
// Make newform 4334.2.a.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4334_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_4334_2_a_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_4334_2_a_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 := [-45, 364, -589, -1160, 2328, 1621, -2966, -1056, 1814, 295, -582, -13, 94, -8, -6, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1206969, -5195161, -54277538, 57532164, 173150090, -80783623, -162535737, 57985681, 64364521, -23566082, -10763764, 4789161, 425109, -368011, 38967], [-6667041, 29020637, 41879694, -105733568, -109647076, 129455101, 99457169, -83109287, -37764223, 29719730, 5331816, -5421409, 83031, 385559, -54577], [9614679, -37132227, -8677928, 131815598, -38061078, -207141833, 46213081, 144520227, -23976157, -48396178, 7561874, 7723627, -1355241, -474903, 98545], [-4212711, 20801875, 4119481, -82511057, 13025086, 137575398, -12475420, -101964871, 6862302, 36652260, -3409969, -6314419, 889566, 418510, -80572], [-8052534, 46589803, -11155560, -160461943, 63141154, 216656261, -63949346, -136201942, 30673450, 42707227, -8512356, -6517250, 1322100, 387355, -86087], [-2783562, -4272852, 60359008, 28493975, -179832582, -108981575, 156946498, 98685486, -63023512, -36106162, 13864988, 5892832, -1717950, -359889, 95161], [-327249, -15173542, 59640216, 67736473, -196917457, -167007722, 177756743, 137700412, -73469164, -48853678, 16736406, 7881734, -2151627, -479233, 122660], [6171147, -61626257, 69913222, 282202990, -250384418, -522481431, 222123893, 394052573, -96384309, -138607224, 26403326, 22996733, -4408503, -1461863, 315587], [9127485, -58103513, 39397850, 219074643, -154808801, -343823822, 140933616, 238297865, -61404685, -79521244, 16167988, 12684189, -2530686, -781019, 171261], [-12825138, 81641645, -66568482, -295970219, 255858365, 463558579, -236646475, -318963461, 102423653, 104958869, -25498530, -16438801, 3660657, 991991, -229297], [23916789, -147425245, 115541358, 600794404, -461061778, -1015675733, 419414591, 732657025, -183244921, -250923130, 49263864, 40845473, -7971699, -2558707, 556079], [-18878145, 134300729, -125603226, -607023698, 509130320, 1144422331, -461314723, -871959305, 202934273, 308276840, -56674248, -51272989, 9641889, 3263243, -698419], [-38939397, 264001171, -227477512, -1078771302, 921906808, 1877086147, -850031139, -1371894949, 371832821, 471467996, -98195870, -76630953, 15465027, 4782031, -1054983]];
