
// Make newform 7942.2.a.bm in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7942_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_7942_2_a_bm();".
// 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_7942_2_a_bm();".
// 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 := [4, 2, -29, 3, 38, -4, -14, 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, 1, 0, 0, 0, 0, 0, 0], [77, -511, -182, 479, 97, -101, -9, 6], [-312, -131, 804, 145, -336, -74, 23, 5], [-159, -126, 398, 200, -251, -121, 17, 9], [74, 191, -735, -400, 502, 242, -34, -18], [20, 261, -14, -789, 45, 316, 4, -23], [-154, -625, 547, 1360, -438, -652, 18, 49]];
    Rf_basisdens := [1, 1, 61, 61, 61, 61, 61, 122];
    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_7942_a();"
function MakeCharacter_7942_a()
    N := 7942;
    order := 1;
    char_gens := [5777, 6139];
    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_7942_a_Hecke(Kf)
    return MakeCharacter_7942_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], [-1, 1, 0, 0, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -1, -1, -1, -1, 0, 0], [1, 0, 0, -1, 1, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-1, -2, 1, 0, -1, 1, -1, -1], [1, -1, -1, 1, 0, 1, -1, 2], [-3, 1, -1, 0, -1, 1, 1, -1], [-2, 0, 1, 1, -1, 0, 0, -1], [-2, 0, 0, 0, 2, 0, -1, -2], [0, 2, -1, -1, 1, 0, -1, 0], [5, 2, 1, 2, 2, -1, -1, 3], [2, 0, 0, 1, -1, -1, 1, 1], [0, -2, 1, -1, 1, 0, -2, 3], [-1, -2, 1, -1, 0, 0, -1, -2], [2, -1, 0, 1, -2, -2, -1, 5], [-1, -4, 2, -1, -3, 0, 0, -2], [-1, 1, -4, 0, 1, -2, 2, 0], [-2, -2, -1, 1, 3, 1, 1, 0], [-3, -1, 0, -1, 1, -1, 0, -2], [0, 3, 2, 1, 2, 0, 0, 1], [5, 1, 1, 2, 2, 2, 1, 6], [3, -1, -1, 2, 0, 0, -3, 4], [0, 0, -3, -1, 1, -1, 1, 4], [-5, 3, -3, -1, 0, 0, 0, -2], [2, -3, -3, -1, -3, -2, -1, 2], [-8, 6, -1, -2, 1, 3, 1, -7], [-2, 0, 0, -1, 2, 0, 0, 2], [3, -2, -1, 1, 0, 1, -2, -2], [-6, -1, -2, -3, -2, -4, 1, -3], [-4, 2, -1, -4, -3, -4, 2, 2], [7, 4, 1, 2, 4, 1, -3, 0], [-4, -1, -4, 0, 0, -2, 0, 2], [-7, 1, -1, -4, -4, 1, 3, 1], [4, 2, -4, 3, 1, 3, -1, 3], [3, -2, 3, 0, 0, -1, 0, -2], [-4, -2, -1, -1, 1, -3, -1, 6], [-5, -1, 3, 0, 1, -1, -2, 1], [6, -2, -3, 