
// Make newform 136.2.h.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_136_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_136_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_136_2_h_a();".
// 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_136_2_h_a();".
// 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 := [64, 0, 136, 0, 268, 0, -156, 0, 240, 0, 36, 0, 20, 0, 8, 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, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-581192704, 0, 110695532, 0, -1043866460, 0, -515961792, 0, -169343412, 0, -66487248, 0, -14068890, 0, -1122577, 0], [151276224, 0, -1494405076, 0, 649223580, 0, -438449040, 0, -46964780, 0, -25288536, 0, -13519726, 0, -1923537, 0], [-96545102, 0, -93359810, 0, -14220716, 0, -130749772, 0, -23013159, 0, -12763476, 0, -4434484, 0, -511505, 0], [450774640, 0, 1098739804, 0, -638782644, 0, 597246024, 0, 69125296, 0, 39114176, 0, 18755430, 0, 2530173, 0], [0, 49963176, 0, -1104732028, 0, 587561828, 0, -826357224, 0, -131353936, 0, -68170444, 0, -26976512, 0, -3391627], [0, -14322168, 0, -686733980, 0, -282204156, 0, -1047214652, 0, -227489312, 0, -107383932, 0, -33502210, 0, -3710839], [997692448, 0, 85178972, 0, -853219156, 0, 941650648, 0, 113648052, 0, 84156112, 0, 35818322, 0, 4466259, 0], [0, -870945096, 0, -889740780, 0, 203899652, 0, -707674738, 0, -117192984, 0, -58720920, 0, -22018279, 0, -2689779], [0, 588735180, 0, -6379140, 0, 47661826, 0, 364403110, 0, 70747866, 0, 37660840, 0, 12471803, 0, 1397209], [-337140920, 0, -1291451648, 0, 559120396, 0, -1087856768, 0, -177380254, 0, -93697396, 0, -35845480, 0, -4414637, 0], [0, 205818936, 0, -3208985236, 0, 2539117108, 0, -1741523828, 0, -185835896, 0, -116440672, 0, -58299578, 0, -7976257], [0, -1023743432, 0, -4071316148, 0, 1818685188, 0, -2385051376, 0, -339496648, 0, -183627060, 0, -76989604, 0, -9891357], [0, 1007435080, 0, 1381761084, 0, -1873346700, 0, 2465582864, 0, 348654544, 0, 214521736, 0, 87123872, 0, 10838191], [-1418449856, 0, -2688584780, 0, 3063239676, 0, -2849496608, 0, -344990604, 0, -221895280, 0, -99013618, 0, -12834559, 0]];
    Rf_basisdens := [1, 1, 388027784, 388027784, 97006946, 388027784, 388027784, 388027784, 388027784, 194013892, 97006946, 194013892, 388027784, 388027784, 388027784, 388027784];
    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_136_h();"
function MakeCharacter_136_h()
    N := 136;
    order := 2;
    char_gens := [103, 69, 105];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_136_h_Hecke();"
function MakeCharacter_136_h_Hecke(Kf)
    N := 136;
    order := 2;
    char_gens := [103, 69, 105];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 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], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1], [0, -1, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 1, 0], [0, -2, 0, 0, 0, 0, 0, 1, 0, -1, 2, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, -1, 0], [0, 2, 0, 0, 0, 0, 0, -1, 0, 2, -1, 0, 1, -1, -1, 0], [1, 0, -1, -1, -1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, -1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, -3, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [1, 0, 1, -1, -1, -3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, -1, 0, 1, -1, 1, 0], [-1, 0, -1, 2, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1], [0, 2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 2, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 1, -2, 0, 1, 0, -2, 0], [1, 0, 1, -1, -1, -3, 0, 0, -1, 0, 0, 0, 0, 0, 0, -2], [0, 0, -2, -3, 2, -1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 0, -1, -1, 1, 0], [0, 0, 0, 2, 0, -2, 0, 0, -1, 0, 0, 2, 0, 0, 0, -1], [2, 0, 0, 2, 2, -6, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 1, 0, 0, -1, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 1, 0, 2, 3, 0, -1, -1, -1, 0], [4, 0, 2, 4, -4, 4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 1, 2, -2, 0], [2, 0, 1, 4, -4, 4, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -1, 0, -1, -1, 1, 0], [2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2], [8, 