    Rf_basisdens := [1, 1, 634962, 634962, 634962, 317481, 317481, 317481, 317481, 634962, 317481, 317481, 634962, 634962, 634962];
    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_4334_a();"
function MakeCharacter_4334_a()
    N := 4334;
    order := 1;
    char_gens := [1971, 199];
    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_4334_a_Hecke(Kf)
    return MakeCharacter_4334_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, 0, 0], [-1, 1, 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, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 1, 0, 0, 0, -1, 1, 0, -1, 1, 0], [1, -1, 0, 1, -1, 0, -1, 0, -1, 0, 0, 1, 0, 0, -1], [-1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1], [-2, 0, 0, -2, 1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 1], [-1, 1, -1, -1, -1, 0, -1, 0, -1, -1, 0, 0, 1, 0, 0], [-3, 0, 1, 1, 0, -1, 0, -1, 0, -2, 0, -2, -1, 0, 0], [-1, 0, 0, -1, -1, 0, 1, 0, 1, 0, -1, 0, 1, 0, -1], [1, -2, 0, 1, 1, 1, -1, 0, -1, 1, 0, 3, 0, 0, -1], [0, -2, 1, -1, 1, -1, 1, 0, 2, 2, -1, -1, 0, 1, 0], [-2, -1, 1, 2, -1, -1, 2, 2, 1, -1, 0, -2, -1, 0, 0], [0, 1, 1, -1, 1, 2, -1, 1, 2, 0, 2, -1, -1, 2, 2], [-4, 0, -1, 0, 1, 1, 0, -1, -3, 1, -2, 1, 0, -2, -2], [-2, 1, -1, 0, -1, 0, 0, 1, -1, 0, 0, -1, 0, 0, 1], [0, -2, 1, 0, 1, 1, 0, -2, -1, 0, -1, 1, 0, -1, -1], [0, -2, 2, 1, -1, -1, 0, 1, 0, 1, 1, 0, -1, -2, -1], [0, -2, -2, 1, 1, 0, -2, -1, -2, 2, -1, 0, 0, 0, -1], [-1, 1, 1, 0, -1, 2, 0, 2, -1, -2, 1, 0, -3, 0, -1], [-4, 3, -2, -3, 0, 0, 1, 3, 0, 0, -1, 1, 2, -1, 0], [-1, 1, 0, 0, -2, -4, 0, 4, 3, -2, 1, -2, 1, 1, 2], [2, -3, -1, 2, 0, -2, 1, 0, 1, 1, 1, 1, 0, 2, 1], [1, -4, 2, 1, 1, -2, 4, -4, 4, 5, -3, -1, -1, 2, 1], [1, 0, -1, 2, -1, -1, -2, -1, -2, 1, 2, 1, -2, 0, 0], [1, -1, 1, 2, 0, -3, 0, -1, 2, 3, 0, -1, 1, 0, -1], [-2, -1, 0, 3, -1, -2, 0, -1, 3, -2, 1, -3, -1, 0, -1], [0, -2, -1, 0, 0, 0, 1, -2, 1, 1, 1, 1, 1, 1, -2], [-4, 5, -1, -2, -2, -1, 3, 0, 3, -1, -1, -3, 2, 0, 3], [-6, 3, 1, -2, 0, 1, 0, 1, 2, -1, 0, 1, 1, -1, 1], [3, -3, 2, 2, 0, -2, 1, 0, 2, 3, 0, 0, 1, -1, 0], [-3, -1, -4, -2, 0, 0, 3, -2, -2, 3, -3, 1, 2, 1, -1], [-5, 2, 1, 1, 1, 0, 1, -2, -1, -2, -2, -4, -1, -2, 0], [-8, 5, 1, -3, 2, 0, -1, -1, 2, -1, -2, -1, 1, -1, 3], [-6, 3, -2, 1, -1, 1, -2, -2, -1, -2, 0, 0, 1, -1, 0], [-4, -1, 0, -1, 1, -1, 4, -2, 1, 2, -3, -1, 2, 0, -1], [-1, 5, -1, 1, -2, -3, -4, 3, -2, -1, 2, -2, 3, -2, 2], [-6, 