3, 3, 0, 2, 11], [9, -3, 2, 0, -1, 3, -2, 1], [-6, -2, 2, -2, 2, 1, 2, -2], [2, -3, 2, 2, 0, 3, -1, -2], [3, -1, 1, -3, 7, 2, -1, 5], [9, -3, 1, -1, -4, -4, 3, 8], [0, -2, 1, -1, 5, 1, 3, -3], [-7, -3, -3, 1, 2, 1, 2, 4], [6, -2, 0, -3, -1, -1, 1, 1], [-1, 2, 0, 6, -1, -1, -5, 4], [-11, -1, -1, -2, -4, 0, 4, -2], [4, -1, -3, 7, 3, 0, 0, 6], [-5, -1, 1, 2, 4, 2, -1, 3], [4, -5, 3, 2, 3, 4, -6, 0], [-2, 2, 3, -5, -1, 0, 2, 2], [-2, -5, 0, -2, -6, 0, 0, 5], [3, -7, 4, 4, -3, 1, -4, 6], [-2, -3, -2, 1, 0, -1, -3, 2], [2, -5, 5, -1, -1, 1, 1, 0], [8, -1, -5, -2, 1, -1, -2, -2], [-11, 0, 5, -2, -6, -1, -1, -10], [-3, -1, 6, 0, -5, -1, 1, 2], [12, 4, -4, -1, 4, -2, 3, 3], [-6, 3, 5, -3, -1, 2, 1, 0], [5, 2, 0, -1, 7, 1, 2, 0], [3, -4, 1, 6, -5, 0, -3, 7], [-11, 4, 0, 4, 5, -1, 0, -2], [2, -2, 4, -1, 4, 3, 2, 0], [2, 5, -4, 0, 6, -1, -5, -2], [-4, 9, -1, 0, -3, 2, 6, -7], [7, 5, 2, -2, 1, 0, -1, -8], [-6, 0, 4, 2, 0, 5, 1, -1], [-16, 1, -1, 1, -1, 2, -1, -6], [18, 0, 3, 1, 3, -1, -2, 2], [-18, 0, 1, 1, -4, 3, 3, -1], [0, 2, -5, 3, 0, -4, 4, 9], [-10, 11, 2, 1, 5, 3, 1, -19], [-12, 2, -2, 0, 5, 2, -3, -5], [0, 8, 3, 3, 3, 0, 5, 0], [-2, 13, -2, 2, -1, 1, 1, -11], [6, 1, 0, -10, -2, 0, 4, 3], [2, -2, -2, 0, -2, 0, -5, 4], [5, -2, -5, 1, 3, 0, -5, -1], [3, 3, -3, 3, 4, 3, -3, -8], [-2, -6, 3, -2, -5, -6, -2, 1], [-3, -4, 7, 2, 0, 3, -3, -1], [7, -3, -5, -1, 0, -2, -3, 2], [-8, 6, -1, 1, -5, -4, 1, -4], [-11, -1, -7, 1, -6, 2, -1, -7], [-14, 0, -3, 2, -2, 2, -1, -7], [-14, -1, 0, -1, -2, 5, 1, 6], [-7, -5, 2, 2, 6, 0, 0, -5], [-6, 0, 1, 6, 5, 0, -4, -2], [5, 6, -3, 8, 10, 5, 0, 7], [-10, 1, -5, 2, -9, 3, 1, -2], [-4, -4, 7, -3, -7, -4, 5, -2], [0, -1, 3, -2, -5, -5, -4, 6], [5, 2, 5, -1, -2, -3, -2, 12], [-7, -4, 4, -6, -15, 3, 2, -6], [-9, 1, -5, -4, -4, 0, 5, -8], [-16, -3, -3, 4, -5, -2, 6, 2], [-1, 7, 0, 3, 15, 7, -7, 0], [6, 7, -3, -1, -5, -5, -2, 4], [-2, 8, 3, -4, -7, 4, 2, -6], [-1, 8, -7, -1, -1, -2, 3, 7], [-13, -1, 5, -1, -6, -2, 1, -17], [-6, 14, -4, 9, 0, -3, -2, -2], [7, -3, -1, 4, 0, -4, -7, -9], [-15, 4, 2, -6, 1, 4, 3, -2], [0, 6, -2, -4, 0, 0, 6, 0], [-17, 11, -8, -1, -1, 2, 4, 2], [-1, -4, 2, 2, 10, -2, -5, 3], [7, 