0, -4, -2, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0], [-4, 0, 1, 3, 4, 3, 0, 0, -1, 0, 0, -1, 0, 0, 0, -2], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, -1, 0, -1, -1, 1, 0], [0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 3, 0, -1, -1, 3, 0], [0, 0, 0, 0, 0, 0, -3, -3, 0, 0, -1, 0, -3, -1, 1, 0], [3, 0, -1, -4, -3, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, -3, 2, 0, 0, -2, 0, 4, -2, 2, 0], [-3, 0, 1, 1, -3, 5, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0], [0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, -6, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, -1, 0, -1, -4, 0, 1, 2, 2, 0], [0, 0, 0, 0, 0, 0, 6, -1, 0, 0, 1, 0, 0, 1, -1, 0], [3, 0, -5, -1, 3, -5, 0, 0, 5, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, -2, 0, 0, 1, 0, 1, 1, -1, 0], [0, 0, -2, -2, 0, 6, 0, 0, 1, 0, 0, -2, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 1, 0, -6, 3, 0, -1, -1, 1, 0], [-11, 0, 3, 7, -3, -1, 0, 0, -3, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -1, 0, 1, -1, -3, 0], [-1, 0, 1, -2, 1, 2, 0, 0, 4, 0, 0, -3, 0, 0, 0, 3], [8, 0, -3, -8, 2, 4, 0, 0, 3, 0, 0, 6, 0, 0, 0, 0], [-2, 0, 0, -2, -2, 6, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 3, 0, 0, 1, 0, -7, 1, -1, 0], [1, 0, 5, -1, 1, -1, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 0, -4, 0, 0, 0], [4, 0, -3, 1, 2, -5, 0, 0, 3, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 2, -2, 0], [-2, 0, 0, 2, 2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, -2], [-3, 0, 1, -3, 3, 7, 0, 0, 3, 0, 0, -6, 0, 0, 0, 2], [0, -2, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, -1, 3, 3, 0], [0, -4, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, -2, 2, 2, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 1, 0, -7, 1, -1, 0], [5, 0, -1, -3, -5, -5, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0], [0, -6, 0, 0, 0, 0, 0, 1, 0, -6, 1, 0, -1, 1, -3, 0], [0, 0, 0, 0, 0, 0, -4, -1, 0, 0, 2, 0, 6, 2, -2, 0], [0, 0, 3, -1, 0, -5, 0, 0, 3, 0, 0, -1, 0, 0, 0, 0], [2, 0, 5, 1, 0, -1, 0, 0, -5, 0, 0, -1, 0, 0, 0, 0], [-9, 0, 1, -5, 7, -9, 0, 0, -1, 0, 0, -2, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, -1, 0, 1, -3, 0, 1, 1, 1, 0], [0, 0, 2, 0, 0, -4, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 6, -1, 0, 0, 1, 0, -4, 1, -1, 0], [14, 0, -2, 4, -2, 0, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0], [-4, 0, 2, 0, 4, 4, 0, 0, 1, 0, 0, -4, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -7, -1, 0, 0, -1, 0, 7, -1, 1, 0], [0, -8, 0, 0, 0, 0, 0, 1, 0, 2, 1, 0, -1, 1, -5, 0], [2, 0, 2, 0, -4, 8, 0, 0, -2, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 2, 0, 0, -4, 0, 0, -1, 0, 0, 0, 0, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 4, 0, -1, -2, 0, 0], [-2, 0, -2, 1, 0, -1, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, -5, -3, 0, 1, 1, -1, 0], [-1, 0, -3, 2, 1, 6, 0, 0, -4, 0, 0, 1, 0, 0, 0, -1], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 4, 0], [-2, 0, -4, 1, 0, -1, 0, 0, 4, 0, 0, -1, 0, 0, 0, 0], [4, 0, -3, 3, -4, -5, 0, 0, -1, 0, 0, 7, 0, 0, 0, 2], [2, 0, 2, -6, 2, 2, 0, 0, -2, 0, 0, 4, 0, 0, 0, 0], [3, 0, -1, 2, -3, -6, 0, 0, -2, 0, 0, 5, 0, 0, 0, -1], [0, -2, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, -1, -3, 0], [0, -6, 0, 0, 0, 0, 0, 2, 0, -1, 6, 0, -2, -2, -4, 0], [-1, 0, 5, 0, 1, -8, 0, 0, -2, 0, 0, -1, 0, 0, 0, -7], [0, 0, 0, 0, 0, 0, 8, 6, 0, 0, -2, 0, -3, -2, 2, 0], [0, 4, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 1, -3, -1, 0], [2, 0, 4, -4, -2, -8, 0, 0, -4, 0, 0, -2, 0, 0, 0, -8], [0, -6, 0, 0, 0, 0, 0, 1, 0, -6, 1, 0, -1, 1, -3, 0], [-1, 0, -1, 4, 1, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 0, -12, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, -1, -4, 4, 0], [0, 0, 6, -6, 0, -6, 0, 0, 3, 0, 0, -6, 0, 0, 0, -3], [-3, 0, -3, 3, 3, 9, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -1, -7, 6, -5, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 10, 1, 0, 0, -1, 0, -10, -1, 1, 0], [2, 0, 1, -4, 4, -4, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [3, 