
// Make newform 6354.2.a.bl in Magma, downloaded from the LMFDB on 08 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6354_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_6354_2_a_bl();".
// 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_6354_2_a_bl();".
// 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 := [-1024, -1024, 1584, 1056, -946, -340, 239, 44, -26, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-158336, -242800, 16352, 209218, 32820, -62967, -8076, 7594, 498, -313], [-38944, -65808, 21826, 57484, -1807, -17400, -470, 2090, 55, -84], [-401088, -679568, 196312, 578866, -18656, -162159, -2548, 18146, 342, -689], [-943616, -1576272, 519808, 1350878, -80772, -376617, -212, 41814, 542, -1575], [1070848, 1701712, -573856, -1511894, 77380, 431837, 2820, -48590, -742, 1843], [64752, 112868, -32686, -97506, 2997, 27474, 518, -3088, -65, 118], [489280, 919248, -232544, -804666, 21180, 228227, 3676, -25634, -474, 973], [-1500160, -2227664, 928992, 1867046, -173668, -512501, 7516, 56030, 374, -2075]];
    Rf_basisdens := [1, 1, 13184, 1648, 6592, 13184, 13184, 824, 6592, 13184];
    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_6354_a();"
function MakeCharacter_6354_a()
    N := 6354;
    order := 1;
    char_gens := [4943, 2827];
    v := [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_6354_a_Hecke(Kf)
    return MakeCharacter_6354_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]];
    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 := [[1, 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, -1], [0, 1, 0, 1, 0, 0, 0, 0, 1, 0], [0, 1, 0, 1, 0, 1, 0, 1, 1, 0], [-1, 0, 1, 0, 0, 1, 1, 0, 0, 0], [1, 0, 1, 0, -1, 0, 1, -1, 0, 1], [2, 0, -1, 0, -1, 0, 0, 1, 0, 0], [1, 0, -1, 0, 0, -1, 1, -1, 0, 0], [1, -1, 0, 0, 1, 1, 0, 0, 1, 1], [0, 0, 0, -2, 2, 0, 0, 1, 0, -2], [-1, 0, 2, 0, 0, 0, 1, -1, -1, 0], [0, 1, 1, 1, 0, 0, 0, 0, -2, 0], [1, -1, 0, 0, 1, -1, 0, 0, -2, 0], [1, 2, -1, 0, 0, -1, -1, 1, 0, 0], [4, 0, -1, 1, 0, 1, -1, 1, -1, 1], [1, -2, 1, -2, 1, -2, -1, -1, 1, 0], [-1, 4, 1, 0, -1, 0, -1, -1, 1, 0], [0, 1, -2, 0, 1, 1, -1, 0, 0, -1], [1, -1, -2, 0, 1, 1, 1, 0, 0, 0], [0, 1, -2, 1, 1, -2, -2, -1, 0, -2], [4, -1, 0, 1, -2, 0, 0, 0, 1, 2], [1, 0, 3, -2, 1, -2, 1, 1, -2, -1], [3, 0, 0, -2, 0, -2, -1, 0, -1, -2], [0, -3, 