5, -1, 7, 2, -1, -2, 1], [1, -3, 5, 2, 1, 0, -8, -11], [15, 4, -3, -1, 5, -1, -3, -1], [3, 1, -4, 5, -3, -1, -3, 7], [-11, 10, 2, -6, -3, 3, 2, -19], [8, 4, 4, 0, -2, -6, -2, 7], [17, -5, 0, 8, 1, -2, -2, 14], [-11, 12, 3, 6, -3, 7, 6, -15], [4, 5, -4, 1, -8, -1, 4, -6], [-2, 3, -3, 8, 5, -3, 1, 12], [-8, -8, 4, -5, -1, -1, 0, 5], [6, 1, 2, 4, 5, 0, -4, -11], [-9, 2, -4, 5, 1, 5, -5, -8], [4, -2, -4, 5, 4, -3, -2, 4], [-12, -3, 0, -4, -2, 0, 4, 5], [7, -4, 8, 1, -1, -6, 2, 7], [-27, 1, -1, -1, -8, -2, 8, -8], [5, -9, 7, 1, -6, -3, -1, 0], [-4, -10, 7, 2, -7, 3, 6, 2], [6, 1, 4, 6, 1, 5, -7, 7], [17, 0, 6, 1, -1, 2, 5, 0], [-2, -9, -8, -7, 2, 0, 0, 8], [6, 10, -1, -3, 5, -2, -1, -2], [5, 2, -4, -6, -9, -1, 5, 3], [-6, -9, 2, 5, 2, 8, -2, -18], [-8, -12, 9, -3, -6, -1, 4, -1], [5, -16, 7, -3, -2, 1, -1, -4], [8, 4, -4, -7, 4, 3, 1, 6], [-4, 4, -1, 4, 5, 0, 5, -2], [-31, 9, 0, -4, -8, -2, 11, -21], [-9, 3, -6, -1, -7, 1, -1, -14], [11, -2, -4, -4, 3, -7, 4, 2], [13, -10, -2, 10, 6, -2, -5, 15], [7, 5, 5, 1, 2, -3, 5, 10], [-27, 16, -4, -8, 5, -1, 9, -26], [6, -11, 3, 4, -12, -3, 0, 19], [2, -3, 0, 14, -3, 4, 0, 9], [-9, -8, -4, -7, -5, -2, 4, -2], [8, 1, -1, -10, -1, 0, 2, 12], [18, 0, 1, -1, -10, -10, -2, -7], [-9, -11, -4, 3, -3, 4, 1, 0], [-16, 6, -1, -4, -8, 0, 1, -13], [-5, 3, -3, 2, 15, 0, -3, 1], [-19, -8, 4, 0, 7, 3, -4, -4], [-13, 4, -8, -7, 0, -1, 10, -11], [-6, -7, 3, 1, -7, -8, 0, 12], [-17, 5, -7, -1, 6, 0, 1, 0], [-4, -3, -7, -7, -3, 2, 6, -2], [-19, 15, 0, 4, 3, 1, 1, -10], [-6, 5, -2, -2, 2, 1, 7, -4], [-9, -8, -7, 2, -6, 1, 3, -17], [-13, -7, 3, 2, -4, 3, -3, -17], [2, 7, -1, -4, 7, -1, 0, -5], [-4, 9, -1, -2, 0, 0, 11, 1], [-22, 3, 2, 6, 6, 6, 1, -6], [27, 2, 0, 0, 7, 3, -4, 16], [-15, 16, -2, 1, 1, 5, 0, -12], [18, 4, -6, 6, 0, 5, -6, 20], [-1, -5, 7, 2, -9, -2, -1, -5], [-7, -8, -6, 2, -1, -6, -1, -2], [16, 0, 8, 1, 0, 3, 4, -4], [5, -11, 2, -6, 1, -3, -3, 13], [-9, -1, 2, -3, -3, -10, 0, -4], [13, -11, -1, 10, -10, -4, -3, 6], [-25, 5, 3, 5, 0, 2, 3, 7], [5, 8, -1, 2, 8, 3, 0, 20], [0, -9, 6, -3, 0, 4, -5, 6], [6, 4, 3, -1, -2, -2, 8, 25], [-9, 13, 8, -3, 1, 8, -1, -18], [-16, 3, -6, -5, -6, -2, -1, 3], [-2, 5, 