
// Make newform 6162.2.a.bj in Magma, downloaded from the LMFDB on 21 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6162_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_6162_2_a_bj();".
// 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_6162_2_a_bj();".
// 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 := [688, -1520, -228, 1684, -300, -596, 149, 84, -22, -4, 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], [-14416, 27580, 21368, -28004, -3966, 6621, -225, -458, 39, 5], [-103152, 93016, 58972, -81048, 5372, 15434, -3993, -483, 297, -31], [15528, 21052, -37460, 1168, 19066, -4429, -3318, 891, 180, -46], [32656, -96568, -75372, 78512, 42816, -21222, -7667, 2531, 407, -109], [8352, 5380, -7496, -12424, -29, 5019, 536, -679, -51, 28], [153488, -262480, -224020, 276080, 101864, -88956, -17783, 10819, 1011, -423], [304240, -359168, -417220, 358608, 174328, -113468, -28635, 13941, 1551, -561], [-304528, 470768, 362236, -461976, -144244, 137928, 23693, -15791, -1317, 591]];
    Rf_basisdens := [1, 1, 3592, 7184, 3592, 7184, 1796, 7184, 7184, 7184];
    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_6162_a();"
function MakeCharacter_6162_a()
    N := 6162;
    order := 1;
    char_gens := [4109, 5215, 6007];
    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;

function MakeCharacter_6162_a_Hecke(Kf)
    return MakeCharacter_6162_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+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 := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -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, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0, -1, 0, 0], [1, 0, -1, 0, -1, -1, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1, -1, 0], [1, 0, 1, 0, 1, 1, 1, 1, 0, 0], [1, 1, -1, 0, -1, 0, 0, 0, 0, -1], [2, -2, 1, 0, 0, 0, 0, 1, 0, -1], [3, -1, 1, -1, 1, 1, -1, 0, 1, 1], [1, 1, 1, 1, 0, -2, 1, -1, -1, -2], [2, -1, 0, 0, 1, 1, 1, 0, 0, 0], [3, 1, 0, 1, -1, 1, 1, -1, 1, 0], [0, 1, 0, -1, 0, -1, 0, 1, 0, 0], [2, 1, 1, 0, 0, 1, 1, 1, 0, 0], [1, 1, -3, 0, -1, -1, 0, 0, 2, 0], [0, -1, 2, 0, 1, 1, 1, 2, -1, -1], [3, 1, 0, -1, -1, 0, 1, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 1, 0, 0, 1, 0, -2, -1, 1], [1, 0, -1, 1, 0, -2, 0, -3, -1, -1], [2, 