2, -1, -1, 3, 2, 3, -1, 1, 1, -1, 1, 1, 2, 0], [0, 0, 0, 1, -1, 4, 2, 1, -1, 0, 1, 3, -2, 1, -1], [-5, 4, 1, -3, 1, 4, 1, 2, 1, 1, 0, 0, 2, 1, 1], [-7, 5, 0, 1, -1, -2, -3, 3, -1, -2, 1, -3, 0, 0, 2], [0, -1, -3, -2, 0, -3, 0, -1, -1, 1, 0, 0, 2, 1, 3], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, -3, -1, -1, -2, 0, 1, 1, -4, -1, -1, 3, -1, -2, -5], [3, -1, 2, 0, 2, 1, 1, -3, 0, 4, -2, 3, -2, 0, -1], [-1, 0, -2, 2, -1, 1, -5, 3, -3, -2, 4, 3, -1, 1, -5], [3, -1, -3, 0, -1, 1, 0, 2, -1, -1, 2, 5, 1, -1, 0], [-8, 1, -1, 0, 3, 4, 1, -5, 3, -1, 0, -1, -1, 2, 1], [9, -2, 4, 2, 1, 0, 1, -1, 1, 2, 0, -1, 0, 2, 1], [2, 0, 0, 4, 0, 3, -4, 0, -4, -1, 2, 1, -2, -1, -5], [0, -9, 5, 4, 3, 0, 1, -4, 1, 5, -1, 0, 0, 1, -3], [0, 0, 5, 3, 1, 3, 0, -3, 2, 3, 4, -2, -3, 4, 2], [0, -5, -4, -1, 1, 1, -1, -5, -5, 2, -3, 7, 0, -2, -3], [0, 0, -1, -2, 4, -2, -3, -4, -1, 2, -1, 3, 2, 2, 3], [-7, 8, 3, -4, -2, -1, 2, 2, 5, -4, 1, -4, 1, 3, 4], [-2, -5, 1, -2, 3, -4, 3, 0, 3, 4, -5, -1, 2, 0, 2], [4, 6, -1, 1, -1, -2, -2, 2, 0, -3, 1, -5, 2, 2, 6], [0, -2, 0, -1, 0, -2, 4, -4, 1, 0, -4, -1, -3, -1, 2], [-4, 1, 1, 2, -4, -1, 3, 2, 1, -2, 1, 1, -4, -1, -2], [-4, 3, 2, -2, -2, -3, 2, -1, 0, -5, -2, -2, -1, -2, 1], [-6, 1, -2, 2, 0, -1, 3, 0, -3, -1, 1, -1, -3, -3, 0], [-9, 4, -1, -4, -3, -3, -2, 1, 2, -4, 0, 0, 0, 2, 0], [2, 7, 2, 4, -4, 0, -2, 6, 4, -5, 8, 0, -2, 0, 2], [7, -6, -2, 2, -1, 4, -5, 1, -1, 3, 2, 5, -3, -1, -3], [2, -2, 1, -1, 0, 3, 3, -2, 5, -1, -1, 3, 0, 0, -2], [7, -1, -2, -2, 6, 4, -4, 2, -1, 2, 0, 5, 4, 1, 2], [3, 0, 1, 2, 0, 2, 0, -6, -1, -1, 1, -1, -4, 2, -2], [-2, -1, 1, 1, 1, 0, 2, -4, 3, 3, 3, -2, -3, 2, -3], [10, -2, -1, 4, -3, -2, -5, 1, -7, 0, 4, 0, -2, -3, -3], [1, 6, -2, -1, -2, 1, -4, 6, -1, -4, 6, -2, 0, -1, 0], [3, -7, 4, 0, 3, -3, 3, -2, -2, 6, -6, -2, 0, 2, -5], [-6, 0, -4, 2, -2, -2, 3, 1, -2, -1, -1, 2, -2, -1, 0], [-2, 4, 1, 0, 1, 1, 2, -3, 2, 2, -2, 0, 0, 2, 3], [-1, 1, 2, 0, 3, 3, -1, 0, 4, -6, 4, -1, -2, 0, 3], [1, 2, -2, 3, 1, 1, -1, -3, -3, 0, 0, 0, 1, -3, 1], [-5, 5, -3, -4, 3, 1, -2, 3, 3, -2, -2, -2, 5, -4, 2], [3, 1, 1, -2, 2, -2, 1, -1, -1, 4, 2, 1, -3, 1, 4], [5, -3, -1, 0, -2, -3, -2, -2, -1, 0, 0, 2, 0, -1, -2], [-1, -1, 1, 3, 4, -2, 0, 1, 1, 0, 1, -2, 2, 0, 1], [-5, -2, 0, -6, 5, 2, 4, -8, -3, 3, -6, 5, 0, 2, -1], [10, 1, 4, 5, 0, -2, -4, 1, -5, 0, -2, -3, 0, -4, -2], [-1, 5, -1, 0, 0, -6, 5, 0, 5, -3, 0, -8, 2, -2, 3], [-5, -3, 4, -1, 7, 1, 2, -2, 