
// Make newform 7942.2.a.bs in Magma, downloaded from the LMFDB on 25 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_bs();".
// 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_bs();".
// 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 := [81, 648, -621, -1596, 972, 1236, -581, -408, 162, 59, -21, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [40812606, -482664987, 67446858, 866719260, -257318814, -451891531, 144702246, 84133296, -27776825, -4612281, 1662627, -36106], [-29700081, 512716329, -96396558, -797198319, 282218643, 377717441, -156938793, -62468688, 29652583, 2182854, -1747173, 128564], [-373530798, -868570083, 874167639, 1877956020, -926512059, -924159005, 421020771, 145872735, -75920590, -3745263, 4432509, -405461], [-176546574, 168738975, 865436523, -325351413, -731228970, 202415434, 232916424, -61047876, -31571206, 8389437, 1512099, -411575], [177627060, 220086846, -216410547, -609951717, -64728633, 430643887, 72284757, -116721489, -14428411, 13443351, 832122, -557519], [61834725, -221533488, -415759962, 174238677, 437072313, -55678748, -171119802, 18010035, 27509099, -3903051, -1494300, 269641], [80775, 10405599, -7333002, -13793571, 11684083, 4114755, -5635923, 124142, 1027749, -163248, -60119, 13107], [224961003, 1499064741, -423785913, -2592189963, -117505860, 1585199306, 209996652, -427430148, -52047371, 51859659, 3438855, -2304427], [-330617970, -654383565, 2028271047, 359417484, -2392653927, 179450527, 1020506352, -169403457, -173550649, 37214754, 9752256, -2364614], [-54356427, -102234564, -340801851, 75683961, 757354704, -141787421, -386577282, 85197561, 70519139, -17332797, -4086648, 1072042]];
    Rf_basisdens := [1, 1, 32651937, 32651937, 97955811, 97955811, 97955811, 32651937, 572841, 97955811, 97955811, 32651937];
    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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, -1, 0, -1, 0, -1, 0, 0, -1, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, -1, 0, 0, 0, 1], [-4, 0, 1, -1, 0, -1, 1, -1, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, -1, 0, -2, -1, -2, -1, -2, -1], [-1, -1, 0, 0, 1, 0, -1, 1, 1, -1, 0, -2], [1, -1, -2, -1, 0, -1, -2, -1, -1, 1, -1, 2], [1, 0, -2, -2, -1, -1, -1, 0, 0, 1, -2, 2], [1, -1, 0, 2, -1, 0, -1, 1, 1, -2, 0, -3], [-2, 3, -1, -1, 1, 2, 2, 0, 0, -3, -1, 0], [0, -1, 