0, 2, 0, 0, 0, 0, 0, -2, 0], [4, 0, 2, 3, 0, -3, 1, -3, -1, -2], [1, -1, 0, 0, 0, 0, -2, 0, 1, 0], [3, 0, 1, -1, 1, 1, -2, 0, 0, 1], [3, -2, -1, 0, 1, 0, 1, -2, 1, 1], [4, -3, 0, -2, 0, 1, -1, 0, 0, -1], [0, 2, 1, 1, 1, -1, -1, 0, 1, 0], [0, 1, 0, -1, 0, 1, -2, -1, 0, 0], [-3, 4, -2, 2, -3, -3, 1, -2, 0, -1], [-1, 3, -1, 1, 0, 0, 1, -1, 1, 0], [2, 0, -2, 1, 0, 1, -1, 0, 2, -1], [4, 2, 0, 3, -2, -1, 1, -2, 2, -1], [4, 3, -3, -1, -2, 2, 0, 3, 1, 0], [-1, 4, -1, -1, 1, 2, -4, -1, 0, 2], [3, 1, -1, 1, -3, -1, 1, -1, -1, 1], [3, -2, 3, -2, 3, 3, -1, 1, -2, 2], [0, 3, -4, 1, -2, 0, -2, -3, 1, 1], [1, 0, -2, 0, -2, -2, 1, 0, 0, 0], [-3, -1, 1, -1, -1, 1, -1, 1, 1, 1], [-1, 1, -1, -2, 1, 0, 0, -2, -1, 2], [0, 3, -2, -1, 0, -1, 0, 1, 0, -2], [6, 0, 1, 3, -1, -1, 1, -2, 1, -4], [-1, 1, 0, 1, 2, -3, -1, -2, 0, -4], [5, 0, 2, 1, 1, 3, 2, 3, -1, -1], [4, 1, -1, 1, 3, 1, 0, 1, 1, 1], [8, 0, 5, 2, 2, -2, 2, -1, -2, -1], [4, 0, 2, -1, 0, 4, 3, 3, 0, -1], [-5, -1, 1, -3, 3, 3, -1, 1, -1, 1], [1, 1, 2, 1, 3, 0, 5, -2, -2, -1], [-7, 1, -4, 2, -1, -3, 0, -4, -1, -1], [6, 2, -1, -1, -2, -2, -3, 0, -1, 0], [1, 0, 0, -1, -1, 2, -2, 4, 0, 1], [5, -1, 5, 2, 1, 0, 2, -1, -1, 0], [-4, 2, -4, 0, 0, -2, 2, -1, 2, 0], [4, 2, -1, -2, -1, 2, -4, 0, -1, 1], [11, -3, 2, -2, 1, 3, 2, 2, 2, 2], [4, 1, 2, 0, 0, 4, 1, 4, 1, 0], [0, 0, 0, -5, 2, 1, 1, -1, -3, 2], [7, -4, 0, -1, 1, 0, -2, 2, 3, 3], [-2, 1, 0, 3, -4, -3, -2, -3, 0, -2], [4, 2, 0, -1, -3, 0, -1, 3, 0, -2], [4, 0, -1, 0, -3, 0, 0, 2, 3, 1], [2, 5, -2, 2, -2, 0, 1, 0, -1, 0], [-8, -2, -1, 2, 1, -4, 2, 1, -3, 0], [0, -4, -2, -2, -2, 1, 0, 0, 3, 1], [5, -2, 4, 0, 1, 2, -1, 1, -1, 0], [-6, 2, -2, 0, 0, 2, -2, 2, 0, 2], [-4, -4, 2, -1, 5, 2, 1, 0, 1, 3], [13, -4, -1, -1, -1, -1, 0, -1, 1, 1], [-2, 4, 0, 1, -2, 1, 3, 1, 0, -1], [3, -3, -2, -4, 1, 5, -6, 6, 2, 6], [3, 1, 1, -1, -1, -1, -1, -5, -3, 1], [1, -2, 1, -1, 5, 3, 0, 1, 2, -1], [1, -2, 1, -2, -1, 2, 5, 5, -1, -4], [-4, -6, 4, -2, 2, 0, 2, 4, 0, 0], [5, 2, -3, -1, -1, -2, 0, -2, -2, 1], [5, -2, 0, -1, -3, -2, 4, 6, 0, -4], [3, -2, 2, 0, 1, -5, 3, -3, -3, -1], [-7, 1, 1, -1, -3, 3, 1, 5, 1, 1], [4, 5, -2, 0, -5, -3, 1, -1, -3, -1], [-3, 1, 0, 0, 3, -7, 2, -1, -4, -3], [3, -1, -5, 2, -1, -2, 2, -2, 3, 2], [-1, 2, 0, 4, 1, 1, -1, -2, 0, -3], [5, -1, -2, -1, 