
// Make newform 8034.2.a.ba in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_ba();".
// 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_8034_2_a_ba();".
// 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 := [-1492, -506, 10504, 2997, -22759, -4126, 18441, 2268, -5438, -451, 722, 36, -44, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [11371595281753640, -1047631892432145166, 422048253089054251, 4255301279762091841, -915363310465441330, -4492827496115707927, 84321828569428448, 1414782771710524552, 24886922325561149, -191585597830484336, -3739404291058554, 11730896555981242, 134001969718133, -266306853678383], [588284015133280532, -780973362285615831, -2701606611286975207, 3669499430064327162, 3051387240952734794, -4033414366634615670, -1262761014021014640, 1324468395061602794, 217017383487373922, -186676696579320143, -15979394388187493, 11831212064366610, 421045515071387, -276561632426969], [-376805384643590757, 908907497419418757, 1346052708231791049, -3995402229552693528, -1303030165431025011, 4397320407290358181, 652528796048380961, -1419834757376023830, -127532267940051610, 196191030805468054, 10147274836518033, -12211701038963858, -281957981628135, 281008689100883], [727260622566264690, 1120836385553257082, -3783713813055134269, -7216360830653724145, 4368493534082155588, 9011134325268674427, -888965469465066868, -2942272434616681480, 41321895819192391, 403199471683301528, 2179601153394382, -24789913096510800, -153049032331183, 563276920787713], [69082076903089780, 1635447715064563388, -1046735607854569041, -7818651825966159249, 1281274510944306808, 9072106839066945823, 169914683015283554, -2960764032107237352, -114850747220457015, 410367492349937474, 12469830127322176, -25559574153488160, -401515966880495, 587894636532799], [1584010040979027426, 983162040010572610, -8543202845942904737, -7220568332423014109, 10041227588618067172, 9656661921220973037, -2683158768977673228, -3175735116060119102, 279533615334483141, 434206046573977704, -12031687706537128, -26573482465291666, 160676916331065, 600547475775077], [87259893240249591, 828635442318437489, -1059281835457961059, -4402465041131927787, 1294647822855345817, 5139130808556126422, -111735197101452200, -1664502571125875459, -32681850003000513, 228696579415282775, 4846129448665246, -14134878235621520, -172381882120881, 323027709062960], [-1698473816153287560, -1800196355230650124, 7853101744503182057, 9392486941861559115, -8583170474917069942, -11175426298123444609, 2101336739470417196, 3607458160860328818, -187136585825562229, -491662813166917868, 5643504601298036, 30122063309684406, -152764996823, -682482407227411], [91607361939384532, 849290650220779131, -1259504021300032236, -4440659955237449610, 1713977797830856848, 5476590592887655990, -231617478407447300, -1799297386675220078, -20208645124353222, 