0, -1, -1, -2, 2, 1, -2, 2], [2, 3, 2, -1, -1, -1, 0, 0, 0, 0], [-5, 2, 1, 0, 1, 0, 2, 1, 2, 1], [5, -1, -1, 2, 1, 0, 0, -1, 0, -1], [-3, -3, 2, -2, 0, 0, 0, -2, 0, 1], [5, -3, 3, -2, 0, -1, 2, -1, -2, 0], [-5, 2, 0, 0, 1, 1, 1, 2, -2, 0], [1, 4, -2, 2, -1, 1, -1, 2, 1, -1], [3, 0, 0, -2, 3, -1, 1, 0, -2, 2], [-2, -1, 1, -2, -1, 0, 1, -1, -1, 3], [4, -4, -1, -2, 3, 0, 1, -1, 1, 1], [-1, -2, 1, -4, 3, 0, 0, -3, -2, -1], [-7, -3, 2, -2, -2, 2, 2, 0, -2, 1], [2, -5, -1, 0, -1, -2, 1, -1, 1, 1], [1, 2, 0, 0, 4, -2, 1, 0, 1, -2], [9, 1, 1, -2, -5, 0, 1, -3, 1, 2], [5, 1, -3, 2, 1, 0, -2, 3, -2, -1], [-5, -1, -2, 0, 0, -2, 0, -4, 0, 1], [0, -2, 3, -2, -2, 3, 2, 3, -1, 2], [0, -2, 1, 0, 0, -1, -2, 1, -3, 0], [7, 5, 0, 0, -1, 1, -2, 0, 3, -1], [0, 3, -3, 4, 0, 1, -2, 1, 1, 0], [2, -2, 1, 2, -4, 3, 2, -3, -1, 4], [-3, 2, -2, 2, 1, 1, 1, 4, -2, 2], [6, -3, -2, 2, -2, 2, 1, 0, -3, 3], [-6, 2, 1, -1, 1, 4, 3, 3, 0, -1], [2, -1, 3, 0, 3, 0, 1, 1, -1, 3], [-7, 6, 1, 2, -3, 2, -1, 0, 1, 2], [-2, 1, 5, -2, -1, 0, -1, 3, 1, -1], [-4, 5, 2, 0, 2, -2, -1, 0, 3, -3], [8, 0, 2, 0, -4, 0, -4, 0, -2, -2], [-1, -2, -1, 0, 1, 4, -3, 2, 1, 4], [8, 0, 1, 0, -2, 3, 2, -1, -1, 0], [0, 3, 3, 4, 3, 4, 1, 5, 1, -1], [-2, -5, -1, -6, 3, -4, 1, -3, -1, -5], [0, 0, 3, -3, -1, -3, 1, -2, -2, -3], [3, 2, -4, 1, 0, -3, -4, -3, 3, -1], [4, -2, 1, -2, 0, -1, 0, -3, -1, 0], [-3, -2, -1, -3, -3, -3, -2, -4, 3, 3], [-9, 2, 1, -2, 4, 3, -1, -1, 4, -2], [-2, 4, -1, -2, 2, -1, -2, 3, 1, 0], [-5, 3, 2, 0, 5, 1, 0, 2, 1, 1], [7, 3, 0, 2, -3, 3, -4, -2, -2, -4], [-4, 2, 0, -2, 2, 2, 0, -2, -2, -2], [-7, 0, 1, 2, -1, 2, 3, 3, -1, 0], [-1, 0, 1, -4, 3, -2, -3, 1, -1, 2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, -2, -3, 2, 1, -2, -3, 3, -1, -1], [-1, -5, 0, -2, 3, 1, 6, -1, -1, 1], [4, 2, -3, 0, 0, -3, -2, 0, -1, 0], [-2, 4, -5, 4, 0, -1, 2, 1, 5, -2], [-1, -2, -7, 2, 2, 3, -5, 3, 4, 0], [3, -5, -3, -6, 5, -8, 0, -1, -2, -5], [-6, -3, -1, 2, 3, 2, -1, 3, 1, 1], [-3, -2, -2, 4, 2, 0, -3, 2, 1, 0], [2, 3, 0, 4, -5, 3, 1, 2, -2, 1], [1, -3, -1, 0, -1, -2, -2, -1, -2, -7], [6, 4, -4, 0, -2, 2, -2, 0, -2, -4], [-7, -2, 0, 0, 2, 0, 3, -2, 3, -2], [-2, 2, 6, -2, -4, 2, 2, -4, 2, 6], [5, 2, -6, 6, 1, 3, -1, -2, 6, -2], [1, 0, -3, 2, -7, -2, 1, -1, 1, 0], [4, -12, -4, -2, -1, 3, 2, -2, -1, 4], [6, -2, 3, 