], [-16, 0, 16, -10, 10, 4, 0, 2, 0, 0], [0, 19, 0, 3, 3, 0, -17, 0, -2, -3], [-34, 0, -18, -2, 2, 10, 0, 4, 0, 0], [-24, 0, -7, -9, 9, 1, 0, 0, 0, 0], [0, -1, 0, -3, -3, 0, -7, 0, -10, 5], [-14, 0, 9, -3, 3, 11, 0, -1, 0, 0], [0, -7, 0, -1, -1, 0, -3, 0, 2, -3], [0, 11, 0, -3, -3, 0, 1, 0, -6, -5], [0, 20, 0, 3, 3, 0, -22, 0, 4, -3], [14, 0, 10, 0, 0, -10, 0, 0, 0, 0], [-22, 0, -5, -7, 7, 3, 0, -5, 0, 0], [0, -2, 0, -6, -6, 0, 18, 0, 6, 2], [10, 0, -1, -6, 6, 15, 0, 1, 0, 0], [0, -12, 0, 8, 8, 0, -14, 0, -8, 5], [0, 41, 0, 5, 5, 0, -5, 0, 0, 0], [2, 0, -11, 3, -3, 7, 0, -3, 0, 0], [0, -4, 0, 0, 0, 0, -8, 0, -3, -3], [0, -16, 0, 2, 2, 0, 6, 0, 4, -7], [-2, 0, 9, 13, -13, -3, 0, 0, 0, 0], [-12, 0, 7, 4, -4, -15, 0, 4, 0, 0], [0, -8, 0, 0, 0, 0, -10, 0, 7, 0], [12, 0, 7, -12, 12, 1, 0, -8, 0, 0], [6, 0, -5, 6, -6, 3, 0, 1, 0, 0], [0, -19, 0, -3, -3, 0, -3, 0, 6, 11], [0, -28, 0, -4, -4, 0, 8, 0, 11, 7], [0, -13, 0, -1, -1, 0, 7, 0, 8, 5], [0, -2, 0, 8, 8, 0, -12, 0, 3, -2], [0, -7, 0, 2, 2, 0, -9, 0, 5, 2], [0, -12, 0, -11, -11, 0, 8, 0, 14, 4], [-24, 0, 6, -3, 3, 0, 0, 7, 0, 0], [0, 6, 0, -2, -2, 0, 10, 0, -7, 5], [-22, 0, 0, 0, 0, 4, 0, 10, 0, 0], [0, 7, 0, 8, 8, 0, -5, 0, -6, -4], [12, 0, 16, -10, 10, 0, 0, 0, 0, 0], [0, -25, 0, -9, -9, 0, 15, 0, 12, 5], [8, 0, 18, 5, -5, -8, 0, 2, 0, 0], [-24, 0, 4, -11, 11, 8, 0, -1, 0, 0], [0, 14, 0, 6, 6, 0, 2, 0, 1, 5], [0, 35, 0, -11, -11, 0, 7, 0, 3, 6], [-2, 0, 1, 2, -2, 5, 0, 9, 0, 0], [0, -28, 0, -2, -2, 0, -12, 0, 8, 0], [0, -12, 0, 2, 2, 0, 16, 0, -6, 0], [0, 25, 0, 5, 5, 0, -15, 0, -6, 5], [-10, 0, -9, 2, -2, 7, 0, -3, 0, 0], [-16, 0, 19, -5, 5, 3, 0, 5, 0, 0], [-10, 0, 7, 2, -2, -3, 0, 0, 0, 0], [0, -1, 0, 9, 9, 0, -15, 0, -13, -1], [0, -3, 0, 5, 5, 0, -3, 0, -4, -1], [18, 0, -20, 6, -6, -4, 0, 4, 0, 0], [0, -41, 0, 1, 1, 0, -7, 0, -2, -3], [0, -2, 0, -6, -6, 0, 28, 0, 11, 2], [-14, 0, 15, -6, 6, 9, 0, 6, 0, 0], [-6, 0, -17, 13, -13, -1, 0, 0, 0, 0], [0, -1, 0, 9, 9, 0, -11, 0, -10, 1], [38, 0, -11, -1, 1, 9, 0, -1, 0, 0], [10, 0, -26, 4, -4, -6, 0, -2, 0, 0], [0, 9, 0, 8, 8, 0, -3, 0, -6, -8], [0, 18, 0, 2, 2, 0, 8, 0, 2, -1], [-14, 0, -14, 0, 0, 4, 0, 5, 0, 0], [8, 0, 0, 1, -1, -18, 0, 4, 0, 0], [-6, 0, -17, 17, -17, -9, 0, 3, 0, 0], [0, 3, 0, 13, 13, 0, -7, 0, -14, 3], [16, 0, 9, 10, -10, -17, 0, 0, 0, 0], [-22, 0, -3, 8, -8, -9, 0, 4, 0, 0], [-34, 0, 0, 0, 0, -8, 0, -6, 0, 0], [0, -21, 0, 11, 11, 0, -9, 0, 4, -1], [2, 0, 1, 13, -13, -15, 0, -6, 0, 0], [0, -28, 0, -5, -5, 0, 6, 0, 6, -3], [-10, 0, -33, 