0, -1, -1, 1, -2, 0, -1, -1, -3, -1], [2, -1, 0, -1, 1, 1, -1, 1, -2, -1, -1, -1], [4, 1, 1, 2, 0, -3, -4, 0, 0, -1, 2, -1], [1, 0, -2, 1, 2, -1, -1, 1, 2, -2, 0, -2], [-3, 0, 2, 1, 2, 1, 2, 0, 1, 1, 2, 1], [-1, 0, 2, -1, 0, 2, 1, -1, -1, 1, 0, 0], [-2, 0, 1, 2, -2, -6, -4, 1, 2, -1, 0, 1], [4, -2, 0, 1, -2, 1, 1, 3, 0, -1, 1, -2], [-10, 1, 0, -3, 5, 2, 4, -1, -1, 3, 1, 1], [2, -1, 2, 0, 0, 3, 3, -1, 0, 1, 0, -1], [4, -1, -2, -3, -2, 0, 2, 0, 1, -1, -2, 1], [-4, -1, 3, 0, -1, 1, 1, 0, 1, 1, 2, 1], [-2, -2, -1, 0, 0, -3, -3, -3, -2, 4, 0, 1], [0, -1, 1, 1, 4, 2, 2, 2, 0, 2, 3, -1], [0, -2, 0, 0, 0, 2, 4, -2, -2, 1, -1, -1], [6, -2, 0, 1, 0, -1, -2, 1, -3, -2, 0, -2], [3, -2, 2, 1, -1, -6, -2, 0, -1, 1, 1, 0], [-4, -1, 0, -2, -2, -3, 1, 1, 3, 0, 1, 2], [1, -2, 2, 3, -3, 3, 2, 1, 1, 1, 1, 0], [0, 0, -3, 0, 2, -3, -2, 2, 0, 1, -2, 1], [-5, -2, 4, 0, 0, -2, 3, -1, 2, 4, 1, 4], [1, 1, 3, -1, 3, 2, 8, -1, -1, -1, -1, -1], [-1, -2, 3, 1, 1, 3, 1, 2, 0, 1, 3, -3], [-2, 2, 0, 0, 3, 4, 5, 0, 0, -2, 3, -1], [3, -4, -4, 0, -3, 3, -1, 1, 1, 2, 1, 1], [5, -1, -2, -1, -2, -3, -4, 1, 0, -1, -2, 0], [-1, -2, -1, 2, 0, -2, -1, 4, 5, -2, 4, 0], [-3, 4, -1, -1, 0, 4, 1, -4, 0, -2, -4, 0], [-4, -1, 2, 1, 7, 2, 7, 2, 0, 4, 2, -1], [5, 0, 1, 1, 2, 3, 3, 2, -1, -1, 1, 1], [-8, 4, 0, 2, 0, 0, -1, 2, 1, -2, 1, 3], [3, 1, -2, 2, -1, -3, -3, -1, 3, -1, 2, -1], [-1, 0, 1, -2, 0, -2, -3, -4, -2, 6, -4, 4], [8, 2, -1, 1, 0, 2, 1, 2, 1, -4, -1, -1], [-6, 2, 1, 0, 2, -1, -2, 3, 4, -3, 6, -2], [-4, -3, -1, 2, 1, 5, 7, 4, 1, 0, 4, 2], [-3, -1, -1, 2, 1, 0, 0, -2, -1, 1, 0, 0], [5, -3, -2, -1, -2, -7, -6, 0, 1, -2, -3, -2], [1, 4, 1, -2, 2, -5, -6, -2, -3, 2, 0, 0], [0, -2, -2, 0, 2, 2, -1, -1, -1, 0, 2, -3], [-3, 0, -1, 4, -3, 1, -3, 4, 2, -2, 5, 2], [-8, 0, 4, 2, -1, 0, 5, 2, 1, 0, 2, 2], [-3, 3, -5, -3, 0, 0, 2, 0, -2, 0, -2, 1], [-5, 2, -3, 0, 0, 6, 2, 2, 2, -4, 2, -1], [-1, -3, -6, -2, -4, -2, -7, -3, -2, 1, -5, 2], [-4, 1, 0, -3, 5, 3, 1, -2, 1, 3, 0, 4], [-1, -2, -2, -1, -2, 0, -8, -3, -4, -4, -4, -1], [-3, 4, 3, 2, 0, 3, -3, -2, 1, -3, 4, -1], [1, 1, 2, -1, 2, -8, 8, 0, 0, 1, 0, 1], [-7, -5, 2, -2, 0, 4, 0, -7, -2, 9, -2, 3], [-4, -7, 2, 4, -1, 0, 1, 2, 0, -3, -1, -6], [3, 4, 1, 1, -1, 10, -1, 1, 2, -3, -2, -2], [-5, -2, 6, 2, 7, -2, 4, 2, 0, 4, 1, -2], [-8, 5, 3, 2, 4, -2, 5, 0, 2, -3, -1, -1], [-9, -1, 1, -4, -2, 0, 2, 0, -3, 