
// Make newform 156.2.h.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_156_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_156_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_156_2_h_b();".
// 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_156_2_h_b();".
// 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, 0, 0, 1804, 0, 0, 0, 517, 0, 0, 0, 43, 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, 0, 0, 0, 0, 0, 0], [0, 0, -728592, 0, 0, 0, -200087, 0, 0, 0, -16513, 0, 0, 0, -383, 0], [0, 0, 142248, 0, 0, 0, 43277, 0, 0, 0, 3651, 0, 0, 0, 85, 0], [0, -363776, -6938696, 3861388, 0, -65314, -1871100, 1097977, 0, -190, -150788, 89831, 0, 62, -3460, 2069], [0, -105076, 0, 1111158, 0, -64885, 0, 332573, 0, -7171, 0, 28355, 0, -185, 0, 667], [0, 373092, 1138224, -3956990, 0, 102545, 367517, -1132921, 0, 9247, 32931, -94407, 0, 229, 797, -2199], [-174968, 373092, 0, 3956990, -119764, 102545, 0, 1132921, -10380, 9247, 0, 94407, -220, 229, 0, 2199], [174968, 373092, 0, 3956990, 119764, 102545, 0, 1132921, 10380, 9247, 0, 94407, 220, 229, 0, 2199], [-161088, -893432, 0, -9498920, 388452, 9148, 0, -2628785, 56988, 17924, 0, -212783, 1604, 644, 0, -4873], [0, -373092, 1138224, 3956990, 0, -102545, 367517, 1132921, 0, -9247, 32931, 94407, 0, -229, 797, 2199], [161088, -893432, 0, -9498920, -388452, 9148, 0, -2628785, -56988, 17924, 0, -212783, -1604, 644, 0, -4873], [0, 1109960, 1166488, -11775368, 0, 270404, 464630, -3363819, 0, 18684, 47178, -278645, 0, 396, 1198, -6467], [-108264, 583244, 0, 6179306, -139776, 232315, 0, 1798067, -18304, 23589, 0, 151117, -520, 599, 0, 3533], [0, -637920, 0, 6775756, 0, -65314, 0, 1898325, 0, -190, 0, 155883, 0, 62, 0, 3601], [-108264, -583244, 0, -6179306, -139776, -232315, 0, -1798067, -18304, -23589, 0, -151117, -520, -599, 0, -3533], [-108264, -947020, 0, -10040694, -139776, -297629, 0, -2896044, -18304, -23779, 0, -240948, -520, -537, 0, -5602]];
    Rf_basisdens := [1, 34268, 5272, 274144, 68536, 137072, 137072, 137072, 274144, 137072, 274144, 274144, 137072, 137072, 137072, 137072];
    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_156_h();"
function MakeCharacter_156_h()
    N := 156;
    order := 2;
    char_gens := [79, 53, 145];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_156_h_Hecke();"
function MakeCharacter_156_h_Hecke(Kf)
    N := 156;
    order := 2;
    char_gens := [79, 53, 145];
    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, 0, 0, 0, 0, -1, 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, 1, 0, -1, -1, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, -1], [0, -1, 0, 1, 1, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 1, -1, 2, 0, -2, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0, -1], [0, 1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0], [0, 0, -2, 2, 0, -1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0], [0, -1, 0, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, -2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0], [0, -1, 0, 1, 1, 2, 0, 0, 0, -1, 0, -1, 0, 3, 0, 0], [0, 5, 0, 0, 0, -3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 2, -4, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, -5, 0, 0, 0, 0, 1, 0, -1, 0], [0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, 3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4, 0, -2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, -2, 0, 0, 2], [0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 0, 0, -4, 0, 4, 0], [0, 0, 0, 0, 0, 0, -2, 2, -4, 0, 4, 0, -2, 0, -2, 0], [0, 0, 0, -2, 0, 1, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, -3], [0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, -1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 2, -2, 2, 0, -2, 0, 1, 0, 1, 0], [0, 0, 0, 0, 7, -5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3, 0, -3, 0], [0, 0, 0, 0, -3, 5, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0], [0, 7, 0, -7, -7, -3, 0, 0, 0, -4, 0, 7, 0, 1, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0], [0, 5, 0, -5, -5, -2, 0, 0, 0, -3, 0, 5, 0, 1, 0, 0], [0, 1, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 4], [0, 0, 0, 0, 0, 0, -2, 2, -2, 0, 2, 0, -1, 0, -1, 0], [4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0], [0, 0, 0, 0, 0, 0, 1, -1, -2, 0, 2, 0, -1, 0, -1, 0], [0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, -3, 0, 0, 0, 0, -3, 0, 3, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, -1, -5, 0, 0, 0, 4, 0, 1, 0, -9, 0, 0], [0, 9, 0, 0, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [0, 0, -6, -2, 0, 1, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -3, 6], [0, -1, 0, 0, 0, -7, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0], [0, 0, 4, -4, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, -3, -2, 0, 2, 0, -1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 3, 0, 1, -4], [0, 0, 0, 0, 0, 0, 4, -4, 4, 0, -4, 0, 2, 0, 2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, -3, 0, -4, 7], [0, -5, 0, 5, 5, 5, 0, 0, 0, 0, 0, -5, 0, 5, 0, 0], [-14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0], [0, 0, 0, 0, 3, 9, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 5, 0, -5, 0], [0, 0, 0, 0, 1, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 3, 0, -3, -3, -2, 0, 0, 0, -1, 0, 3, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -5, 5, 2, 0, -2, 0, 1, 0, 1, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0], [0, 0, 0, 0, 3, -5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 1, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -3, 0], [0, 0, 0, 0, 0, 0, -1, 1, 4, 0, -4, 0, 2, 0, 2, 0], [0, 0, -2, 2, 0, -1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, -13, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, -4, 