0, -1, -2, -3, -2, 0, 0, 4, 0, 0, 1, 0, 0, 0, 5], [-14, 0, 3, 9, -4, -1, 0, 0, -3, 0, 0, -5, 0, 0, 0, 0], [6, 0, 2, 6, -2, -2, 0, 0, -2, 0, 0, -4, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, -3, 0, -6, -9, 0, 3, 3, 1, 0], [0, 8, 0, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, -3, 5, 0], [2, 0, 2, 2, -2, -10, 0, 0, 6, 0, 0, 4, 0, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 5, 0, -1, -3, -1, 0], [0, 0, 0, 0, 0, 0, -8, -3, 0, 0, 0, 0, -4, 0, 0, 0], [-7, 0, -1, 7, -7, 7, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 8, 0, 0, 0, 0, 0, -3, 0, 6, -9, 0, 3, 3, 5, 0], [0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 6, 0, 3, 6, -6, 0], [19, 0, -7, -9, 3, 3, 0, 0, 7, 0, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 0, -4, 0, -4, -4, 4, 0], [1, 0, 3, -5, -1, -3, 0, 0, 1, 0, 0, -4, 0, 0, 0, -2], [2, 0, -6, 4, -2, 4, 0, 0, -4, 0, 0, 6, 0, 0, 0, 2], [0, 6, 0, 0, 0, 0, 0, -2, 0, 4, -4, 0, 2, 0, 2, 0], [0, 0, 0, 0, 0, 0, -3, 1, 0, 0, 1, 0, 17, 1, -1, 0], [0, 0, 0, 0, 0, 0, -6, 1, 0, 0, -1, 0, -6, -1, 1, 0], [0, 0, 0, 0, 0, 0, 6, 4, 0, 0, 1, 0, -3, 1, -1, 0], [8, 0, -4, -8, -8, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, -1, -2, 0, 0, 2, 0, 6, 2, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [17, 0, 1, 5, -7, 9, 0, 0, -1, 0, 0, 2, 0, 0, 0, 0], [2, 0, 0, -6, -2, 2, 0, 0, -6, 0, 0, -4, 0, 0, 0, -6], [-1, 0, -1, 0, 1, 4, 0, 0, -2, 0, 0, -1, 0, 0, 0, -1], [0, -6, 0, 0, 0, 0, 0, 2, 0, -3, 3, 0, -2, 1, -1, 0], [0, 6, 0, 0, 0, 0, 0, 1, 0, -7, -1, 0, -1, 3, 11, 0], [0, 0, 0, 8, 0, -8, 0, 0, -3, 0, 0, 8, 0, 0, 0, -3], [12, 0, -3, 4, -2, 0, 0, 0, 3, 0, 0, -2, 0, 0, 0, 0], [-8, 0, 3, -4, 10, -16, 0, 0, -3, 0, 0, -6, 0, 0, 0, 0], [-8, 0, 2, 10, 8, 2, 0, 0, 5, 0, 0, 2, 0, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, -9, 0, 0, 2, 0, 6, 2, -2, 0], [-6, 0, 4, 4, 6, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -8, 0, 2, 4, 0, 0], [0, 0, 0, 0, 0, 0, 10, -4, 0, 0, 1, 0, 9, 1, -1, 0], [0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 3, 0, -7, 3, -3, 0], [0, -10, 0, 0, 0, 0, 0, -1, 0, -5, -4, 0, 1, 2, -10, 0], [0, 0, 0, 0, 0, 0, -2, 1, 0, 0, -5, 0, -4, -5, 5, 0], [4, 0, -11, 5, -4, 9, 0, 0, -7, 0, 0, 9, 0, 0, 0, 4], [0, -6, 0, 0, 0, 0, 0, 5, 0, -3, 11, 0, -5, -1, 3, 0], [0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 8, 2, -2, 0], [0, 12, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, -4, 4, 0], [-11, 0, 5, 7, 11, 5, 0, 0, -1, 0, 0, -4, 0, 0, 0, -6], [-20, 0, -2, -2, 4, -6, 0, 0, 2, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 6, 0, 0, -2, 0, 12, -2, 2, 0], [0, 2, 0, 0, 0, 0, 0, -1, 0, -2, 1, 0, 1, -3, -3, 0], [3, 0, 5, -8, -3, -8, 0, 0, -2, 0, 0, -5, 0, 0, 0, -7], [0, -2, 0, 0, 0, 0, 0, 1, 0, 7, 6, 0, -1, -4, -4, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 3, 0, -5, 3, -3, 0], [0, 6, 0, 0, 0, 0, 0, -2, 0, 7, -2, 0, 2, -2, 0, 0], [5, 0, 9, 7, -3, -1, 0, 0, -9, 0, 0, -4, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 3, 0, 6, 7, 0, -3, -1, 1, 0], [12, 0, -12, -2, 6, -10, 0, 0, 12, 0, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -11, -3, 0, 0, 1, 0, 13, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 2, 0, 2, 2, -2, 0], [-20, 0, 4, -7, 6, -5, 0, 0, -4, 0, 0, 1, 0, 0, 0, 0], [9, 0, 3, 7, -7, 7, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 5, -1, -5, 0, 0, 5, 0, 0, 6, 0, 0, 0, 6], [-16, 0, 9, 3, -6, 9, 0, 0, -9, 0, 0, 3, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, 0, 1, 0, -9, 3, 0, -1, -1, -3, 0], [0, 12, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, 0, -3, -2, 0, 0, -2, 0, 12, -2, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_136_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_136_2_h_a();".
// 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_136_2_h_a(:prec:=16)
    chi := MakeCharacter_136_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(997) - 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_136_2_h_a();".
// 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_136_2_h_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_136_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;