2, -2, -1, -2, 1, -1, 6], [-5, -1, 1, -3, 4, 1, 1, 3, -3, -2], [3, -2, -4, -1, -3, -5, -4, -4, 2, -1], [2, 3, 4, 2, -6, -2, 3, -4, -1, 7], [-12, 6, 2, -1, -4, 2, -1, 2, -2, -1], [-1, -7, 6, -4, -1, -1, 4, -5, -5, 1], [2, -3, -4, 0, -2, -4, -5, -2, -1, -3], [8, -2, 6, -3, 0, 1, -3, -3, 0, 3], [3, 0, -1, 0, -1, 2, -1, -2, 5, 2], [-4, -2, 3, -8, 0, 1, 4, -3, -1, -4], [1, -4, -4, -4, -3, -1, -1, 0, -3, -1], [4, 3, 2, 6, -1, 5, -1, 2, 0, -1], [-3, 0, 3, -4, 3, -2, 5, 1, -1, 2], [-1, -2, 3, -4, -5, 4, 1, 1, 3, 4], [5, -1, 3, -3, 4, -1, 3, 1, 1, -2], [0, -3, 0, 0, -4, 0, -5, -2, 3, 1], [0, 0, -4, 4, -5, -1, 0, -4, -5, 2], [12, -1, -3, 8, 2, 5, -5, 3, 7, 4], [0, 2, -7, 6, 0, -3, -8, 1, 3, 0], [4, -11, 0, -6, -2, -4, 3, -6, -5, 9], [4, -2, 5, 1, -5, -3, 5, -2, -4, 1], [-2, -5, -1, -7, -1, 0, 0, -5, -1, -4], [6, 1, 3, 2, -5, 10, 3, 3, 1, 9], [-10, 0, 6, -2, -2, -2, 2, -2, -2, 2], [-9, -1, 4, 0, 0, 0, 0, -4, 8, -1], [5, 6, 0, 8, -3, 5, 1, 6, 2, -4], [-14, 10, 6, 2, 4, 4, 2, 2, 2, 0], [2, 2, -9, 6, 2, 1, 0, 8, 1, -4], [10, 1, -7, 6, 1, 4, -5, 1, 5, 1], [-4, -10, -5, -2, 0, -5, 2, -1, -3, 2], [-4, -7, -1, 1, 3, 2, 2, -5, -1, 2], [-12, -1, 6, 0, 0, 0, 4, -2, 0, 6], [10, 0, -6, 2, 2, 2, -4, 0, -2, -4], [0, 3, -9, 4, -4, -3, -2, 1, 1, 0], [-6, 1, 6, 0, 1, -5, 5, -6, 2, 1], [9, -7, -5, -2, 7, 0, 2, 1, 2, -7], [6, -5, -9, 0, -1, -2, -3, 1, 5, 3], [-1, 2, 5, 2, 3, 0, -1, -3, 1, 0], [11, -5, 3, -2, -2, -3, 4, -4, -1, 7], [-11, 0, -2, -8, 6, -6, -1, -1, -1, -4], [-1, -1, 1, -3, -6, -1, 3, 1, -5, 4], [2, -12, -2, 0, -2, 0, 2, -2, -2, -4], [5, -7, -1, 0, 0, -5, -3, -3, -2, 2], [0, 2, -4, -2, 4, 2, -2, 2, -4, -6], [-7, -1, 2, 0, -1, 3, -3, 4, -2, -4], [5, 0, -4, -2, 5, 1, -10, 0, 7, -3], [-13, -7, 6, -6, 2, -2, -1, -8, -1, 4], [-8, 8, 0, 4, -7, 5, -4, 4, -1, 0], [-14, -3, 4, -10, 5, -1, 3, -2, -6, 1], [5, -1, -1, -4, 2, -3, 2, -10, 1, -3], [-2, 4, 5, 2, -3, 2, 2, 1, -4, 0], [7, 1, 2, 6, -3, 3, 0, 4, 1, -1], [1, 2, 6, 2, -2, 2, -7, 6, -5, 4], [-10, -4, 2, -4, 6, -6, 2, 2, -8, -2], [-2, 1, -2, 3, -5, -5, -6, -4, 2, -2], [-1, 6, 3, -1, 7, -2, 2, 3, 5, -1], [0, 11, 4, 4, -5, 7, 5, 0, 6, 5], [-9, -3, 2, 2, 0, -2, -3, 4, -1, 6], [9, -4, -7, -2, -1, 0, -11, 0, 5, -4], [-3, 2, 2, 3, -5, 1, 2, 0, -4, 9], [12, -2, -2, 10, -7, 3, 0, 0, 5, -2], [-7, 4, 4, -1, -4, -3, 4, 1, 1, 1], [-2, -2, 9, 2, -2, -3, 8, -7, -1, 