248673668063896890, 4333005041742547, -15413816930355473, -166497424213521, 352781396910762], [1424281961283445274, 2137547859976077336, -6870285231682939387, -10254865555632034835, 7459946733255625440, 12173804052853690563, -1643994001829427568, -3944695232805112230, 110071620789847963, 540215049753776070, -62661878222990, -33253610912031224, -144783307200521, 756784517059657], [-641231492639617501, -1371444613064346176, 3488559562581261272, 6656963138830352527, -4027119627133068275, -7583560590369499978, 870810312435771415, 2429698868506903576, -52645645511792452, -330994820970097058, -683299742724623, 20312989919152389, 101629522210970, -461379618788969], [812680240968073994, 4702972689934548944, -4247182212949563787, -20309684730415768281, 4786003452335976276, 22595322669123401215, -234855379984642544, -7249462015732823470, -170706210676448595, 992798322812597212, 22466875749627886, -61270437020665254, -773564887767339, 1399081209291127]];
    Rf_basisdens := [1, 1, 43692742328669662, 21846371164334831, 21846371164334831, 43692742328669662, 43692742328669662, 43692742328669662, 21846371164334831, 43692742328669662, 21846371164334831, 43692742328669662, 21846371164334831, 43692742328669662];
    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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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], [-1, 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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0], [0, -1, 0, 0, 0, 1, 0, -1, 1, 0, 0, 0, 1, 0], [1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, -1, -1], [0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1], [3, -1, 1, 1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0], [-1, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0], [1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, -1], [0, 0, 1, 0, 0, 1, 1, -1, 0, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 1, 0, 0, -1], [-3, 0, 0, 0, -1, 0, 1, -2, 0, 1, 0, 1, 0, 0], [0, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [-2, 1, -1, -2, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0], [0, 2, -1, 0, -1, 0, 1, 0, -1, 1, -1, 0, -1, 0], [0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, -1, -1, 1], [0, -1, -1, 0, 0, 0, -1, -1, 0, 0, 1, -1, 0, 1], [2, 1, -1, 0, 1, 0, -1, 0, -1, 1, 0, 1, -1, 0], [0, -1, 1, 0, 1, -1, 1, 1, -1, 0, 0, 0, 0, 0], [0, -2, 0, 0, -1, 0, -1, -2, 0, -1, 2, 0, 1, 0], [2, 0, 1, 2, 1, -1, -1, -1, 0, -1, 0, 1, 0, 0], [-2, -1, 0, 0, -1, -1, 0, -1, 0, 0, -1, 1, 1, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -2, -1, -2, -1, -1, -1, 1, 1, 0, -2, 1, 1, 0], [2, 0, 1, 1, 0, 1, 1, -1, -1, 0, 0, 0, -1, 0], [0, 0, 0, -1, 2, 0, 0, -1, 0, 0, 0, 0, 0, -1], [3, -1, -1, 0, 1, 0, -1, 0, -1, -1, 1, -1, 0, -1], [-2, 0, -1, -1, -1, -1, 0, -1, 1, -1, -2, 