3, -6, -9, -10, 0, -12], [16, -7, -2, 2, 10, -9, -3, 14], [2, 4, -2, 8, 2, -8, -2, 15], [-1, -2, -1, -7, -8, -14, -1, 4], [16, -2, -8, 13, 6, 1, -1, 1], [8, 14, -4, 5, 10, 6, 3, -30], [-3, 3, 1, 9, -2, 2, -2, 2], [-9, 0, -2, -2, 8, 0, 6, -11], [6, -4, -7, 9, 3, 3, -1, -4], [12, 8, -1, 9, -3, -1, 0, -10], [17, -6, 3, -4, 1, 0, 2, -5], [-4, 8, -15, -6, 7, -5, 3, -8], [-21, -5, 5, -15, -8, -1, 2, -4], [20, 4, 5, 8, 19, 4, -3, 6], [-16, -7, 0, -2, -3, 1, -3, 17], [4, -7, -5, -2, -1, 0, -8, 10], [23, -8, -4, 9, 9, 0, -6, 26], [-7, -4, 1, 15, 6, -7, -5, -2], [-26, 11, -4, -8, 2, 1, 1, -28], [4, 10, -4, -5, 9, 1, 9, -15], [6, -7, 6, 1, -2, -1, 9, 14], [2, -4, 1, -9, 6, 0, -1, -7], [-9, -7, -2, 1, -13, -2, -1, 18], [35, -3, -2, -1, -8, -9, 6, 17], [16, -7, -5, -4, -9, 1, -3, 4], [-14, -7, -2, 6, -2, 2, 6, 2], [19, 6, -3, -6, -2, -3, 5, 25], [11, -9, -5, 14, -2, 1, -11, 8], [12, 4, 1, -9, -7, 6, 4, -1], [-1, 5, -4, 11, 5, 6, 2, 16], [-2, 14, -3, 11, -3, 8, 1, 5], [-22, -6, -2, 5, -10, 4, 5, -2], [18, -11, 12, 1, 1, 7, 4, 17], [-2, -5, 6, -11, -2, -6, 6, -30], [-11, 8, 3, 6, -2, 3, 5, -8], [-10, 12, 11, 10, -1, 2, 3, -11], [-21, -7, 2, -7, -19, 1, -4, -10], [-25, 4, -8, 3, -11, -3, 5, -16], [20, 14, -5, 2, 19, -5, 1, -4], [15, -12, 11, -3, -1, -5, -8, -17], [-29, 0, 4, -6, -3, -3, 4, -13], [-16, -2, 5, -9, 3, -4, 0, -14], [9, 6, 1, -9, -4, -3, -4, -2], [-30, 6, -2, 3, 5, 4, 1, 13], [18, -1, 3, 7, -1, -7, -5, 10], [10, -4, 8, 2, 6, -1, -1, 22], [23, 4, 0, -11, 5, 3, 0, 8], [-13, 9, 0, 0, 17, 1, 0, -7], [-3, 7, 7, -10, -8, -1, 5, -16], [37, -8, 8, 17, -5, -3, -6, 18], [-24, 8, 3, -3, 7, 3, -1, 10], [10, -6, -4, -5, -9, -14, -2, -7], [14, 0, -5, -6, -11, -7, 10, 14], [-9, 4, 0, -2, -4, -3, -7, 13], [13, -5, -4, 3, 13, 0, -15, 14], [-4, -3, -6, 0, 2, -3, 7, -18], [-15, 5, -4, -10, 6, 4, 1, -39], [-6, 5, 1, -9, 2, 15, 2, -9], [23, -7, 10, 1, 15, 8, 3, 12], [4, 10, -2, -8, -2, -11, 6, 2], [-20, 2, -4, 8, 1, -9, -1, 17], [-1, -7, -3, 1, 6, 6, -5, 14], [13, 5, 4, 9, 9, 6, 7, 2], [-4, 14, -2, -8, 12, 3, -1, -11], [-20, 8, 1, -2, -8, -9, 6, -17], [14, -2, 3, -17, 9, -7, -6, 6], [19, 23, -8, 7, 1, -6, 4, -12], [-2, -6, 6, 0, 12, 5, -9, 4], [10, 1, -8, 