10], [7, -1, -6, -4, 6, -4, -6, -4, -2, -11], [5, -4, 4, -8, -5, 1, 11, -2, -4, 0], [-2, 0, -6, -1, -6, -5, 1, 1, -4, -3], [-9, -4, -3, -2, 8, 7, 1, 11, -8, -6], [3, -2, -2, -1, -8, 3, 4, -1, -1, 3], [-2, -11, -3, -3, 1, -2, -4, 5, -3, -10], [1, -2, -4, -4, -7, -1, -3, 2, 0, 0], [6, 0, -2, 6, 2, 4, -4, 8, 4, 0], [-1, -4, -6, 6, -5, -5, -1, -4, 0, 8], [10, 8, -5, 8, -4, 1, -4, 6, -1, 0], [2, -9, -2, -4, 2, -8, 1, -2, 1, -3], [20, 2, -4, 0, 5, -1, -8, 2, 1, -10], [-8, -6, 2, -4, 8, -8, -2, 2, -4, -2], [10, -3, -2, -2, 3, -1, -1, -4, 0, 3], [-8, 0, 2, -5, 9, 0, -3, 1, -3, 3], [-19, 4, 12, -6, 0, -2, 1, 2, -1, 2], [1, -5, 1, -4, -4, 3, 2, 1, -3, 1], [-3, 0, -5, 9, 5, -4, 2, 1, 3, -7], [-13, -4, 0, -6, 3, -7, -5, -2, -2, 0], [21, -3, -9, -2, 3, -2, 2, -3, 4, -1], [8, 3, -4, 2, 0, -4, -7, 2, -7, 1], [-2, -1, 3, 0, 1, 0, 1, -1, 5, -11], [-9, 2, -7, 10, -4, 1, -2, -5, 3, 5], [10, 5, 2, 6, -7, 1, 1, 4, -2, 11], [-21, 4, 10, -6, -1, -1, 8, -2, 1, 9], [-5, 8, -2, 0, 1, -1, 1, 8, 0, -2], [6, 1, -2, 4, -1, -1, 1, 6, -1, 2], [11, 5, 2, 2, -3, 3, -6, 2, 9, -1], [-2, -1, -5, 4, 3, 8, -9, 1, 5, 1], [-3, 10, -4, 14, 0, 2, 1, 7, 5, 4], [13, -11, -5, 0, 5, 0, -2, 0, -2, 5], [3, 3, -1, -2, -2, -1, 4, 1, 0, 8], [-3, 3, 0, 8, 4, 2, -8, 8, 0, 1], [3, -6, 5, -2, -1, 0, 3, -7, 2, -1], [3, -2, 1, 4, -2, 1, 1, 1, -2, -4], [-5, 1, 0, 0, 3, 7, 0, 2, 7, 3], [1, 0, 1, -4, -3, -2, -7, -5, -1, -2], [-11, -2, -1, 3, 9, -1, -4, 2, 3, -11], [5, -2, 4, 2, 0, 8, 5, 0, 2, 3], [-13, 10, -5, 2, 6, -1, 1, 1, 4, -6], [-16, 3, -1, -2, 8, -1, -1, -5, 8, 5], [-2, -1, 8, 2, -4, 2, 5, 6, 1, 7], [22, -7, -9, -2, 1, -12, -3, -5, -1, 3], [-11, 3, -2, 4, -1, 1, 0, -8, 3, 7], [8, -5, 3, -3, 2, 0, 2, -4, 0, -4], [4, -6, 0, -6, 1, 3, 4, 4, -1, 2], [10, 7, 10, 4, 1, 9, 9, -2, 4, 11], [17, -1, -1, 4, -4, -3, -3, -3, -2, 2], [-12, 2, 0, -2, -3, 1, -4, 0, -3, 4], [-7, 2, -6, 4, 2, 2, -5, 0, 7, -8], [16, -8, -5, 2, -4, -11, -2, -8, 1, 6], [13, -10, -2, -8, -1, -7, -3, -6, -6, 6], [-2, 3, -2, 2, -2, 8, -8, 4, 6, 2], [10, -3, 6, -4, -7, -7, 5, -4, -6, 1], [2, -13, -5, -5, 6, -8, -8, -4, -8, 0], [-8, -8, 2, -2, -8, 4, 2, -8, -8, 6], [6, -1, -9, -1, 6, -7, -6, -1, -4, -8], [-5, -2, 1, -4, 5, 4, -1, -1, -5, -6], [4, 2, 14, 0, -5, 5, -4, 2, -1, 4], [13, 5, -12, 10, -3, 3, -8, -2, 10, -2], [10, 5, 0, 8, -7, 9, 3, 2, 2, -1], [-9, 19, 