4, -2, 5], [-3, 0, 4, -1, -2, -4, 2, -1, -2, 3, 0, 1], [-7, 0, 0, 1, -1, -2, -2, -1, -3, 5, 2, 0], [-5, 3, 2, 3, 2, 6, 7, -3, -2, 3, 5, 1], [-2, 0, -5, -2, -2, -6, -3, 0, 1, 1, -4, 5], [-8, 0, 0, 0, 0, 0, 2, 1, 0, 5, 3, 1], [-2, 5, 2, 2, 1, -3, -1, 2, 2, -5, -4, -3], [12, -3, 3, 5, -6, -4, -2, -1, 3, -2, -1, -2], [11, 0, 3, -3, 2, 6, 0, 3, -3, 2, -2, -1], [-4, -4, 0, 0, 1, 1, -2, -2, 2, 1, 2, -2], [1, -2, -2, -4, -3, 3, -1, 0, -4, 5, 0, 2], [-10, -1, 1, 2, 2, 1, 1, -6, -2, 7, 2, 4], [-4, -3, 4, 2, 0, 3, 0, -1, 0, -1, -1, -6], [-6, -3, -5, -1, 0, -2, 4, 6, 1, 3, 1, -1], [-13, 1, 2, 3, 5, -1, 3, 4, 4, 2, 2, 1], [7, 0, -3, 0, -9, 1, -6, 0, 0, 0, 2, -2], [9, 5, -5, -2, -4, 2, -3, 1, 4, -4, -3, 0], [-2, 0, 9, 6, 3, -3, 4, 3, 2, -3, 2, -6], [-3, 2, -5, -3, 3, 2, 0, -3, -5, 0, 1, 0], [-10, 2, 5, 6, 5, 0, -4, 3, 1, 0, 3, -7], [-10, 2, 2, -1, -2, -1, -3, -4, 4, 2, 4, 0], [-2, -6, 4, 0, 2, -1, 4, -4, -2, 6, -3, 2], [-3, 2, -7, -2, 1, 3, 0, 5, -2, 0, -1, -1], [-3, -3, -2, -6, 4, -2, -6, 1, -2, 6, -4, 1], [2, 0, -7, -4, 2, 10, 4, -4, 0, 0, -3, 4], [-5, 5, -3, 5, 0, -9, -7, 4, 1, -4, 4, -4], [7, -9, 3, 3, -1, 4, 2, -2, 1, 2, -3, 0], [1, -5, -2, 4, -2, 0, 2, -2, 0, 4, 0, 1], [4, 2, 2, 0, -1, -12, -10, -3, 0, 1, -3, 3], [7, 7, -4, 2, 4, 1, -2, 5, -2, -7, 1, -3], [3, 4, -4, 5, 3, 5, 2, 3, 0, -1, -1, -4], [4, 4, -4, 3, 0, -2, 1, 3, 4, -9, 2, -1], [8, 4, 4, 3, 5, 7, 12, 4, 4, -1, 5, -1], [-8, 7, 4, -1, 5, -4, -1, -1, -1, 1, 5, -2], [-5, -5, -3, -3, -1, -9, -8, -1, 2, 10, 1, 2], [-1, 2, -4, 2, -4, -9, -19, 4, -2, -3, -1, 1], [-1, 1, 0, -1, -5, 5, -1, 4, 1, -1, 0, 7], [-4, 2, -3, -1, 11, 2, 1, 6, 2, -1, 5, -3], [-8, 4, 8, 5, 8, 4, 5, 4, 2, -5, 7, -2], [-7, -1, -7, -4, 1, -5, -4, -4, 0, 3, 1, 3], [0, -5, 1, 1, -5, 1, -4, -3, -2, -3, -8, -6], [-5, 3, 1, -1, 3, -5, 5, 6, 7, -4, -1, -3], [4, -4, 9, 10, 3, 5, -4, 0, -5, 0, 6, -9], [-6, 2, 4, 1, 3, 17, 15, 1, 4, -3, 5, -6], [-6, 3, 6, 4, -4, -4, 11, -1, 0, 3, -1, 3], [-1, -3, -2, -4, 1, -3, 0, -5, 3, 5, 3, -2], [16, 5, -2, 3, 1, -6, 0, 3, 0, -3, 1, -1], [2, 7, 1, -2, 5, 10, 9, -4, -5, -6, -1, 1], [-3, 2, 6, -1, -2, -3, -6, -1, -3, 3, -8, 0], [-2, -6, 1, -2, 0, 6, -2, -4, -3, -2, -5, 0], [-1, -2, -7, -7, 0, 6, 1, -3, -2, -1, 1, -1], [-4, 3, -4, -2, -1, 9, 7, 2, 0, -1, 4, 3], [-11, 10, 3, 8, 5, 3, 0, 4, 0, -1, 3, 0], [-12, 5, -8, -6, 2, -4, 2, 6, 4, -2, 3, -1], [-15, -2, 7, 3, 