2, 2, 0], [-1, 2, 1, -1, 1, 1, -1, 2, -1, -1, -1, -2, -2, 0], [0, 0, -2, 0, -2, 0, 2, -2, 0, 2, -1, 0, 0, 1], [-2, 2, -2, -1, -1, 0, 2, -2, 1, 1, -2, 0, -1, 0], [5, 1, 1, 0, 1, 0, 1, 4, 0, 0, 0, -2, -2, 0], [-3, 2, 0, -1, -1, 0, 0, -1, -1, 0, -1, 1, 1, 1], [-1, 0, 0, -2, -1, 0, -4, 2, 0, 1, -1, -1, -1, 2], [-1, -2, -2, -1, -1, -2, 1, 0, 1, 1, 0, -1, 1, 0], [-1, -2, -1, 2, -1, 1, 1, -3, 0, 1, 1, 0, 1, 0], [2, -2, 0, 0, 0, 0, 1, 0, -1, -1, 2, -1, 1, 0], [-3, 0, 0, -2, -3, 0, -1, 0, 1, 0, -1, 0, 2, 0], [2, -1, 0, -1, 0, -3, -2, 1, 0, -2, 1, -1, 2, 0], [1, -1, -1, -1, 0, 0, 1, -1, 0, 0, -1, -1, 1, 2], [2, -1, 0, 1, 0, -3, 2, 0, 1, 0, 1, 0, 1, -1], [2, -2, 1, 0, 0, 1, 1, -2, 0, 0, -1, 0, 2, 0], [5, -2, 1, -1, 3, 1, 0, 1, 2, -2, -1, 1, 1, 0], [12, -1, 1, 2, 3, -2, 1, 1, -1, -2, 2, 1, 0, -3], [-1, 0, 2, 2, 0, 0, 2, -2, -2, 0, 2, 1, -1, -2], [-2, 1, 0, -1, 0, 1, -1, 3, -1, 1, -3, 1, 1, 1], [-3, 3, 0, -1, 2, 1, 0, 3, 0, 0, 0, 1, 1, -1], [3, 0, 0, 3, 1, 0, 0, 0, 0, -2, -1, 1, 0, 0], [5, 1, -2, -1, -3, -3, 0, 1, -1, 0, 0, 2, 1, -1], [-1, -2, -1, -1, 1, 2, -2, 0, 2, -1, 1, -2, 0, 2], [-9, 1, 2, 0, -1, 1, -2, -2, 0, 0, -1, -1, 0, 2], [-3, 1, 2, 1, -3, -1, 0, 1, -2, 1, 0, 0, -1, 0], [-2, 3, 1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 0], [6, -2, 0, -2, 0, 0, -5, 2, -1, -1, 2, -2, 1, 0], [2, -1, 2, 1, 4, 3, -2, 2, 0, 0, -1, 1, 0, 2], [-2, 4, -2, 1, -2, 2, 4, 0, -3, 4, 1, 0, -3, 0], [4, -1, 1, 2, -2, 0, 1, 0, 0, 1, 0, -1, -3, 0], [2, -1, 0, 0, 4, -1, -3, 3, 2, -1, 0, 1, 0, 1], [2, 1, 0, 0, 1, -1, -1, 3, 0, -2, 0, -1, -2, -1], [-2, 1, -1, -2, -1, 3, -1, -1, 1, 2, -2, 2, 4, 2], [-4, 0, 1, -1, 1, 1, 1, 0, 1, -1, -3, 0, 1, 2], [-4, 3, 1, 0, 2, 2, -1, 2, -2, 1, -1, -2, -3, 0], [4, 0, -1, 1, 0, 3, 3, -2, 1, -1, 0, 0, 2, 1], [0, 0, 2, 0, -4, -4, -2, 0, 0, -2, 1, 2, 2, 1], [1, 4, 0, -1, 2, 0, 1, 0, -1, -2, 2, -1, -1, 0], [10, 1, 1, 2, 4, -2, -1, 4, -2, 1, -1, 3, -3, -4], [-6, 2, -3, -1, -1, 1, 1, -5, 2, 0, -3, 1, 3, 2], [-2, 0, 1, -2, -1, 3, 0, 3, 0, 3, -3, 1, 1, 2], [6, 1, 0, 4, 1, 1, 1, -1, 1, 1, 1, 6, 0, 0], [-2, 2, 0, 0, 2, 2, 1, 0, 2, 2, -4, 1, -1, 0], [12, 4, 1, 1, 2, 1, -1, 0, 2, -1, -3, 1, -1, 0], [-3, 0, 0, 1, 0, 2, 1, -3, 1, 2, 0, -2, -1, 0], [5, 2, -1, 2, 3, -1, 0, 3, 0, -1, -1, 1, -2, 0], [14, 2, 2, 0, 3, 0, 2, 2, 0, -3, 2, -3, 0, -2], [5, 3, 0, -1, 1, -3, 4, 0, -1, 2, -3, 3, -1, -4], [3, 2, 1, -2, 1, -1, 3, 1, 1, 2, -1, 2, 1, -2], [7, -1, -1, -1, 0, -4, 2, 2, -1, 1, -1, 3, 0, -2], [3, 0, -1, -5, 0, -1, -2, 4, 1, -1, -1, -1, 0, -1], [11, 1, 0, 0, -2, -3, 1, -3, 0, -1, 1, -2, -1, -2], [0, -3, 2, 1, 4, 1, 1, -1, -1, 1, 1, -2, -1, 0], [4, 0, 0, 0, -1, 2, 1, 1, 1, 0, 0, -3, -1, 0], [4, -3, 2, 2, 3, -3, 1, 2, 0, -1, 1, 0, -1, -3], [5, 3, -2, 1, -4, -1, 3, -1, 0, 0, 0, 2, 2, 0], [14, 0, -3, 2, -2, -5, 0, 0, -2, 1, 1, 4, 0, -1], [-3, 3, 4, 2, 1, 1, 2, -1, 2, -2, 2, -2, 0, 0], [-4, 