2, 10, -3, 7, 2, 8, 7, -3], [1, -4, -2, -10, 3, -9, -1, 2, -6, -12], [-8, 1, 8, 4, -4, 4, -3, 0, -7, 3], [-11, -4, 4, -2, 0, -2, -1, 2, -3, 8], [-5, -9, 3, -6, 9, -6, 4, -1, -6, 1], [5, -8, 4, -6, -5, 1, 8, 0, -5, 11], [-6, -2, 2, -8, 1, -9, 4, -2, -1, -4], [-15, 13, 2, 4, 4, 8, 4, 9, 6, -3], [-20, -4, -2, -2, 4, 4, 0, -2, 0, -6], [26, 4, -8, 6, -6, 0, -6, 3, 6, 4], [-18, 1, 2, 1, -6, -2, 0, -2, -13, 0], [-8, 10, -4, 2, 3, 1, 10, 0, 1, 2], [-11, -8, -4, -12, 11, -5, 5, 4, -6, -4], [1, -7, -1, -2, 0, -7, 4, -3, 2, 2], [-17, 9, -2, 10, -10, 10, 2, 4, 2, 15], [-4, -1, 8, -3, 6, -5, 6, 7, 1, 8], [26, -4, -5, 2, 1, -8, 1, 5, -3, -9], [5, 8, 6, -4, -1, 1, -1, -6, -2, -2], [-9, 4, -7, -2, 3, 2, -1, -9, 8, -5], [5, 4, 5, 8, -3, 0, -1, -3, 5, 8], [-11, -10, 5, -9, 5, -5, -2, -6, -5, -5], [-11, 7, -8, 12, 3, 5, 4, 11, 5, 7], [-13, -8, 0, -8, 3, -9, 3, -8, 0, 2], [2, -1, -1, -2, 1, -12, -1, 7, 1, -3], [-8, 2, -5, -2, -10, -1, -4, -8, 1, -6], [6, 4, -1, 8, 0, 5, 2, -1, 9, -4], [-10, -6, 2, 0, -2, -4, -6, 2, 0, 8], [6, -19, -2, -8, 8, -8, 9, -4, -5, 5], [7, 12, 1, -8, -2, 3, -5, -5, 4, -10], [21, 0, -10, 4, -5, -3, 1, 4, -3, -3], [10, -2, -3, 8, -6, -3, -4, -5, 3, -4], [-20, -1, 9, -8, 5, 8, 9, -3, 3, 1], [-1, -2, 0, -11, 5, -1, 0, -2, -6, -5], [9, 2, 2, -6, -2, -12, -3, -8, -5, -4], [8, 10, -3, 2, 5, 0, -1, -1, -3, -5], [-13, -2, -2, 2, 1, -1, -9, 0, 4, -6], [-18, -2, -6, -3, 3, -4, -1, 1, -5, -3], [6, -8, -4, -7, -4, -5, 1, -1, -6, 5], [-4, 0, -7, -2, 4, -3, 12, -5, -1, 2], [-8, 9, 11, 0, -1, -4, 11, -1, 9, 1], [-20, 4, 11, 4, 4, 13, 2, 5, -5, -10], [0, 3, 2, 0, -6, 2, -4, 0, 0, -2], [4, 1, -4, 7, 3, 10, -2, -7, -2, -2], [-1, 1, 5, 10, -3, 8, 6, 3, 0, 5], [8, -8, 9, -1, -13, 9, 11, -8, -8, 13], [3, -1, 4, -4, 10, 0, -2, 6, -8, -7], [14, 2, -12, 10, 0, 8, -6, 6, 12, 8], [15, 1, -5, 2, 7, -4, -4, 9, 2, -9], [-2, 0, 0, -12, -4, -6, -2, 0, -10, -6], [-12, -6, 0, -8, 2, 4, -10, -1, 2, 2], [-16, 2, 4, -8, -2, -2, -6, -12, 8, -2], [-20, 0, 9, -1, 3, 4, 5, 7, -6, 1], [-21, -2, 9, -4, 13, 0, -3, 1, 5, -6], [-15, 3, -8, 0, -6, -4, -4, 4, 7, 2], [-4, -3, 2, -2, 5, -5, -5, -6, 0, -3], [3, 2, -4, 0, -2, 4, -2, -4, -4, -9], [4, -9, -1, -1, 0, -6, -6, 2, -4, -2], [6, 0, 2, 5, -4, 3, -5, 3, -6, -1], [17, -13, -3, 4, -9, -4, -4, -5, -8, 7], [-1, 4, 11, -3, 1, -10, 10, -3, 1, -3], [1, 10, -7, 5, 