10, -10, -3, 0, -8, 0, 0], [2, 0, -4, 12, -12, -8, 0, 2, 0, 0], [0, 3, 0, -2, -2, 0, 21, 0, 10, 7], [0, -9, 0, -3, -3, 0, -7, 0, -4, 0], [0, 28, 0, 12, 12, 0, -26, 0, -16, -1], [0, -1, 0, 1, 1, 0, -17, 0, -10, -5], [0, 20, 0, 2, 2, 0, 4, 0, 12, -4], [0, 53, 0, -1, -1, 0, -5, 0, 1, -3], [0, -28, 0, -12, -12, 0, -6, 0, 14, 3], [40, 0, 2, -1, 1, 6, 0, 5, 0, 0], [0, 20, 0, -12, -12, 0, -8, 0, -3, 1], [-16, 0, 8, 0, 0, 8, 0, 2, 0, 0], [0, -20, 0, 4, 4, 0, 8, 0, 8, -2], [0, -19, 0, 4, 4, 0, -5, 0, -4, -9], [0, 15, 0, 1, 1, 0, -11, 0, -8, 7], [-4, 0, 6, -6, 6, 2, 0, 6, 0, 0], [0, 7, 0, -7, -7, 0, 17, 0, 0, -7], [0, 0, -3, 5, -5, 5, 0, -1, 0, 0], [0, -36, 0, 8, 8, 0, 4, 0, 0, 0], [-6, 0, -9, 8, -8, 5, 0, -2, 0, 0], [-8, 0, -13, -9, 9, 3, 0, 2, 0, 0], [0, 21, 0, 3, 3, 0, -7, 0, -4, -1], [0, 0, 0, 3, 3, 0, -16, 0, 6, -2], [-10, 0, -12, -10, 10, -4, 0, 0, 0, 0], [0, 6, 0, -10, -10, 0, 0, 0, 8, -5], [0, 51, 0, -2, -2, 0, 1, 0, -7, 2], [0, 25, 0, -4, -4, 0, -1, 0, -2, 5], [18, 0, -31, 8, -8, -1, 0, 2, 0, 0], [0, 26, 0, 8, 8, 0, -2, 0, 5, 5], [0, -25, 0, 3, 3, 0, -19, 0, -8, -4], [0, 10, 0, 4, 4, 0, -12, 0, 2, 3], [0, -7, 0, -3, -3, 0, 13, 0, -2, 1], [58, 0, 8, -12, 12, 0, 0, 0, 0, 0], [44, 0, 9, -4, 4, -1, 0, 0, 0, 0], [-48, 0, -14, 17, -17, -4, 0, -4, 0, 0], [-32, 0, 18, -4, 4, 8, 0, -1, 0, 0], [-22, 0, -14, -8, 8, 0, 0, -3, 0, 0], [-2, 0, 7, -6, 6, -1, 0, -3, 0, 0], [0, -19, 0, 19, 19, 0, -9, 0, -12, 5], [-30, 0, 19, -2, 2, -7, 0, -2, 0, 0], [0, 15, 0, -4, -4, 0, 1, 0, -13, 0], [0, 0, -12, 8, -8, 12, 0, 2, 0, 0], [0, 0, 11, 2, -2, 9, 0, 6, 0, 0], [0, 30, 0, 7, 7, 0, -18, 0, -10, -10], [0, -19, 0, -1, -1, 0, -9, 0, -14, 1], [4, 0, -8, -4, 4, 16, 0, -4, 0, 0], [0, -22, 0, -7, -7, 0, -4, 0, -6, -7], [0, -31, 0, 6, 6, 0, -29, 0, -14, -7], [28, 0, -2, -1, 1, -8, 0, 2, 0, 0], [0, 23, 0, -3, -3, 0, -17, 0, -4, -3], [0, 48, 0, -10, -10, 0, 10, 0, 2, 5], [16, 0, 8, 2, -2, -8, 0, -6, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4368_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4368_2_h_s();".
// 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_4368_2_h_s(:prec:=10)
    chi := MakeCharacter_4368_h();
    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_4368_2_h_s();".
// 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_4368_2_h_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 0, 336, 0, 496, 0, 212, 0, 29, 0, 1]>,<11,R![4096, 0, 12944, 0, 11200, 0, 1576, 0, 73, 0, 1]>,<17,R![256, -128, -96, 18, 11, 1]>],Snew);
    return Vf;
end function;