5, 5, 0, -6, 1, 5, 3], [-14, 5, -1, 0, 0, -3, -1, -1, 1, -4, -4, -11, -1, -4, 3], [-5, 4, -4, -4, -3, -5, 0, 5, -5, -5, -2, -4, 3, -5, -1], [-12, 4, -2, -1, 0, -5, 0, 2, 1, -10, -2, -9, 3, 0, 4], [-8, 8, -8, -6, -1, 4, -2, 5, -10, -6, 3, 8, 6, -2, -1], [5, -5, 1, -1, 6, 0, -5, 1, 2, 0, -3, 0, 1, 2, 2], [-2, -8, 4, -1, 5, 0, 6, -5, 2, 4, -4, 2, -3, 3, -2], [-3, 2, -2, 0, -1, 1, -2, 1, 3, 0, -3, 4, -2, -4, -2], [4, -6, 8, 6, 0, -7, 3, 3, 6, -1, 5, -10, -3, 3, 3], [8, -7, 3, 4, -2, -2, -5, 4, 1, 0, 6, -2, 0, 4, -1], [-4, 7, 0, 4, -1, -3, -3, 1, 0, -6, 4, -9, 2, -4, 2], [3, -1, -2, -1, -1, 2, -2, -1, 1, 1, -3, 4, -2, -1, -5], [-3, 1, -2, -7, 3, 2, -2, 0, 1, 3, -3, 5, 3, 2, 5], [3, 4, -1, -1, -2, -7, 2, 3, 4, 0, 0, -1, 2, -1, 4], [3, -1, 1, 6, -4, -5, 2, 1, 5, 2, 1, -3, -2, 0, 0], [4, -4, -1, 1, 0, -3, -4, 4, 1, 1, 4, -1, 1, 3, -1], [6, 2, 0, -2, 2, 3, -5, 7, -2, 3, 3, -3, 4, 4, 4], [5, -4, 2, 3, -3, 2, 0, 3, -3, -3, -1, -1, 2, -4, -4], [0, 0, 2, -4, 5, 5, -1, -3, -6, -3, -1, 0, 2, 4, 0], [13, -1, -6, 2, -1, 0, -6, -3, -12, 3, 4, 6, 0, -5, -3], [-14, 8, 2, -4, 1, 0, 4, 1, 6, -5, 2, -1, 0, -1, -1], [0, 1, -3, -1, -4, 5, -7, 11, -4, -6, 1, 5, 2, -7, -5], [0, 0, -2, 0, -2, -1, -7, 4, -3, -1, 4, 6, -3, -7, -2], [1, 0, 1, -5, -2, 2, 5, 0, -1, 3, -3, 3, 2, -1, -1], [-4, 4, -4, -3, -1, 7, -6, -6, -10, 1, 2, 8, -3, -4, 0], [3, -2, -3, 3, -6, 2, -2, 4, 4, -1, 5, 3, 2, 0, -3], [-8, 0, 2, -3, 4, 5, 1, 1, -1, -2, -1, 0, 0, -1, -5], [-14, 5, -9, -9, 4, -1, 4, -2, -2, 0, -5, 0, 4, -2, 2], [20, -12, 8, 5, 3, 2, 3, 2, 9, 5, 2, -1, 1, 4, -1], [-6, -3, -1, 1, 2, -1, 0, -10, -8, 4, -1, 1, -1, 3, -1], [3, 9, -4, -2, -6, -3, -3, 6, 4, -1, 5, 0, 1, -4, 6], [3, -8, 0, -2, 7, 5, 2, -5, -2, 11, -3, 4, 3, -1, 0], [7, -3, 5, -1, 2, 10, -2, 3, -4, 5, 1, 11, -3, 6, -4], [-12, 13, -1, -2, -6, 3, -2, 2, 0, -11, 2, -4, 0, -3, 0], [2, -5, 1, 0, -3, -2, 10, -11, 0, 2, -3, -3, -4, -1, 1], [3, -7, 5, -5, 2, -2, 8, 1, 11, 3, -3, 3, -1, 5, 4], [-11, 4, -8, -1, -2, 2, -1, -6, -10, 0, -1, 2, 0, -8, -1], [6, -4, 2, 10, -9, 1, 0, 1, -2, -1, 1, 4, -5, 0, -6], [1, -3, 4, -3, 0, 2, 7, 2, 7, 2, -1, -2, -1, 5, 3], [-5, 6, -1, 0, -5, -6, 4, 5, 2, -5, -3, -5, 6, -7, 1], [-5, 0, -2, 2, 0, 1, -4, -3, -3, -5, 2, 2, -2, 2, -2], [-17, 13, -1, -2, 0, 1, 3, 2, 7, -10, -1, -4, -1, -2, 2], [4, -4, -1, 3, -1, 1, -3, -1, -5, 1, -3, 5, 1, -2, -3], [-3, 0, -2, -6, 3, -2, 4, -1, 9, 2, 0, -2, 5, 4, 11], [7, 2, 2, 0, -4, 0, -6, 6, 2, -2, 5, -3, -4, 