3, 7, -8, 12, -3, -11], [22, 13, -7, 2, -3, 2, -3, -1, 7, 3], [18, -7, -4, -2, -10, -6, 0, -8, -6, 2], [-6, -9, 7, -6, -9, 8, 9, -5, -11, 1], [-13, 9, -5, 2, -14, 1, -10, 1, 5, -3], [-13, -11, -8, 2, -4, 0, -8, -8, -3, 4], [16, -8, 7, -11, -7, 2, 5, -3, 2, 7], [5, 0, 14, 0, 0, 6, 5, 0, 0, 1], [-2, 0, 12, -8, 2, 6, 2, 4, -4, 0], [-13, 0, -2, 6, 0, 6, -11, 2, 7, 2], [6, -1, -5, 1, -2, 7, 2, -9, 12, 6], [8, -4, -1, -2, -2, -13, -4, -1, -1, -2], [12, -2, 2, 3, 5, 5, -3, -4, 5, 7], [10, -1, 0, -4, -3, -1, 11, 2, 0, -5], [25, -3, -8, 4, 5, -3, -13, -2, -4, -10], [6, 11, 3, 0, -13, -2, -3, -11, 5, 3], [-17, 5, -3, 2, 9, 8, 9, 9, 3, 4], [-6, 13, 7, 7, -4, -1, 0, 3, 4, -14], [22, -6, -2, -10, -8, -2, -2, -2, -2, 10], [10, -8, 13, -10, -4, -9, 4, -11, -11, 0], [-14, -7, 0, -6, 14, -4, 3, 2, -7, -7], [-6, -4, 0, -8, -1, -9, -8, -8, -7, 2], [-32, -6, 4, -8, 8, -4, 2, -4, -10, -6], [5, 11, 2, 6, 2, 6, 0, 6, 10, -11], [10, 12, 0, 4, -2, 0, -8, 0, 2, 0], [25, -14, 6, -12, -7, -1, 3, -14, -6, 0], [-3, 2, 0, 8, -9, 5, -5, -7, 12, 6], [16, -1, -10, 12, -4, 0, -7, 8, 7, 7], [8, 1, 1, -2, 6, -3, 3, 9, -13, -10], [10, -12, 0, -4, 12, -2, 0, -6, -2, 2], [-22, -2, 3, 4, -4, 3, 0, 3, 3, 14], [-25, -1, 14, 0, 9, -3, 10, 1, 7, 7], [14, 7, -6, 2, 0, 4, -9, 8, -5, -5], [20, 9, 0, 12, 1, 1, -5, 14, -2, 7], [2, -8, 2, -2, 10, -2, 6, -6, -8, 6], [4, 1, 6, 1, -4, -3, 6, -3, 1, 14], [-9, -6, -10, 2, 4, -4, -1, -5, -3, 4], [-9, -14, 0, -5, -5, 1, 8, 4, -12, 11], [1, -6, 3, -2, -7, 8, 7, 9, -11, 8], [3, -1, 4, -6, 0, -6, -4, -3, 8, -7], [22, 4, 8, 2, 4, 6, 0, 2, -4, 14], [5, -10, -2, -10, -6, 0, -11, -6, 7, 8], [-7, -4, -4, -6, 4, -6, -7, 6, -7, -10], [4, 2, -5, 6, 5, -2, -1, 9, -1, -5], [-19, -12, -1, -7, 2, -2, 6, 0, -6, 15], [7, 10, -5, 4, 9, -2, -5, 15, 2, -5], [-6, 6, -4, 3, -6, 2, -3, 4, -4, 3], [0, 8, -9, 10, -6, 5, -8, 5, 1, 2], [-4, 18, -8, 14, 2, 2, -16, -2, 14, -6], [10, 9, -2, -4, 7, -11, -7, 8, 4, -1], [13, -2, 1, -7, -2, 0, -4, -6, 2, 9], [12, -2, 9, 0, -6, -1, 10, -11, 5, 6], [7, -12, 0, 4, -7, 3, 5, -2, -2, 4], [-12, 16, -1, 14, -2, -1, 6, 3, 5, 2], [12, 1, 5, -8, -8, -15, -3, -13, -4, 3], [-1, -3, 0, -6, 1, -1, 2, -2, 0, -2], [16, 6, -2, -4, 18, -12, -4, 10, 4, -16], [-6, 4, -13, -8, 10, -1, -4, 8, 5, -18], [11, -12, -6, 3, 13, -3, -4, -4, -6, -1], [10, 5, 11, 4, -4, 7, 0, 1, -5, 