1, -2, -3, -4, 2, 3], [2, 1, 0, 2, -2, 4, -1, 4, 5, 2], [2, -7, -1, -1, -2, 0, 2, 1, 1, -2], [-1, 3, -2, -5, 1, 2, 3, 0, 0, 3], [-4, 3, 2, 3, 4, -1, 2, 1, 2, -2], [-1, -4, 0, 2, 5, -4, -1, -1, -3, 0], [-6, -3, 4, -2, 0, 0, 1, 2, -3, 2], [1, 6, -2, 3, -1, -3, 4, -6, 1, -4], [-8, -3, 2, -3, 4, 3, -2, 3, 0, 6], [-3, 2, 3, 4, 3, 0, 1, 4, 3, -1], [0, 0, -4, -3, 0, -3, -3, -5, -3, 2], [0, 2, 0, 1, -4, -1, 1, 0, 0, 1], [11, -7, 6, -1, 1, 2, 1, 0, -1, 3], [0, 0, 2, 3, 2, -3, 1, -4, 2, -3], [-7, -3, 2, -2, 4, 0, 0, -4, 0, 0], [-1, 5, -2, 5, 3, -1, -1, -8, 1, 0], [4, 1, -1, 1, -6, 2, 0, 6, 1, 2], [-7, -4, 8, 1, 7, -2, 2, 0, -3, 1], [-1, -3, -2, -4, 1, 6, -2, 2, 3, -1], [2, -1, 4, 1, -2, -1, -2, 3, 2, 0], [9, 4, -7, -1, -4, -2, 0, -4, 2, 0], [-3, -7, 2, -3, 3, 6, 5, 6, -1, 3], [1, 5, 0, -1, -1, 0, -5, -2, -2, 1], [-17, -2, 2, -1, 4, 5, 0, 4, 0, 5], [1, -2, 4, 6, 1, -6, 5, -7, 1, -4], [2, 3, -3, 4, -3, 2, 1, -4, 2, -3], [0, -4, -2, -3, -2, 5, -1, 10, 2, 1], [-4, 4, -3, 4, -6, -5, 0, -2, 1, -6], [-6, -1, -8, -4, 0, 6, -5, 4, 0, 2], [3, 4, -7, 1, 1, -4, -6, -3, 0, -2], [-9, -1, 1, -3, 3, -3, -1, 1, -5, -3], [1, -1, 3, 1, 3, -3, -1, 1, -1, -5], [-13, -1, -1, 4, 1, -4, 0, -6, 1, 0], [1, 6, -2, 0, 1, 0, 3, -1, 3, -4], [1, 1, 4, 3, 1, -2, 3, -2, 0, -1], [17, -5, 3, 0, -1, 2, 4, 7, 5, -2], [-6, 3, 3, 2, -2, -5, 1, -4, -3, -1], [-6, 5, 0, 0, 4, 2, -1, -4, 1, 2], [-12, 5, -3, 5, 0, -4, -6, -2, -1, -2], [0, -2, 6, -2, 4, -4, 0, 2, -2, 2], [-7, 4, 3, -3, -3, 6, 0, 5, -3, -2], [6, -2, -4, -2, -4, -8, -2, -6, -2, -2], [3, 2, 2, 0, 1, -1, 1, -2, 0, 0], [-6, -3, -2, -4, -2, 2, 3, 0, 1, 0], [-2, 1, -2, -3, 2, -3, -4, -5, -2, 6], [-1, -9, 6, 2, 3, -1, 4, 1, 2, -3], [6, 0, 3, 4, 1, -2, 10, -4, -3, -3], [-10, -4, 5, -3, 5, -1, -1, -3, 2, 1], [-6, 10, 1, 1, -6, -3, 1, -1, -5, -3], [1, 1, 2, 2, 0, 0, -4, 0, 0, 0], [-2, -1, -1, 0, -6, -7, 3, -5, -3, -4], [0, -1, 5, 1, 4, -4, 4, 0, -1, 0], [-10, 5, 2, 2, 2, 0, -1, -2, -4, 4], [-5, -5, 1, -1, 1, 4, -1, 1, -2, 6], [-15, 3, 0, -8, 0, 4, -4, 4, -2, 4], [-12, -2, -1, -3, 1, 1, 3, -2, -1, 0], [-1, 0, 7, 0, 2, 5, -3, -1, -3, 2], [6, -1, -7, -6, 2, 5, -3, 6, 6, 4], [-14, 3, 4, -5, 2, 11, -4, 7, 0, 6], [6, -2, 0, -2, -6, -4, -2, -2, -2, 2], [4, 1, 2, 0, -2, 2, 7, 0, -4, 0], [11, -8, 3, -2, 6, 7, 7, 4, 1, 2], [-10, 3, 0, -1, 0, 1, 4, 5, 