4, -3, -1, -5, -3, 1, -3, 2], [4, 0, -4, -8, -5, -5, -2, -8, -1, 6, -4, 6], [-3, 1, 2, -3, 4, 1, 5, -1, -3, -1, -4, 4], [3, 5, 0, -3, -3, -6, 4, 2, 6, 1, -2, 4], [6, -11, -3, -2, -7, -3, -8, 3, 0, -4, -6, -1], [5, -2, -1, -4, -3, 2, 3, 9, 4, 3, 2, -1], [1, 2, 2, 2, 0, 2, 1, 5, 3, -5, 9, -4], [-1, 4, 5, 1, 0, 0, -6, -3, 0, -5, -7, -5], [2, 8, -1, 4, 0, -1, -7, -1, -6, -8, 0, -4], [7, 4, 1, 7, -1, -9, -1, 4, 6, -7, 0, -6], [3, -4, 9, 0, -3, -6, 4, -9, 5, 8, -3, 5], [-10, 9, 1, 5, 7, -1, 3, -4, -1, -2, -2, 5], [0, 11, 1, 8, 2, -2, -4, 1, 1, -3, 4, -1], [10, -3, 2, 3, 1, 5, -15, 4, -4, -8, 1, -11], [-13, -6, 1, 2, -4, -3, -10, 9, 4, -2, 1, -5], [-11, -1, 0, -5, 1, -4, 13, -1, 2, 1, -1, -3], [-12, 0, 1, 2, 6, 0, 5, -5, -2, 3, 10, -1], [3, -5, 0, -4, 7, -3, 9, -7, -5, 2, 0, -5], [10, -1, 2, 1, -4, -6, 1, 4, -4, 2, 4, 0], [-13, 10, -1, 3, 4, -5, -1, 3, 2, -10, 1, -5], [-8, 1, 4, 0, 7, 14, 19, -6, -2, 7, 6, 6], [-13, -6, 9, -5, 5, 4, 16, -5, 4, 8, -1, 2], [-15, 0, 3, 0, 1, 6, 5, 1, -2, -2, 4, -3], [-5, 2, -5, 1, 6, 4, 7, 4, -5, -6, 1, 0], [-18, 6, -4, -2, 5, 14, 17, -8, -3, 10, 1, 4], [-8, -6, -3, -6, 0, -2, 8, 0, -2, -1, -9, 0], [-1, -1, 9, 3, 1, -5, 4, -6, 2, 18, 5, 7], [-4, -10, 2, -3, -6, 3, 5, -3, -3, -1, -1, 1], [-11, 6, 3, 2, 5, -1, 6, 8, 11, -8, 0, 0], [-2, 0, -5, -1, -6, -2, -12, 1, -7, -1, -4, -2], [-5, 5, 4, -4, 6, 4, 17, 7, 6, 1, 2, -4], [-18, 17, 1, 3, 4, 5, 10, 5, 3, -9, 7, 0], [-4, -3, 0, -3, 6, 0, -9, 1, 1, -1, -3, 1], [2, 4, 0, -1, -3, -10, -15, -1, -2, 0, -3, 9], [4, -5, -4, 0, -9, 0, 5, 0, 2, -4, 0, 0], [13, -10, 6, 2, -6, 0, 6, -3, 0, -1, 0, 2], [-5, -7, -7, -1, -4, 6, -9, -5, -8, 4, 4, 4], [-10, -5, 1, -3, -6, -2, -5, -2, 0, 8, -2, 5], [-24, 1, 4, 4, 4, 0, -7, 6, 4, -2, 6, -11], [-5, 2, 1, 5, 4, 7, -5, 5, 0, -9, -1, -11], [-2, 2, 9, -2, 5, -8, 13, -6, 2, 5, 1, 6], [6, 0, 1, 6, -8, -11, -7, 1, 8, -3, -7, 3], [-13, 3, 8, 5, 1, -8, 10, -4, 3, -5, -3, 5], [-5, 3, -7, -1, -7, 0, -16, -6, 0, -6, -4, 2], [-4, -4, -2, -9, 7, 16, 14, -8, -6, 9, -2, -5], [-21, 1, 1, -5, 0, 11, 9, -7, -3, 2, 1, 1], [-5, 6, -3, 0, 10, -5, -13, 1, -4, 5, 7, 2], [-21, 2, -4, -10, 5, -2, 13, 1, 0, 7, 1, 5], [-17, 1, -2, -5, 3, -3, 5, -3, -5, 4, 9, 3], [-11, 0, 6, -4, 1, -6, 6, 6, 5, -5, -3, -4], [0, -7, -5, 2, 0, 3, -13, -2, 2, 4, 0, -3], [13, -3, -5, 2, -2, -13, -11, 9, 4, -13, -6, -11], [1, 2, 8, -3, -4, -5, 13, -6, 5, 