11, 8, 11, -2, -2], [31, -4, 4, -3, -3, 2, -13, 16], [0, 12, 0, -15, 0, -7, 7, -5], [-13, 2, 0, 8, 15, 7, 1, 4], [-8, -12, -3, 0, 7, -2, -9, 0], [16, -6, 3, 3, 8, -10, -6, 9], [-17, 1, -5, 0, -8, 5, -7, -10], [7, -2, 6, -2, -11, 5, 2, 15], [15, 1, -4, -3, -1, 9, -2, -2], [2, 13, 7, -4, -3, -2, 0, 0], [-23, 2, -2, -5, 9, 10, 10, -2], [-22, 1, 6, -11, 8, -1, 1, -4], [-11, 10, 4, -3, -3, -2, 9, -4], [15, -10, 4, -9, 1, 5, -1, -5], [-27, 9, -3, -15, -2, -5, 5, -22], [18, 11, 7, 2, -2, -1, -1, 11], [10, -12, -10, -4, -15, -5, 2, 3], [-24, -14, -1, -7, -15, 3, 1, -12], [-6, -6, 17, 10, -13, -3, -8, -2], [-6, 13, -2, -5, 6, 3, -4, -42], [-19, 0, -12, 2, 3, -8, 3, 13], [-34, 2, -3, 0, 4, 3, -8, -19], [17, -13, -7, -3, -18, -5, 1, -4], [-12, 5, 7, 1, 7, -1, -6, 6], [13, -10, 9, 6, -16, -1, 2, -3], [9, 5, -6, -6, -8, -15, 2, 5], [4, -5, -5, -13, 1, 0, -3, 2], [-18, -12, 0, 11, -4, 8, -1, -10], [-6, -8, 1, -3, -1, 3, 13, 0], [14, 7, -2, 1, 12, -5, -5, 38], [8, -2, 1, 2, -6, -9, -5, -5], [4, 17, -9, 13, 3, -1, 4, 16], [-15, 17, 12, 9, 11, 15, -3, -12], [4, -3, -9, -9, 1, -6, 0, 16], [-20, 5, -5, -9, 6, 7, 3, -19], [0, -12, 7, -4, -10, -5, 5, -5], [32, 5, 0, -1, 6, -6, -6, 34], [7, 12, -5, 11, 10, 1, -4, 21], [-11, -13, 2, 4, -11, 2, -1, -28], [-5, -4, 15, -4, 0, 3, -3, -33], [-7, 0, -7, -5, 1, -3, -12, -11], [-7, 7, 8, -5, 7, 9, -4, -16], [25, -13, 10, 2, -3, 5, -3, 32], [5, 3, 22, -1, 19, 5, -4, -6], [1, 8, 6, 1, -3, 5, -3, -20], [31, -3, 14, 5, 7, -2, -2, 5], [-1, -12, 1, 4, -7, 3, -7, 9], [10, 11, 1, -2, 3, -3, -3, 18], [26, 19, 11, 5, 3, -8, -3, 6], [22, -8, 1, 1, 17, 6, 5, 22], [8, 3, -5, 9, -3, -5, -5, 30], [-10, 9, -13, 2, -9, 5, 2, -24], [-1, 8, -3, 7, 10, -1, -2, 13], [22, 3, -5, -1, 6, 7, -5, 1], [19, -14, 4, 3, -8, -10, -5, -13], [29, 14, -2, -1, 13, -6, -7, 4], [23, 7, 7, 6, 9, 7, -10, -5], [50, 1, 7, -4, 12, -3, 3, 13], [-33, -1, 14, -4, -13, -7, 7, -22], [34, 10, 1, -5, -3, 5, -2, 10], [-26, 11, -3, -10, -11, -5, 7, -3], [-13, -1, -4, -8, 3, 8, 1, -32], [28, -6, -2, 1, -3, 2, -12, -1], [-23, -13, 5, 0, 8, 1, -11, 6], [11, -2, -3, 8, 10, -2, -3, -12], [11, 3, 1, -2, -5, -4, 5, 11], [-27, 14, 5, 7, -6, -1, -4, -24], [2, 