2, 0], [5, -1, -4, -5, -7, 6, 1, 6, 1, 1], [-19, -4, -4, 1, -1, -1, 2, 4, 0, 0], [2, -6, -4, -6, 1, 9, 0, 9, 4, 3], [15, -1, 9, 3, 3, 0, 3, -3, -2, -2], [-8, 4, -6, 9, -6, -5, -1, -4, 2, -5], [-8, 2, -1, -4, -2, 0, 4, -3, -9, 1], [-25, -1, -4, 0, -5, -5, -2, -3, 0, 1], [-1, -1, 5, -5, 5, 3, -1, 2, 3, 3], [16, -12, 8, -2, 2, 0, 4, 2, -4, 0], [-6, -1, -2, 1, 1, 6, -6, 4, 2, 3], [-7, 8, -2, 0, -1, -1, 1, -1, 2, -3], [0, 5, -3, 7, -4, -6, 2, -4, -1, -6], [-8, 1, 2, -2, 8, 6, 1, 2, 1, 2], [-9, -3, -6, -5, -3, -2, -3, -4, 2, 3], [-3, 7, -5, 5, -7, -3, 7, -2, 1, -7], [4, 11, -6, 1, 1, -4, 2, -10, -2, -1], [-7, -1, -2, 0, 0, 2, -2, 2, 5, 8], [-8, 0, -1, -4, -3, 10, -8, 8, 5, 5], [11, -2, 2, 3, 3, 1, 0, -6, 0, -2], [3, 1, 3, -2, 1, 2, 0, 2, 2, -3], [-17, 4, -5, -4, 4, 3, -3, 1, -3, -2], [3, 6, 2, 2, -1, 5, 7, 4, 2, 2], [-9, -1, -2, 4, -1, 4, 0, -5, 6, 2], [9, 1, -1, -7, 5, 7, -1, 8, 5, 5], [6, -5, 8, -6, -2, 0, 3, 8, -1, -2], [14, -1, -5, -1, -2, 4, 2, 0, 5, -2], [-3, 1, 6, -1, 5, 4, 5, 4, 2, 5], [-8, 2, 2, 7, 0, -5, 3, -3, -3, -2], [-1, -13, -1, -4, 3, 2, -6, 8, 3, 3], [12, 8, -6, 0, -2, -4, -2, -6, 0, -4], [4, -4, -1, -6, 2, -1, -6, -5, -1, 4], [2, 8, 4, 2, 0, 5, 2, 0, -1, -1], [4, 5, -5, 0, -2, -3, -5, 5, -2, 2], [-7, 2, 2, 0, 5, 1, 1, -4, 0, -1], [-11, -1, -3, -1, -7, -3, -1, -3, 1, -1], [-13, 6, -6, 4, 1, 0, -3, -13, 1, 2], [4, -4, 5, 4, 1, 0, -8, 2, -1, -1], [-8, 0, -6, 0, -2, 0, -4, 4, -2, 2], [-6, -2, 2, -6, -2, -4, 0, 2, -6, 4], [-14, -6, 2, 0, 4, -2, 8, 4, 2, 0], [-6, 6, -6, 0, -2, -4, 2, -4, 6, 2], [-5, -10, -2, -8, 1, 1, -3, 5, 2, -3], [2, -3, 2, 4, 4, -6, 3, -4, -5, -2], [-17, 3, -11, -2, -1, 0, -4, 1, 3, 0], [-11, 0, 6, -2, 9, 4, 1, -7, -5, 0], [-2, -5, 2, -4, -2, 4, -5, 2, -2, 4], [0, 4, -10, 4, -8, -4, 2, -6, 2, -4], [-10, -5, -1, -4, 4, 7, -7, 6, 1, 3], [14, -6, 4, 3, 0, 3, -3, -3, 1, 2], [8, -9, 3, -5, 8, 5, -6, 9, 6, 6], [3, 0, -6, -5, -5, 2, 0, -2, 2, 3], [-18, -3, 4, -1, 0, 1, -2, 3, 0, -2], [-5, 0, 3, -6, 5, 10, -11, 5, 1, 10], [12, 11, 2, 4, 1, -1, 3, -3, -5, -3], [-9, -1, -7, 1, -8, -6, -3, -2, 5, 4], [1, 0, -3, -1, 1, 3, -2, 1, 2, 11], [-4, 4, 3, 8, 0, -7, 4, 0, 3, 0], [0, 4, 2, 0, -2, -5, -6, -10, -3, -3], [-7, -4, 5, 1, 4, 8, -4, -5, 5, 9], [-5, 0, -3, 2, 3, -4, 5, -7, 3, -3], [-20, 6, -4, 4, -2, -4, 0, -2, -2, -4], [1, -1, 