12], [-8, -9, 6, -13, 1, -1, -4, -8, -4, 4], [-9, 13, -1, 4, -2, -5, -2, 5, 5, -9], [13, 7, 3, 2, -1, 10, -10, 13, -4, -11], [12, -12, 1, -2, 14, -3, 0, 1, 5, -6], [33, 9, 1, 8, -1, -2, -6, 5, 8, 5], [8, -7, -2, -2, -10, -2, -2, 0, -12, -6], [-5, -7, 23, -8, -2, -3, 14, 1, -12, 0], [14, -15, -7, -2, 1, 6, 5, -1, 1, 1], [3, -16, -8, -4, -1, -17, -9, -8, -10, -2], [0, 3, -9, 8, -4, 3, 2, 1, 1, 4], [26, -10, 4, 0, 0, -2, -2, -6, 4, 0], [19, 7, 2, 4, -7, 5, 8, 4, -1, -5], [5, -9, 3, -12, 9, -4, 0, -4, -8, -13], [21, 6, -8, 5, -5, 11, -4, 14, 14, -1], [20, -1, -8, 6, 6, 2, 3, 12, -5, -3], [15, -1, -8, 4, -10, 8, -8, 10, 4, 1], [24, 1, 3, -6, 11, 0, -5, -5, 1, -11], [1, 8, 6, 8, -2, 18, 3, -6, 11, 12], [-12, -1, 14, 0, 1, 11, -5, 7, -6, 1], [-4, 17, 0, 4, -1, 9, 7, 0, 11, -2], [-20, 23, 3, 12, 4, 3, -1, 9, 4, 1], [12, -3, -6, -4, -2, -4, 6, 2, -8, 4], [-10, -5, -3, 6, -5, -4, 1, -1, 7, 7], [4, 9, 3, 0, 9, 8, -1, 9, 1, -13], [5, 2, -13, 14, 5, 0, 1, 7, 4, 3], [9, -8, -4, -14, -2, -12, -3, -18, -7, -6], [12, 1, 6, -2, -1, -1, 9, 0, -2, 15], [10, 4, 6, -4, -9, -15, -4, -16, 1, -4], [10, -1, 6, 10, -3, 7, 5, -7, 12, 5], [25, -4, 5, -2, -5, -10, 11, 5, -5, -2], [5, -2, -9, 2, 9, -4, 1, 7, -8, -5], [33, -9, 0, -4, -8, -6, -6, 1, -10, -9], [-18, -9, -4, -2, 8, -4, 7, 2, 3, -3], [7, -9, 12, -6, -7, 7, 16, 2, -6, 16], [32, -12, 0, -6, -13, -3, -4, -4, 6, 7], [20, 10, -1, 10, -3, 0, -7, 7, -5, 9], [-30, 12, -2, 10, -4, 10, -8, 2, 4, -6], [21, -11, 6, -13, -2, -1, -5, -7, 0, 8], [18, 6, 20, 0, -8, 0, 12, -4, 0, -2], [-24, -7, -11, 8, 4, 3, 5, 3, -7, 4], [-4, 0, 8, -4, 10, -4, 0, 6, 2, 8], [23, 1, -6, 8, 3, 7, -8, 14, 1, -1], [17, 0, 6, 3, 5, 7, -6, -6, 8, 5], [11, 3, 6, 2, -14, 2, 6, -10, -2, -5], [-4, 1, -6, 6, 0, 8, -3, -4, 13, 11], [-21, 2, 4, 4, -6, 6, 1, -6, -3, 10], [29, 3, -21, 12, -6, 9, -8, -1, 2, -4], [22, 2, 14, -4, 7, 1, 0, -6, -3, 4], [-23, -13, 5, -4, 3, -2, 2, -9, -8, 1], [32, -11, -4, 5, 5, 1, 0, -2, 8, 8], [-3, -4, -9, -2, 7, 2, -13, 3, 4, 1], [-7, 21, 5, 16, 4, 7, -4, 15, 7, -9], [18, 1, -10, 11, 6, 1, -2, 13, 1, -6], [0, -1, 5, 6, 4, 3, -7, 5, -1, -12], [-7, 17, 2, 12, 2, 10, 0, 3, 12, 1], [27, 1, 11, 2, 0, 3, -2, 7, -3, 3], [-5, 8, 0, 5, -9, 5, 6, 6, 18, 1], [22, -5, 14, -6, -3, -5, 7, -4, -4, 15], [11, -18, -5, -14, 3, -2, 5, -4, -13, 6], [2, 26, 2, 12, -1, 7, -2, 4, 