7, -2, 4], [-6, -4, 7, 0, -7, -4, 4, -3, -2, 4, -2, 6], [-10, 7, 0, 4, 1, 12, 5, 8, 9, -6, 4, -7], [10, 5, -10, 0, -3, 7, -4, -4, -5, 3, 2, 7], [-9, 4, -3, 5, -3, 12, 1, -3, 2, 5, -3, 7], [16, -2, 2, 4, -11, -1, -1, -2, 6, -7, -3, 0], [20, -7, -1, -1, -5, -13, -15, -1, -4, -4, -2, 2], [-6, -7, 9, 0, 0, -4, 11, -5, 11, 4, 3, 3], [-2, 0, 3, -4, -1, 0, -8, -9, -9, 1, -12, 3], [-11, -6, 12, 9, 5, -4, -9, 6, 6, 2, 6, -10], [0, 5, -1, -1, 13, 10, 6, 13, -1, -2, 11, -8], [-10, 0, -1, 0, 2, 18, 23, 4, -4, 3, 3, 5], [-11, -2, 3, 7, -4, 11, -5, 6, 6, -3, 7, -11], [18, -3, 1, 5, -3, 8, 1, 6, -3, 3, -1, -6], [-9, 2, 5, -2, -1, 6, 6, -1, -2, 9, -1, 9], [7, 13, -8, -1, -1, 3, -1, 4, 4, -2, -2, 1], [5, 9, 6, 7, 2, -18, -1, -1, 9, 7, 8, 8], [-19, 3, 11, 1, 15, -16, 14, -1, 4, 3, 9, -1], [19, 0, 2, 0, 7, 18, 2, -2, -9, 2, -8, -7], [-1, -8, -10, -5, -6, 7, -6, -4, -5, -3, -4, 0], [8, 6, -4, 0, 0, -2, 7, 1, 4, -6, -4, 8], [9, 4, -3, -2, -6, -17, -10, -3, 3, -3, -2, 7], [11, -2, -6, 4, -1, -7, -8, 9, 3, 0, 10, -5], [0, -3, 0, -1, -2, -7, 8, -13, -3, 14, -4, 12], [-22, 0, 4, 1, 2, 8, 4, 2, 4, 1, 0, 4], [-7, 1, 5, 7, 0, 2, 19, -3, 0, 2, 3, -2], [-4, 1, 2, -4, 6, 10, 21, -1, -6, 8, 6, 4], [19, 2, 0, 1, -6, 8, -7, 0, -3, 1, -1, 5], [-9, -1, 1, 7, 10, -2, -11, 11, 3, -7, 8, -11], [3, 8, 0, -5, -8, -7, 3, -3, -2, 5, 2, 12], [13, -1, 0, -1, 2, 5, -9, -7, -12, 12, 2, -1], 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-9, -4, -11], [-40, 13, -3, -9, 5, -15, 24, -3, 2, 5, -5, 15], [-33, 4, 10, 6, -2, 3, 2, 0, 0, 2, 6, -7], [-14, -2, -1, 9, -1, 0, 4, 4, 14, -6, 13, 6], [20, -11, -9, -7, -13, 6, -11, -3, -16, 1, -10, 6], [8, 13, -6, -8, -9, -31, -20, 1, -4, -4, -11, 4], [-23, 10, 8, 5, 10, -20, 0, 14, 15, -4, 12, -4], [6, -9, -18, -6, -2, -4, -4, -8, -6, 5, -14, 14], [-12, 2, -17, -16, 2, -4, -20, -11, 0, 3, 4, 11], [32, 4, -16, -12, -20, 11, 1, -3, -6, -2, -15, 6], [-6, -2, 7, 9, -6, 1, -5, 8, 12, -3, 11, -11], [4, -10, -2, 6, 1, 4, -11, -9, -12, -11, -13, -18], [22, 1, -15, 1, -23, 6, -23, 4, -2, -15, -2, -11], [-3, -9, 10, -4, -3, 5, 5, -11, -5, 18, -7, 3], [7, -1, -5, -3, -10, -11, -8, -14, -7, -5, -16, 2], [18, -4, -1, 3, -2, 14, -15, -1, 4, -5, 10, -14], [24, -5, -10, 1, -12, -1, -15, -5, -20, 4, -15, -11], [24, 6, 1, 9, -18, -6, -13, 0, -3, -1, 3, 8], [0, -2, -5, -7, 2, 19, 31, 7, -4, 5, -6, 15], [-32, 1, -9, -2, -6, -6, -8, -4, -3, 