4, 4], [-5, -11, -2, 3, 3, -9, 7, -6, -3, 4, -4, -4, 0, 0, 1], [3, 5, 0, 1, -7, 2, -4, 2, -6, -3, 6, 5, -4, 0, -5], [12, 2, -2, 4, 0, 5, 0, -2, 1, 2, 7, 7, -4, 0, -6], [-11, 5, 1, -8, 0, 7, -1, 4, 4, -7, 3, 5, 0, -1, -1], [9, -10, -3, 5, -2, -4, 5, -5, 4, 7, -7, 0, -3, 1, -6], [0, 4, 0, 3, -2, -2, 2, 0, 6, 0, -4, -10, -2, 0, -2], [-12, -3, -2, 3, -2, -4, 2, -4, 3, -2, -2, -5, 1, -2, -4], [-7, 0, -3, 4, 1, -2, -4, -3, -7, 1, -8, -1, 0, -5, -6], [-5, -4, -5, -4, 0, 8, 3, -7, -6, 1, -5, 11, 0, 0, 0], [9, -6, 5, -3, 9, -2, -2, -2, -1, 6, -6, -3, -1, 6, 2], [15, -1, 7, 6, -5, 0, -3, -3, -2, -8, 6, -3, -4, 2, 1], [-22, 11, -15, -11, 3, -2, -3, 4, -3, -1, -6, 1, 8, -3, 9], [1, -2, 10, 0, 3, 10, 9, -4, 8, -1, -2, -2, -7, 9, 3], [-4, 8, -5, -4, -4, 5, -1, 5, -7, -9, 6, 1, -1, 1, -1], [3, 4, -3, 1, 2, 2, 4, 0, -1, -4, -2, -2, 0, -3, -1], [-7, 14, -7, -2, -5, 4, -1, 7, -4, -9, 3, -4, -1, -7, 3], [-17, 9, -6, -3, -2, -5, 0, 2, 0, -1, -8, -6, 6, -5, 3], [-2, 9, -7, -3, -9, 8, -9, 19, -6, -8, 8, 7, 3, -1, -3], [14, -8, 4, 1, 0, 0, 9, -3, 7, 5, 0, -2, -1, 9, 1], [-8, -3, 6, 2, 2, -3, -4, 9, -2, 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6, 1], [4, 13, 6, -13, 9, 6, 0, 14, 9, 5, 5, -1, 2, 5, 11], [-3, -1, 1, -11, 3, 9, 6, -3, 1, 1, -7, 6, -3, 5, 0], [17, 0, 14, 4, -1, 0, -1, 4, -2, 6, 5, -2, -3, -2, -2], [-5, -17, 19, -1, 8, -1, 8, -6, 13, 7, -1, 4, -4, 6, 1], [8, -24, 3, 4, 10, -4, 16, -13, 6, 17, -12, 2, -8, 2, 0], [-4, -16, -7, 10, -4, -7, 1, 3, -3, 3, -1, 8, 2, 0, -5], [27, -14, 19, 9, 9, -1, -3, -2, 8, 1, 4, -4, 1, 11, 4], [-3, -3, 12, 4, 0, 3, 6, -5, -1, -7, 1, -8, -2, 9, -7], [-28, 1, 6, -4, 8, 4, 12, -13, 9, 1, -8, -4, -4, 5, 4], [-10, 13, 1, 4, -6, 0, 8, -3, 8, -7, 5, -7, -2, -1, -3], [3, 0, 8, -5, 3, -1, -14, 16, -2, -5, 3, 0, -4, 8, 6], [-5, -5, 2, 1, -12, -12, 11, 16, 1, -1, 6, 1, 5, 7, -2], [-4, -9, 0, 8, 10, 1, -5, -11, -11, 14, -7, 2, -9, -2, -6], [-11, 30, -6, -17, -1, 8, -7, 8, -1, -12, 12, 2, 7, 2, 4], [6, -1, 2, 2, 5, 4, -1, -3, 5, 3, 6, -3, -1, 4, 11], [0, -27, 0, 1, 13, 3, -7, -10, -8, 11, -5, 12, 1, 0, -6], [20, 5, 5, 3, 1, 7, -19, 15, 4, -7, 21, -1, 8, 4, 0], [15, -6, -3, 8, 2, -5, 8, 0, 2, 4, -2, -6, 3, 9, -4], [2, -25, 2, 2, 3, 0, 12, -12, -3, 4, -13, 3, -7, 6, -11], [15, -9, 0, 0, -11, -11, 6, -2, -5, 5, -4, 4, -3, -11, -4], [15, 1, -10, 10, -4, 2, -12, 2, -1, -10, 13, -1, 5, -3, -2], [9, -14, 7, 7, 12, 9, 2, -1, 9, 14, -6, 2, -5, 12, -5], [-8, 15, 5, -2, -2, 