5, -1, -1, 8, 3, 3, 4, 6], [-2, -1, -2, 0, -6, -4, -3, 2, 1, -2], [-19, 4, 0, 0, 3, -3, -3, 1, 2, -1], [-13, 7, -3, -1, 1, 7, 1, 1, 1, 5], [7, 7, -8, -3, -1, 1, -1, -2, 5, -2], [-5, 0, -3, 2, -7, -4, 1, 0, 3, -1], [-14, 5, -3, 5, -2, -6, 2, 2, -1, -2], [-4, 4, -9, 5, -9, -3, -11, 4, 9, 0], [12, -1, 4, -2, 0, -8, 7, -6, -9, -2], [-1, 8, 4, 2, 0, -4, 5, -6, -8, -4], [4, -3, -2, -3, -2, -3, 2, -2, -8, 0], [-20, 10, -4, 6, 0, -2, 2, -4, 6, -4], [14, -8, 6, -1, 0, -1, 1, 8, 0, -1], [12, -6, 0, -12, 4, 6, 0, 8, 2, 4], [12, -8, 3, -5, 2, 0, 3, 9, 1, -5], [-2, -2, 3, -1, 8, -2, 5, 0, -9, 2], [-23, -9, 3, -1, -1, 3, 3, 5, -3, -1], [-3, 10, 3, -1, -3, -1, 2, -11, -1, 1], [-7, 6, 0, -4, -1, -3, -1, 5, -2, 3], [-4, 6, -5, 2, -3, 0, 0, -2, 7, 1], [-19, -1, -2, 1, -3, -8, 1, -1, -10, -5], [-25, -6, -2, -7, 3, -7, 2, -3, -5, -1], [-20, -4, 4, -3, 6, 10, -1, 10, 7, 8], [23, -2, -2, -4, -1, 1, 9, 0, -6, -7], [2, 3, 2, 1, 0, 1, 2, 1, 4, -4], [25, -5, -2, 3, 5, -8, -3, -4, 0, -1], [4, 5, -8, 2, -12, -2, -1, -7, 2, 1], [-15, 1, -5, -8, 1, 2, 4, 2, -1, 0], [9, -1, -1, 4, -1, -4, 2, 2, 1, -9], [-2, 9, 4, 3, 6, 1, 4, 0, -6, 1], [1, -2, 5, -10, 4, 3, -5, 4, 1, 4], [-3, -2, 1, 5, -1, 3, 4, -7, -1, -1], [-3, -2, -10, -8, 1, 4, -5, -3, 5, 4], [-4, 6, -6, 1, 0, -9, -1, -2, -2, 5], [-11, 3, 5, 3, -1, -9, -1, -11, -1, -5], [-6, -6, -8, 2, 2, 0, -2, 11, 6, 0], [10, -2, 9, 2, 1, 6, 0, -2, 1, -1], [15, -1, 0, 2, -3, 5, -10, 4, 8, 4], [2, -5, 0, 5, 0, 1, 2, -3, 10, 4], [-9, -2, 3, 8, 1, -4, 9, 2, -3, -3], [-10, 6, -8, 5, -8, -7, -1, 3, 7, -6], [-2, 0, 7, -2, 0, 5, 0, 0, -9, 2], [17, 7, 2, 11, -5, -6, 1, -4, 4, -7], [29, -10, -1, -4, -3, 1, -9, 7, 2, 2], [-6, 0, 0, 0, 2, 6, -2, 4, -2, 6], [-10, 0, 3, -9, 6, 0, -5, 4, -2, 2], [10, 6, -2, 0, -8, -2, -4, -4, -2, -4], [4, -1, -6, 3, -6, -3, -6, 0, 6, 6], [18, 9, 6, 2, 4, 7, 7, 6, 5, -2], [-5, -9, -3, -2, 1, 0, 0, 0, 5, -1], [-7, -2, -2, 6, 3, -9, 7, -5, 2, -3], [12, -13, 10, -6, 10, 8, -3, 7, 0, 5], [26, -6, -5, -5, 2, 8, -1, 7, 1, 1], [-11, -2, 3, 1, 4, 10, -2, 7, 11, 5], [11, -5, -4, 2, -3, 5, -2, 6, 6, 4], [-13, 10, -3, -1, 0, 2, 0, 0, 5, 0], [-7, -2, -2, -7, -7, 2, -2, 8, -1, 3], [9, -7, 8, 4, 7, -2, 4, -9, -7, 0], [-23, -12, -3, 0, 1, 3, -11, 3, 4, 4], [8, -11, -1, -4, -4, 9, 5, 3, -1, 7], [4, 2, -3, 2, 4, 3, -2, -5, 