10, -11, -9, -1, -2, 11, -3], [3, 3, 2, -8, 3, 0, 5, -36], [11, 8, 5, -8, -4, -1, -1, -4], [7, -1, -6, -7, -22, -8, -5, 3], [-28, 2, -1, 7, -13, 1, 7, -18], [-30, 6, -4, -16, 0, -7, 11, -23], [16, 17, 1, -2, 17, -8, 3, 0], [17, 0, -4, -10, -7, -4, 7, -1], [28, 10, 5, -1, 11, 1, -7, 13], [23, 4, 0, 4, 1, -3, -10, 6], [-19, -7, -1, -5, 6, 0, -6, -2], [-14, -2, 4, -1, -20, -15, -7, -6], [-7, -18, 5, -6, -18, 3, 2, 12], [-21, 4, -8, -7, 5, 2, 12, -5], [52, 5, -1, -11, 3, -4, 6, 2], [11, 10, -3, -16, 4, 7, -5, -22], [-32, 0, -5, -6, -13, -6, 9, -14], [-4, -8, -17, 3, 11, 0, -5, 10], [20, -1, 6, 12, -10, 1, 0, 33], [-52, -2, 1, -12, -13, -9, 9, -2], [-16, -3, -1, -2, -11, -7, 1, 8], [-11, 5, 1, -7, -12, -7, 19, -12], [13, -7, -4, 24, 14, 4, -11, -11], [-23, 1, -1, -5, 5, -3, -8, 11], [-9, -1, -2, 5, 5, 11, -4, -20], [-10, -6, -2, 3, -2, -3, 6, -25], [30, -15, -3, 4, 8, 4, 1, 33], [-17, 1, -2, 2, 1, 13, -6, -2], [7, -9, 11, 1, -1, -13, -1, 21], [-23, 12, -5, 4, -8, -4, 15, -12], [-12, -9, -4, -8, 2, -11, 2, -14], [4, -3, 6, -13, -2, 1, -1, 11], [-27, 14, -4, -5, -3, -3, 11, -16], [14, 0, -2, -8, 8, 1, -5, 16], [28, 6, 4, -12, -5, -9, 15, -1], [-15, -14, 7, -12, -14, 4, -8, -14], [-16, 9, -4, -12, -2, -2, 3, -10], [-76, 5, -7, -11, -15, -7, 6, -24], [25, 16, 4, 13, -1, 3, -5, 14], [-7, 4, 4, -9, -13, -19, 7, 2], [-5, -8, 8, -3, 9, 3, -10, -16], [15, 2, -10, -3, -11, -4, 2, 3], [-6, -2, -6, -10, 2, 0, -5, -2], [1, 5, -5, -13, -6, 1, 8, -2], [-17, -1, -10, 9, -4, -7, 12, 5], [-31, 3, -11, -11, 2, 3, 6, -36], [11, 2, 9, 4, 5, 14, -7, -3], [36, 9, 0, 17, 14, 11, -4, 14], [-22, -11, -12, 8, 6, 6, 9, 10], [-13, 13, -7, 8, 26, 0, -6, 2], [-32, -9, 2, -6, 0, 0, -8, -8], [15, -7, -6, 19, 15, 2, -16, 8], [12, -7, -3, -2, -15, -4, 4, 7], [15, -3, 18, -4, 7, 5, -8, -12], [-3, 5, 0, -4, -13, -4, 13, -14], [-11, -13, 4, -14, -17, -10, 9, -10], [-33, 13, 13, -1, -4, 6, 8, -9], [48, 6, -3, 24, 7, 4, -2, 19], [-11, -3, 5, -3, -24, -8, 8, -10], [14, -11, -4, -1, -4, -7, 6, -10], [-1, 2, 11, 7, -4, 5, 1, -16], [-16, 6, 22, 0, 8, 12, -14, -12], [4, 1, -6, -11, 2, 2, 19, -5], [35, 10, -4, 17, 9, 10, -10, 22], [-14, -5, 23, -6, -3, -2, 2, 6], [25, 4, -1, -8, 2, -11, 