3, -8, 3, 14, -12, 15, -1, -2, -2, 5], [15, -8, 7, 9, 4, 6, 0, -11, -4, 9, -2, 4, -1, 9, -2], [12, 2, 2, -1, 4, -6, -8, 14, 4, 12, 2, 11, 6, -10, 9], [-7, -18, 7, 12, 0, -8, 2, -2, -1, -6, -3, -10, -9, 5, -3], [-30, 22, -8, -1, 4, -4, -2, -2, -7, -8, -1, -1, -2, -5, 3], [15, 12, -7, -9, -3, -3, -5, 4, -17, -5, 2, 4, 5, -3, 8], [10, 1, 17, -1, 3, 9, -2, 0, 1, -1, -2, -4, 1, 0, -5], [6, -25, 6, 6, 4, 1, -4, 3, -3, 1, 1, 4, -8, 1, -4], [17, -19, 6, 14, 5, -6, 1, -5, -2, 3, -2, 2, -7, 4, 0], [15, 10, 4, -2, -3, 5, -12, 21, -5, -6, 10, 11, 1, -1, 3], [29, -9, 7, 8, -5, -7, -3, 5, 11, 2, 5, -6, 0, 5, -6], [9, -8, -4, -5, -5, 3, 4, 5, 8, 1, 0, 0, 3, 0, 1], [-22, 11, -3, -7, -3, 2, 2, 0, 1, -12, -5, 0, 4, -8, 5], [3, -6, -8, 3, -4, -18, 4, 15, 8, -2, 6, -5, -1, 0, 12], [10, 10, -8, -6, 3, 8, -7, 5, -8, -7, -4, 7, 2, 4, 4], [11, -12, -1, 9, -8, -13, -14, -2, -2, 7, -3, -4, 3, -13, -1], [-21, 4, 3, -5, -9, 0, 4, 7, 3, -12, 11, 6, -5, 3, -6], [-23, 9, -17, -16, 0, -3, 9, -1, -4, 9, -20, 1, 7, -8, -1], [-4, 4, -7, -8, -7, -4, 4, 7, 4, 0, 1, 0, 7, 3, 12], [11, -19, 17, -1, 8, 0, 9, 4, 14, 7, -7, -2, -7, 6, -6], [-1, 7, -2, 7, -4, -4, -1, 0, -4, -13, -3, -9, 4, -8, -6], [-17, 1, 0, 0, 4, 10, -1, -9, -7, -1, 4, -9, -8, 1, 3], [-2, 11, 9, 6, -1, 7, -6, 9, 4, -14, 8, 3, -17, 6, -2], [9, 5, -3, -1, -3, -11, 9, -6, -8, 1, -9, -3, 5, -6, 5], [-17, -8, 11, 8, 8, 12, 0, -9, 2, -1, 8, -6, -24, 15, -6], [-16, 3, -20, -16, -1, -6, -13, 9, -21, 2, -14, 9, 16, -14, -5], [10, 0, -1, -3, -4, -10, -11, 13, -4, -1, 7, -3, 14, 2, 6], [3, 10, 1, -4, 1, -1, -2, -7, 5, -5, -5, -8, 10, 1, 10], [2, 1, -1, -16, 14, 7, -9, -6, 14, -2, 0, -1, 7, 9, 11], [3, 3, 13, 2, -1, -5, -6, 5, 8, -12, 13, -7, -12, 6, 18], [8, -2, -7, 3, -1, 13, -5, -11, -16, 3, 7, 12, -11, 2, -9], [6, 1, 5, 3, -8, -7, 3, 15, 3, -2, 5, 6, 4, 1, 5], [-9, 4, -1, -16, 6, 1, 9, -7, 16, 3, -1, 2, 3, -2, 5], [-1, -6, -10, 12, -14, -14, 6, 4, -5, -16, -5, -7, 0, -7, -13], [17, -22, 11, -2, 2, -4, -3, -3, 16, 3, 1, 0, -6, 10, 0], [-20, 0, 3, 2, 2, 3, -1, 2, -5, -6, -9, -14, -1, -4, 2], [-29, 3, 3, 0, -4, -10, 4, -1, 6, 5, 2, -14, 0, -1, 5], [7, -5, -1, -6, -1, -1, 5, 1, -3, 3, 1, 1, 5, -10, -4], [-27, 21, -11, -12, 2, 3, -6, -6, 0, -12, -2, -7, 6, -5, 9], [0, 10, -6, -1, 0, 7, -12, -8, -20, 3, -7, 2, 1, -2, -4], [0, -15, -2, 1, 10, 8, 0, -4, -12, 2, -12, 2, -3, 0, -11], [10, -14, -9, 2, -5, 8, -4, 1, -16, 10, -3, 17, 3, -11, -18], [19, -11, 7, 