1, 4], [19, -6, -4, 4, 7, -2, 1, -7, 3, 4], [-9, 5, 1, 6, -3, -4, 2, -8, -7, -9], [-10, 6, -8, -2, -2, -6, 0, 2, 0, -2], [15, -7, 2, -4, -3, 10, -2, 1, 2, 0], [8, -3, -1, 6, 2, -5, 1, -10, 5, 1], [15, -3, -3, 1, 5, -5, 3, -5, -2, 3], [14, 7, -2, 0, -8, 7, 3, 8, 3, 0], [-13, 0, 12, 2, 3, 0, 3, -3, -3, 2], [32, -11, 6, -5, 4, 7, 4, 3, -8, 0], [0, -9, 5, 1, -6, -4, 0, 0, -3, 0], [31, -4, 1, 6, -1, -2, 5, 6, 5, -9], [13, 7, 2, 7, -3, -6, -3, -8, -3, 3], [-3, -6, 0, -5, 3, 7, 8, 2, 5, 6], [-11, 6, -7, -8, -5, 12, -1, 6, 7, 7], [11, 3, 4, 1, -1, -2, -3, 6, -4, 3], [-5, -6, -1, -1, -1, 5, 0, 4, 7, 12], [15, 6, 2, 11, -5, -3, 4, -7, 5, 1], [16, -2, 5, -1, 0, -6, -1, -4, -1, -6], [2, 11, -8, 1, -14, -5, 0, -1, -4, -6], [0, -2, 6, 2, 0, 2, 2, 0, -6, 2], [-5, -4, 12, -2, 9, 1, -1, -8, -2, 7], [-2, 7, 5, 5, 11, -1, 0, -3, -1, 1], [0, 2, 2, 1, 4, -5, 1, 1, 5, -4], [-13, -1, -4, -3, 3, -3, -5, -7, 1, 6], [9, 16, -3, 5, -9, -7, 6, -13, 1, -5], [7, -2, 15, -4, 3, 9, 9, 5, -4, -2], [-10, 3, -6, 9, 4, 4, 4, -3, 5, -1], [14, 12, 8, -5, 4, 7, 3, 4, -4, 1], [10, -4, -1, -6, 1, 12, -2, -1, 5, 10], [19, 10, -2, 7, -1, -6, -2, -6, 2, -5], [-18, 4, -6, 1, -4, -1, 5, -2, -2, 3], [17, -4, 0, 3, -5, -6, 6, -6, 2, -2], [-6, -4, 4, -1, -6, -3, 7, 2, -2, 1], [7, -5, 16, 1, 11, -2, 9, -8, -8, -7], [-6, 11, 0, 1, 0, 8, 4, -1, 5, -3], [28, -7, 10, -11, 4, 5, 4, 8, -4, 8], [11, 4, -1, 0, -1, 0, -1, 0, -5, 1], [26, -11, 4, -3, 0, 1, -2, 2, -4, -2], [-5, 12, -8, -1, -3, -9, 2, 7, -3, -7], [-9, 10, -1, 10, -7, -6, 9, -2, -7, -13], [13, 7, -16, -3, -1, 1, 1, 0, 5, 2], [11, 0, 6, -6, 0, 2, 3, -3, 0, -2], [-18, 1, -3, -2, -7, -4, -7, -8, 1, 4], [-8, 2, -6, -2, -2, -2, 2, 2, 4, 8], [28, -15, 6, -5, 2, 13, 2, 0, 2, 9], [-6, 6, -5, 5, -6, -2, -5, 0, -1, -1], [-9, 8, -6, 1, 5, 5, 10, 1, 5, -5], [4, -1, 2, -1, -4, -3, 10, -3, 0, -6], [-3, 6, -15, 4, -7, 0, -5, -5, 13, 9], [-5, -3, 0, 5, -7, -4, -1, -12, 4, -5], [-8, -17, 12, 0, 14, 2, 5, 4, 1, 0], [9, 4, 8, 1, 1, 3, 2, 3, 7, -1], [11, 6, -10, -7, -7, 6, -14, 3, 6, 4], [5, -5, 2, -5, 3, 10, 1, -2, 12, 9], [16, -8, 5, -2, 4, -1, 0, 3, 3, -4], [38, 0, -4, 6, -10, -10, 0, 6, 6, -8], [-9, -1, 7, -7, 5, 3, -5, -5, -7, 9], [21, 4, 7, 1, 1, 3, -4, -1, -2, -5], [41, 0, 1, -1, -3, -3, -2, 3, -5, -3], [8, 3, 2, -3, -4, -5, 10, 9, 