20, 13, 19], [-2, 9, -5, -4, -2, 30, 6, -15, -8, -5, -2, 1], [1, 10, -15, -4, 1, -9, -7, 7, 13, 1, 11, 6], [14, -19, -3, 9, -15, 6, -5, -9, -18, -4, -4, -7], [19, 4, -5, 3, -8, 3, -29, 0, -5, -23, 7, -21], [22, 2, 3, 3, 8, -11, -7, 10, -10, -14, 5, -11], [0, 14, -5, -5, 4, 13, 9, 2, -6, 0, -11, -4], [43, 2, -5, -1, -4, -15, -22, 5, 2, -21, 0, -2], [-31, -5, 6, -8, -1, 9, 6, -10, -5, 6, 2, 11], [9, 7, -3, -5, -13, -1, 0, -13, 8, -6, -13, 5], [-6, 7, -5, 5, -6, -19, -36, -7, 0, -6, -10, -5], [10, 0, -7, -13, -6, -6, -23, -18, -12, -6, -4, 0], [-11, 14, 8, -4, 9, 18, 10, 8, 5, 5, 7, 2], [-10, -2, 10, -1, 4, -3, 11, 4, -5, 6, 22, -2], [21, 10, 1, 14, 3, -3, -8, 7, 4, -8, 10, -10], [41, -11, -8, 5, -13, 9, -28, 6, -1, -20, -1, -13], [31, 4, -19, -1, -4, 10, 10, -8, -18, -9, -6, -4], [-20, 0, 17, 2, 13, 15, 16, 7, 1, 21, 19, 11], [18, -8, -2, 0, -9, 5, 21, 0, 5, -3, -6, -7], [5, -4, -8, -9, 1, -3, 3, -5, -12, 6, -19, -1], [-9, -2, 0, 9, 2, -23, -14, -11, 5, 3, 8, 3], [-1, -8, -5, 3, -6, 16, 2, -5, -6, -10, -4, -9], [-2, 5, 7, -6, 5, 20, 5, -14, -8, 0, -18, 2], [7, -3, 1, -5, 22, 0, -3, 7, -4, -2, -5, -13], [-17, -11, 16, 4, 11, 15, 14, 1, 11, 3, 9, -16], [7, 20, -1, -9, 11, -3, 3, 4, -1, -2, -2, -1], [2, 3, 3, 14, -2, 1, -41, -3, 0, 2, 0, -5], [-9, -12, 8, 8, -24, -9, -15, 5, 0, -2, -1, 3], [14, -1, -17, 6, -11, -10, -26, -1, -7, -1, -3, 2], [20, -5, -11, -4, -9, 11, -6, 2, -1, 3, -11, -4], [-14, -1, -4, -7, 9, 6, 27, 1, -2, 10, -15, 9], [-4, 3, -8, -14, 9, 16, 3, 5, -6, 2, -7, 7], [12, -1, -9, 3, 4, -8, 15, 12, 9, -4, 0, -5], [-6, 3, 3, 8, 20, 13, 21, 23, 6, 0, 31, -9], [15, 4, -5, 9, -9, -8, -20, -4, 3, -4, -11, -2], [-18, -3, 0, -8, -5, -18, -19, 4, 2, 9, 0, 0]];
    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_bs();".
// 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_bs(:prec:=12)
    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_bs();".
// 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_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7942_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![81, 648, -621, -1596, 972, 1236, -581, -408, 162, 59, -21, -3, 1]>,<5,R![-57, -1080, -4995, -9042, -5604, 1968, 3449, 663, -465, -183, 3, 9, 1]>,<13,R![-46567, 174321, 17967, -190063, 45258, 49008, -17028, -4080, 1857, 100, -75, 0, 1]>],Snew);
    return Vf;
end function;