10, -4, -8, -6, -1, 9, 1, -2, -10, -5, -2, 2], [1, -9, 8, -1, 8, 2, 21, -16, 8, 7, -11, -4, 0, 5, -1], [5, 14, -3, -5, 9, 7, -5, 2, -17, 9, -3, 14, 8, -7, -5], [26, -25, 5, 5, -9, 3, 7, 0, 10, 9, 1, 9, -6, -1, -3], [-10, 9, -2, 2, -1, 6, 3, -8, 7, -9, 5, -6, -4, 4, 14], [25, -23, 6, 7, 12, 14, -10, -5, -11, 16, 4, 14, -2, 2, -15], [-5, 5, -6, 3, -12, -6, 6, -2, 4, 1, -1, -5, 2, 6, 12], [-11, 19, -16, -12, 3, -7, 9, -1, 6, 2, -14, -13, 12, 1, 7], [-8, 6, -1, -6, 6, 5, -1, 8, 6, 0, -4, 6, 4, -8, 3], [-1, 15, -15, -5, -7, 3, 3, -6, -11, -2, -10, 3, 8, -9, 5], [-7, 9, 4, -6, 4, -6, -7, 22, 5, 3, 5, 2, 4, -3, 4], [-10, 20, -12, -5, -15, -5, -1, 2, 1, -22, -6, -13, 4, 0, 12], [-1, -4, 5, 1, 9, -1, -11, 1, -13, 7, -6, 9, 7, -3, 2], [-24, 11, -6, -13, -6, 3, 2, 4, 7, -16, -1, -4, -3, 2, -4], [-4, -9, 12, 9, -9, -6, 8, 15, 3, -13, -2, -7, -5, -3, -5], [22, -17, 22, 11, 4, -1, 7, 10, 17, -5, 13, -17, -16, 8, 2], [15, 12, 5, -10, -3, 3, 7, 6, 14, -5, 4, 3, 7, 18, 4], [-8, -15, 6, 4, -5, 4, 0, 9, -1, 7, -6, 3, 5, 4, -8], [-4, -2, 10, -4, -3, -2, 3, -1, -6, -1, -6, -2, 2, -1, -9], [24, -28, 11, 4, 3, -8, 9, 9, 20, 12, -4, -10, -2, 15, 1], [7, 4, -2, -6, 14, -13, -3, -3, 1, 3, -2, -4, 9, -7, 13], [-1, -4, 3, 9, -13, -13, 17, 2, 10, 5, -6, -7, -4, 3, -10], [-12, 26, 12, -1, -6, 5, 9, 11, 13, -15, 6, -4, -7, -2, -2], [-3, -11, 14, 12, 9, 1, 12, -3, 8, 10, -8, -11, -8, -2, -2], [-12, 3, -6, 17, 3, 2, -15, 3, -8, 0, 9, 4, -1, 3, -8], [-10, 13, -24, -12, -2, 7, -8, -4, -17, 4, -2, 20, 10, -6, -10], [-1, -3, 2, 9, 1, -3, 8, -7, 8, 11, -8, -2, 11, 7, 3], [15, -2, 5, 6, 4, 10, -7, 5, -5, -9, 7, 2, -12, -3, -7], [-5, 6, 1, 0, -8, -3, 3, 13, 13, -12, 5, -10, -8, 3, 11], [-10, 6, 8, -5, 9, 8, -6, 1, 3, -9, 2, 2, 1, 1, -10], [4, -18, 4, 4, 8, -7, 8, -6, 15, 16, -12, -3, -3, 6, 4], [-10, 2, -3, 0, 9, -10, 6, 4, 5, 11, -15, 4, 12, 1, 2], [-6, 5, 4, 7, -1, 11, -2, -11, 9, -6, 6, 2, -2, 0, -5], [-4, -3, 11, 0, 8, 18, -8, 7, -3, 0, 7, 6, 3, 9, -6], [35, -30, 14, 6, 9, 14, -6, -3, -4, 15, -2, 13, -3, 18, -11], [14, 14, 0, 3, 4, -6, -13, 5, -2, -12, 6, -6, 6, -6, 7], [-22, 8, -19, -11, 0, -13, -2, 16, -2, 7, -5, 2, 9, -5, 10], [6, -19, 13, 1, 7, 12, -11, 0, -3, 6, 9, 11, -5, 3, 2], [-19, -4, -13, -3, -3, -17, 12, -9, -5, -1, -17, 2, 4, -12, 2], [-11, 11, -2, -8, 8, 1, -11, 4, -6, -18, 2, -4, 2, 2, -3], [-2, 0, 6, -1, 10, -14, 0, -9, 2, 12, -7, -8, 6, 5, 3], [-11, 17, 4, -3, 0, -11, -7, -2, 1, -2, -22, -22, -3, -14, 