0, 4, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [0, -1, 0, 1, 1, 2, 0, 0, 0, -1, 0, -1, 0, 3, 0, 0], [0, -11, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 3, 0, 6, -9], [0, 0, 4, -4, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, -1, 15, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0], [0, 0, 6, -4, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, -2, 10, 0, -10, 0, 5, 0, 5, 0], [0, 0, 0, -2, 0, 1, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0], [0, -11, 0, 0, 0, 5, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 11, 0], [0, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0, 0, 6, 0, -6, 0], [0, 0, 0, 0, 0, 0, 2, -2, 2, 0, -2, 0, 1, 0, 1, 0], [0, 0, 0, 0, -5, -3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, -12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0], [0, -5, 0, 5, 5, 0, 0, 0, 0, 5, 0, -5, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 11, -11, 6, 0, -6, 0, 3, 0, 3, 0], [0, -5, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 1, 1, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, -7, 7, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 12, -12, 4, 0, -4, 0, 2, 0, 2, 0], [0, 0, 0, 0, 7, 15, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 3, -5], [0, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0, 0, 4, 0, -4, 0], [0, 0, 4, 2, 0, -1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -18, -18, 0, 0, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 3, 10, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 6, -6, 2, 0, -2, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 3, 0, 1, -4], [0, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, -6, 0, 6, 0], [0, 0, 2, 4, 0, -2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0], [0, -5, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 3, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 0, -8, 0, 4, 0, 4, 0], [-10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 5, 0], [0, 0, 0, 0, 0, 0, -6, -6, 0, 0, 0, 0, 2, 0, -2, 0], [0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7, -3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0], [0, -11, 0, 11, 11, 8, 0, 0, 0, 3, 0, -11, 0, 5, 0, 0], [0, -7, 0, 7, 7, 1, 0, 0, 0, 6, 0, -7, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -3, 0, 5, 0, 2, -7], [0, 13, 0, -13, -13, -10, 0, 0, 0, -3, 0, 13, 0, -7, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 8, 0, -4, 0, -4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 5, 0, 6, -11], [0, 0, 0, 0, 0, 0, -15, 15, -6, 0, 6, 0, -3, 0, -3, 0], [0, 0, -4, -4, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [-4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, -3, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, -4, 0, -6, 10], [0, 1, 0, 0, 0, -16, 0, 0, 0, -16, 0, 0, 0, 0, 0, 0], [0, -7, 0, 7, 7, 10, 0, 0, 0, -3, 0, -7, 0, 13, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, -5, 0, -6, 11], [0, 0, -8, 12, 0, -6, 0, 0, 0, 6, 0, 12, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, 7, 2, 0, -2, 0, 1, 0, 1, 0], [0, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0, 0, 1, 0, -1, 0], [0, 0, -6, 6, 0, -3, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0], [0, 11, 0, -11, -11, -12, 0, 0, 0, 1, 0, 11, 0, -13, 0, 0], [0, 5, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -22, -22, 0, 0, 0, 0, 4, 0, -4, 0], [16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, -8, 0], [0, 0, 0, 0, -7, -7, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 14, -4, 0, 2, 0, 0, 0, -2, 0, -4, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, -3, -3, -8, 0, 0, 0, 5, 0, 3, 0, -13, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 5, 0, -5, 0, 0, 5], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -5, 0, 6, 0, 1, -7], [0, -7, 0, 7, 7, 2, 0, 0, 0, 5, 0, -7, 0, -3, 0, 0], [0, 0, 0, 0, 1, -10, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 17, 0, 0, 0, -8, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0], [24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 4, 2, 0, -1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -6, 0, -6, 0, -2, 0, -8, 10], [0, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, -1, 0, 1, 0], [0, 1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -2, -2, 0, 1, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 8, 0, -3, 0, 5, -2], [0, 0, 0, 0, 0, 0, -9, -9, 0, 0, 0, 0, 12, 0, -12, 0], [0, 0, 0, 0, 0, 0, 3, -3, -6, 0, 6, 0, -3, 0, -3, 0], [0, 0, 0, 0, 0, 0, 21, 21, 0, 0, 0, 0, -2, 0, 2, 0], [0, 0, 0, 0, 0, 0, 5, -5, 6, 0, -6, 0, 3, 0, 3, 0], [0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, -15, 0, 15, 0], [0, 0, 0, 0, 13, -15, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0], [20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 14, 0], [0, 0, 0, 0, 31, -14, 0, 0, 0, 14, 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, 0, 0, 0, 0, 6, 0, 6, 0, -3, 0, 3, 0], [0, -11, 0, 11, 11, 3, 0, 0, 0, 8, 0, -11, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, -7, 7, -4, 0, 4, 0, -2, 0, -2, 0], [0, 0, 0, 0, 13, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 7, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, 0, 14, 0, -14, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_156_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_156_2_h_b();".
// 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_156_2_h_b(:prec:=16)
    chi := MakeCharacter_156_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_156_2_h_b();".
// 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_156_2_h_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_156_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 0, -5, 0, 1]>],Snew);
    return Vf;
end function;