2, -8], [-21, -2, -1, -3, 7, -1, -4, 2, -7, 4], [-2, 6, -4, -4, -2, 4, 0, 10, -7, 8], [16, -11, 8, 0, -2, -4, -3, -4, 5, 4], [-16, 2, 4, 2, 8, 8, 0, -2, 7, 6], [16, -8, 6, -6, 8, 4, 2, 0, -2, 8], [11, -5, 9, 7, -1, 1, 7, 2, -5, -7], [16, -12, 1, 3, -4, 2, 3, 4, 8, -4], [-21, -2, 2, 6, 2, -2, -3, 0, 4, -2], [11, 6, -2, -2, -9, -13, 9, 1, -3, -7], [13, 1, -4, -1, -5, -5, -7, -2, -3, -2], [21, 6, -3, 3, -2, -4, 6, 6, -3, -6], [9, -5, -8, -8, 5, 13, -4, 11, 8, 7], [3, 12, -7, 11, -6, -4, -2, -16, -3, 0], [-2, -9, 2, -8, 4, 8, -3, 14, 7, 6], [10, 12, -10, 2, -8, 4, -2, 4, 8, 4], [5, -10, 10, 1, 15, 8, -4, -10, -3, 3], [6, -11, 2, 0, 2, -2, 1, 4, 8, -7], [-47, 12, -2, -2, -1, -6, -5, 1, -3, 0], [-5, 5, -3, 2, 1, 1, -4, 8, -2, -4], [-6, -3, -8, 1, -2, 0, -4, 10, -1, -3], [0, -6, 0, -10, 10, 4, -2, 6, -6, 4], [4, -9, 2, 7, 6, -3, -2, -3, 6, -6], [3, -6, -2, -4, -10, 2, 1, 8, -2, -2], [22, 3, -6, 0, -2, -2, 3, -4, -2, 4], [14, -9, 3, 0, -4, -3, -1, -8, -3, 1], [1, -6, -9, 2, -8, -7, -1, -5, 1, -2], [2, -4, 2, -12, 6, 7, 6, 8, -1, 13], [-4, 5, 5, -2, -6, 5, 3, 10, -2, 0], [35, 9, -8, 5, -5, 0, -3, -7, 2, 4], [-10, 11, 2, 4, -2, -8, -3, 2, -1, 0], [-3, -10, 7, -4, 9, 6, -9, 4, 9, 13], [2, 2, 6, -8, 0, 0, 0, 2, -11, 6], [12, -2, -12, 9, -8, -11, 9, -14, 2, -9], [6, 2, 4, -4, 0, 6, -2, 6, -4, -2], [-19, -6, 5, -1, 1, -1, -4, -3, 0, -3], [12, -9, 3, 9, 4, -14, 0, -14, -5, -6], [8, 4, 5, 10, -2, -5, -4, -3, -8, 0], [-22, -8, -1, -5, 4, 4, -7, -6, 4, 6], [20, -3, -8, 0, -4, 2, 5, -4, 5, -4], [30, -7, -3, 5, -5, -11, 6, -4, 1, -5], [7, 5, 7, 3, 9, -1, -7, 5, -3, 3], [-1, 0, 5, 3, 3, -1, 6, -11, -7, -5], [8, 8, -3, 5, 0, 0, 9, -3, 5, -7], [10, -1, -7, -3, -6, -10, 2, 0, 0, -8], [3, -3, 3, -1, -1, -1, 3, -3, -7, 5], [19, 5, -4, 3, -3, -4, -3, -6, -8, -3], [-10, 1, -3, -7, -6, -6, -10, 8, -7, 4], [2, 15, 3, 7, -1, -1, 16, -7, -2, -5], [-9, 14, -5, 5, -7, -11, 8, -1, 0, -9], [-8, 0, -1, 3, -1, -11, 7, -1, -2, -9], [-3, 12, 5, -1, -3, 7, 4, 0, 1, 4], [-32, 15, 1, 12, 3, 0, 5, -3, 1, 2], [-9, 7, 9, 4, 5, 1, 4, 4, -10, -8], [-12, 1, -6, 4, 1, -1, 1, -4, 2, -6], [35, -8, 3, -14, 1, 4, -1, 8, 1, 7], [-12, -2, -5, 0, -8, -17, -4, -9, -9, -4], [29, -4, -2, -3, -9, -7, 0, 10, 1, -4], [9, 12, 3, 6, -1, 6, 9, 2, -4, -7], [23, 0, 0, -11, -3, 5, -10, 3, -3, 5], [9, -4, -3, -5, 3, 9, 