4, -22], [19, 16, 0, -3, 11, -5, 0, -20], [-49, 7, -7, -22, -18, -2, 13, -3], [26, 2, -7, -7, -3, 0, 12, 29], [-7, 8, 2, 18, -2, -4, 1, -1], [6, 7, 5, -10, 19, 13, -8, -14], [5, 6, 12, -4, 5, -9, -5, 37], [34, 4, 1, -6, 0, -3, 10, 17], [-36, 5, -8, 9, -2, 2, 10, -5], [18, 11, 16, 0, 2, -2, -12, -13], [-14, 1, 2, 6, -3, 9, -2, 15], [-4, 3, -2, -2, -15, -4, 0, 9], [17, 0, -2, 9, 23, 2, -12, 16], [-28, -2, 1, -16, -21, -3, 3, -14], [0, -9, -1, 2, 2, -6, -5, 29], [12, 21, -4, 0, 14, 4, -11, -15], [17, -10, -13, 12, 0, 2, -7, 28], [9, -5, 7, 4, -4, -16, -12, 36], [-6, -22, 14, 6, -7, 7, 0, -25], [56, -16, -2, 1, 6, 2, -11, 42], [11, -8, 6, -19, -6, -15, -1, -23], [19, 11, -3, 4, 10, 12, -5, 2], [-7, 22, -10, -2, -7, 7, 16, -10], [19, 6, 4, 13, 9, 0, 0, -9], [18, 14, -4, -11, -10, -18, 3, 8], [16, -9, 0, 4, -13, 2, -9, 47], [4, 1, 2, 2, -10, 0, 6, 13], [-15, -16, 5, 0, -6, -11, 2, 18], [4, -15, -10, -4, -4, -5, -8, -8], [10, 2, 12, -4, -24, -7, 12, -9], [23, -23, 7, 1, -6, -9, -11, 2], [2, 0, 0, 19, 4, -11, -6, -2], [8, -7, 5, -7, 1, -4, -3, -7], [-11, 2, -4, -6, 5, 20, 3, -34], [-3, 4, 3, 3, 2, 15, 3, -21], [-11, 6, 2, -8, -2, 2, -7, -7], [-3, -9, 10, 15, 9, 9, -1, -10], [8, 5, -6, 2, 14, 8, -4, 33], [-19, -23, 14, -10, -21, 4, 4, 12], [31, -10, 5, 1, 11, -7, 1, 35], [-48, 18, -2, -9, 4, 7, 4, -18], [30, 0, 11, -2, 16, 13, 2, -1], [3, 3, 14, 14, 0, -9, 1, 17], [-8, -18, 0, 6, -26, -2, 5, 2], [18, -9, 11, -19, 7, 10, 1, 10], [11, 8, -14, 9, 12, 8, -2, -19], [33, -24, -4, 0, -13, -4, -7, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7942_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7942_2_a_bm();".
// 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_7942_2_a_bm(:prec:=8)
    chi := MakeCharacter_7942_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_7942_2_a_bm();".
// 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_7942_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7942_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-19, -58, 2, 87, 13, -34, -7, 4, 1]>,<5,R![9, 126, -167, -154, 134, 25, -22, -1, 1]>,<13,R![10604, 31494, 19105, 1281, -1903, -486, 3, 12, 1]>],Snew);
    return Vf;
end function;