1], [-1, -12, 0, 1, -9, -2, -8, 14, -8, 9, 5, 26, 6, -6, -13], [16, -13, -6, -2, -2, 2, 19, -13, 7, 17, 4, 1, 4, 6, -3], [7, 5, -2, -1, 3, 4, -3, -6, 8, 6, 4, 2, 4, 11, 12], [-3, 5, 2, -9, 13, 5, 7, -3, 12, 13, -4, -10, 3, 10, 12], [16, -7, 1, 15, -4, -3, 6, 1, 2, -6, 4, -2, -20, 6, 1], [-9, 10, -1, -7, -13, -2, 11, 12, 0, -8, 0, -7, 3, -8, 7], [35, -22, 14, 9, 2, -4, 1, 10, 12, 6, 15, -2, -6, 12, -1], [-2, -22, 8, 7, -3, -16, -4, 4, 5, 10, 0, 10, 0, 0, -9], [-38, 8, -1, -4, 12, -3, 2, -7, 10, 4, -14, -3, 10, -9, 4], [-6, 8, -6, -6, -2, 1, 8, 5, 0, -5, 13, 1, -1, 3, 3], [-22, -9, -12, -5, -5, 5, 7, 0, -10, 6, -9, 25, 8, -12, -18], [2, -15, 12, 0, 10, -1, 3, -7, -3, -1, -6, -6, 5, 13, -8], [40, -17, 16, 5, 8, 6, 6, 15, 18, 6, 5, 7, 2, 13, -2], [9, 3, 3, -11, 4, -4, -1, 1, 15, -3, -2, -1, 6, 6, 11], [0, -16, 5, 14, 9, 6, 2, -20, -3, 15, -5, -12, -2, 8, -6], [15, -2, 17, 8, -4, -3, 1, 3, 17, -5, 14, -22, -16, 17, 19], [6, 2, -1, 5, 2, -8, 8, 0, 5, 3, -6, 0, 11, 4, -7], [-4, 10, 6, -1, 4, -1, -6, 16, 2, 4, 3, -5, 2, 2, -4], [-5, -1, -1, -11, 9, 3, 13, -8, 11, -6, -1, -15, -8, 7, 13], [14, 8, 4, -8, 1, 15, -11, 8, -9, -8, 14, 20, -2, -6, -2], [5, 9, 3, 5, -8, 11, 2, 9, 4, -10, 19, 2, -7, 3, -4], [-3, -4, 6, -14, -1, 9, 14, -2, 17, 7, -2, 11, 2, 8, -4], [17, -10, 2, -3, -7, 5, 0, -2, -11, -2, 3, 4, -6, 18, -1], [5, 18, 6, 6, -5, 0, -16, 3, 5, -13, 15, -8, -12, 7, 9], [-9, -5, 4, -3, 5, 9, 0, 16, 5, -16, 16, 5, -10, 2, 6], [-1, -6, -8, -4, 13, 13, -4, -22, -7, 0, 3, 0, -10, 6, -1], [-6, -4, -4, -5, -2, 5, 8, -3, -14, 7, -11, 4, -1, -14, -15], [-3, 13, 7, 5, -6, -1, -5, 6, 4, -9, 6, -18, -6, -3, 5], [-6, -7, -1, 1, -5, -12, 9, 3, 0, 6, 4, -10, -1, 5, 9], [-7, 4, -15, 2, -6, 0, 9, -5, 2, 7, -13, -1, -2, -1, -7], [-16, 1, -5, 9, -2, -11, -8, 1, -14, 5, -3, 9, 1, -14, -18], [1, 4, 8, -7, 7, 12, 2, -8, -5, 1, 1, -9, -4, 3, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4334_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4334_2_a_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_4334_2_a_b(:prec:=15)
    chi := MakeCharacter_4334_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_4334_2_a_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_4334_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4334_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![92, 94, -911, -876, 2792, 2876, -2897, -3491, 785, 1713, 188, -328, -101, 13, 9, 1]>],Snew);
    return Vf;
end function;