16, -5], [15, 1, -7, 6, -17, 2, 2, -3, 2, 5], [-5, -3, -3, -8, 7, 4, 14, -9, 2, -7], [14, -5, -2, -2, 7, 3, -9, 2, 0, -7], [5, 8, 6, -4, 4, 2, 3, 2, -5, -2], [-1, 0, -21, 9, -6, 0, -6, 10, 10, 1], [-19, 11, -3, 4, 5, -2, -4, 8, -6, -1], [30, 10, 0, 8, -9, 5, -8, -4, 11, 4], [12, 2, -1, 6, -13, 6, -6, 3, -8, 6], [7, -7, 7, 0, -5, 4, 4, -5, 2, 13], [-16, 4, -6, -4, -2, 2, -12, -3, 2, -6], [3, -8, 0, -8, -2, 2, -5, 6, -19, 0], [-14, 1, 10, -14, 0, 2, -2, -10, -4, 4], [17, -9, 0, 2, 10, 0, -2, 6, 7, -8], [17, -7, -8, -10, 5, -5, -8, -8, 1, -5], [0, -2, -10, 8, -2, 10, -10, -5, -4, -2], [13, 10, -15, 17, -3, -6, -4, -1, 11, 3], [-23, 6, -2, 2, 3, -1, -9, -8, 5, 1], [-48, 0, 10, -8, -8, 6, -4, -10, -2, 12], [6, -1, 6, -8, 0, -6, 9, -10, -5, 13], [12, 7, -1, -10, 8, -5, -1, 7, -12, -15], [25, -5, -5, -5, 8, -19, -9, 1, -11, -4], [-3, 7, 12, -8, 6, -2, 6, 6, -9, -12], [2, 0, 2, -10, -6, -6, 6, -10, 6, 2], [8, -7, -5, -2, 7, -2, 1, 7, 5, -1], [-2, 11, 4, -5, 1, -9, 0, 2, 0, -8], [6, 5, -14, 8, -14, 2, -6, 4, 14, 14], [4, 4, 1, 6, -18, 7, 0, -4, 7, 2], [20, -8, -7, 12, 2, -3, 2, -1, 3, -6], [10, -15, 13, -9, -7, -8, 20, -7, -15, 8], [1, 17, -7, -6, 7, 2, -7, 1, 9, -6], [-2, -15, -9, 2, 1, -2, 5, 9, -3, -1], [-8, -7, 0, 0, 14, 8, 14, -6, 0, -4], [-44, 5, 6, -2, 4, -2, 5, -2, -5, 1], [-10, 5, -1, 14, -11, 8, -3, 3, 9, 3], [-15, 8, -6, -8, -1, -5, -7, -7, 4, -4], [20, 17, -6, 24, -4, 12, 1, 12, 15, -9], [1, -8, 4, -4, -3, -3, 5, -4, -6, 16], [0, 5, -7, -2, 7, -4, 11, -7, 9, -5], [23, 22, 14, 17, -11, 11, 0, -2, 12, 13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6354_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6354_2_a_bl();".
// 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_6354_2_a_bl(:prec:=10)
    chi := MakeCharacter_6354_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_6354_2_a_bl();".
// 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_6354_2_a_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6354_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![12, 44, -596, 430, 953, -332, -301, 116, 15, -10, 1]>,<7,R![1488, 6752, 8000, -312, -4200, -416, 812, 18, -51, 0, 1]>,<11,R![-256, 8576, -25264, 22912, -3130, -3662, 1005, 168, -60, -2, 1]>,<13,R![-16, -1600, 374, 7288, -8351, 1414, 1125, -190, -61, 4, 1]>],Snew);
    return Vf;
end function;