0, 2, 3, 6], [8, -12, 0, -10, -8, -6, -6, -2, -5, 4], [-5, 0, 8, -5, 13, 14, -6, 4, 3, 7], [-18, 9, -10, 3, -2, 1, -6, -3, 4, -2], [-22, 2, 0, 0, -4, -4, -2, -4, 2, 4], [-16, -1, -1, 5, -10, -2, -4, 1, 3, -4], [-7, 6, 12, 2, 3, -12, 5, -3, -13, -20], [-5, 6, -6, -5, 9, 0, -4, -10, -7, -1], [-10, 6, 2, -2, 0, 16, -4, 8, 10, 8], [28, 6, -5, -6, -6, -2, 4, 1, -8, -3], [-6, 8, 7, -3, 16, 12, 9, 4, 1, 0], [-12, 10, -2, 9, -2, -9, -1, -6, -4, -1], [17, 9, -3, 9, -5, 1, 1, 4, 11, -3], [-3, -7, 1, -6, 7, 6, -4, -1, -7, 2], [-17, -2, 1, 1, 7, 7, 6, -1, -1, 5], [-5, 6, 4, -4, 5, -1, 1, 4, 4, -2], [-9, -3, 3, -4, 2, -1, 4, 1, 0, 2], [-25, -4, 1, -6, 1, 3, -13, 8, 0, 10], [-6, 3, -1, 7, 3, -13, 2, -13, -9, -3], [24, -10, -2, -4, -14, 2, -4, 14, 10, 0], [17, 5, -7, -11, 3, 7, -3, -3, 5, 11], [19, -15, 6, -2, -1, -1, 2, -2, -7, 3], [21, -1, 5, 5, -2, 8, -1, -3, 3, 2], [-30, 3, -13, -9, -11, 5, -4, 7, 4, 9], [-9, 5, -2, -7, 7, 15, -3, 15, 1, 6], [-16, 2, 6, -2, -1, 9, 6, 8, -4, -3], [-26, 4, -5, 10, -5, -4, -2, 2, 3, -11], [-29, -9, -1, -9, -1, 17, -5, 18, 5, 9], [7, -7, -3, -5, 7, 7, -5, 9, 19, 9], [-3, -8, -1, -5, -1, 7, 0, -6, -1, 0], [10, -2, 8, -3, 10, 5, 3, 5, -7, 6], [-23, -7, 1, -3, 3, -3, -1, -9, -7, 3], [27, -5, 12, -5, 7, -3, 7, -2, -9, 0], [-2, 4, 0, 11, 4, -4, -5, -19, 1, 2], [9, -7, -3, 6, 1, -8, 0, -7, -7, -6], [11, -7, 4, -1, 13, 8, 5, -8, 0, 5], [-12, -12, -8, -7, -4, 5, -7, 10, 12, 13], [-3, 3, 2, 11, -1, -10, 3, -8, -2, -9], [-19, 2, 5, 0, 11, 10, 5, 2, -4, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6162_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6162_2_a_bj();".
// 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_6162_2_a_bj(:prec:=10)
    chi := MakeCharacter_6162_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_6162_2_a_bj();".
// 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_6162_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6162_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-64, -176, 348, 1004, 61, -578, -5, 116, -13, -6, 1]>,<7,R![-2560, -928, 5568, 2740, -2290, -923, 429, 98, -36, -3, 1]>,<11,R![4096, -3200, -12512, 16014, -2021, -3591, 919, 196, -59, -3, 1]>,<17,R![1576192, -967104, -710096, 368684, 48812, -34168, 56, 1135, -64, -12, 1]>],Snew);
    return Vf;
end function;