1, 0, -2, 0, 3, 1, 1, 0, 3, 0, 0, 2, 0], [3, -5, 5, -2, 3, 2, -7, 4, -2, -2, 0, -1, 2, 4], [6, 0, 0, 1, 4, 2, -4, 1, -1, -4, 4, -3, -1, 0], [1, 2, 1, 1, 0, 1, -3, 1, 0, -1, 1, 1, 0, 0], [9, 0, -1, 1, 2, -1, 4, -3, 0, 3, 1, 1, -3, -2], [6, -2, 1, 4, 2, 1, -1, -3, -1, 1, 1, 4, -2, -1], [-2, 3, 2, -1, -1, 2, 0, -2, -1, -1, -2, -1, -1, 0], [3, 1, 0, -1, 0, 5, 3, -1, 1, 1, -2, -3, 0, 0], [2, -1, 2, 3, 2, 1, 4, -6, 2, -2, 1, 1, 0, 0], [16, -3, 0, 4, 1, -5, 0, 0, -3, -2, 3, -1, -2, -1], [4, -2, 4, -1, -2, -1, 2, -3, 0, -4, 2, -5, 2, 0], [1, -1, -2, 1, -2, 2, 2, -4, 0, 5, -2, 1, -4, 0], [5, 7, 2, -2, -2, 1, 3, -1, 1, 0, -1, 1, 1, 0], [2, 1, 4, -1, 2, 0, 2, 0, 1, 0, -3, 2, 1, -2], [7, 2, -1, -2, 2, -5, 1, 2, -2, 0, -4, 3, -1, -6], [8, 0, 0, -2, 1, 0, -1, 5, 1, -1, -1, -2, 2, 3], [6, 0, 0, -1, 2, 3, -1, 3, -2, 0, -1, -4, -1, 2], [0, 0, -1, -4, -1, -1, -1, 2, -1, -1, -2, -3, 1, 3], [-5, 1, -3, -5, 1, 0, -4, 4, 1, -2, -1, -2, 2, 0], [-1, 0, 1, 1, -5, 3, 2, -5, -1, -3, 3, -3, 1, 2], [2, -5, 3, -3, -1, 2, -5, 2, 2, -5, 4, -3, 4, 2], [4, -1, 2, 2, 1, 3, -3, -1, -2, -2, 4, -1, 2, 0], [12, 2, 1, 1, 2, -1, 0, 3, 1, 0, 0, 1, 2, 1], [4, 2, 1, -2, 2, -5, 0, 3, -1, -1, 1, 2, -1, -4], [-1, -2, -2, -3, 0, 0, -2, 2, 2, 1, -3, -1, 0, -2], [14, 2, 2, -2, 1, 0, 1, 0, 0, 1, -2, 3, 1, -2], [4, 1, -2, 1, -3, -5, 2, -1, 0, -3, 0, -2, -2, -2], [-7, 0, 1, -5, -3, -1, -6, 5, -3, -1, 0, -2, 3, 2], [1, 4, -4, -3, -6, -4, -1, 3, -3, 0, -1, 0, -1, 0], [2, -3, 0, 1, 2, 5, -3, 1, 0, 0, 2, -1, 3, 2], [-5, -5, 2, -1, -1, -1, -1, -3, 3, -2, 5, -4, 2, 2], [13, -3, 4, 4, 0, -1, -1, -3, 2, -1, 0, 2, -1, 0], [4, -8, -1, -4, -4, 0, -2, 2, 0, -1, 4, -5, 2, 2], [3, -3, -3, -1, 2, -4, 0, 1, 0, 0, -2, 2, -2, -2], [4, -2, 1, 1, -1, 5, -2, -1, 1, 1, 2, -3, 4, 4], [11, 0, 0, 0, -5, 1, 1, -1, -2, 2, -2, -1, 1, 2], [11, 0, 2, 2, 6, -1, -2, 3, -1, 0, -1, 6, -2, -2], [4, -1, -2, -1, -2, 4, 0, -2, 3, 2, -3, 0, 1, 2], [-3, 1, 1, -4, 0, 0, -2, 0, 2, -3, -1, -3, 1, 2], [8, -4, 3, -2, 4, -3, -5, 6, -2, -2, 3, -2, -4, 0], [5, -6, 0, -3, 3, 0, -5, 6, 3, -3, 0, -4, 3, 2], [11, 5, 1, -1, 0, 0, 2, 4, 0, 0, 1, 1, 2, 0], [12, 1, -2, -2, 1, -1, -2, 2, -2, 0, -2, 1, -1, -2], [2, 0, 1, -3, -5, 3, -4, 7, -1, -1, 0, -6, -2, 6], [-5, -3, -5, -3, -2, 0, 1, -4, 2, 4, -5, 6, 3, 0], [14, 4, -3, 1, 1, -5, 4, -3, 1, -1, -1, 2, 0, -2], [14, 4, 6, 1, 4, 4, 1, 5, 0, -1, 0, 0, -2, -2], [1, -1, 0, 3, -2, -1, 0, -3, 0, -2, 3, 3, 1, 0], [15, -3, 4, 1, 4, -3, -4, 1, -4, -2, 4, 2, 1, 0], [9, -7, 5, 7, 5, -1, 1, 1, 0, -4, 4, 4, 0, -2], [13, 3, 2, 4, 2, -2, 1, 4, -1, 1, -2, 2, -6, -4], [11, -7, 0, 2, 0, -4, 2, -4, 0, -2, 0, 2, -1, -2], [5, -2, 1, -2, -2, 1, -3, -1, -2, -2, 0, -3, -3, 2], [5, -4, -2, -4, 5, -5, -4, 3, 2, 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2, 2], [-9, 13, 2, 2, -1, 7, 4, -5, -1, 3, -4, 3, -4, 2], [-15, 0, -2, 2, -6, 8, 5, -12, 2, 6, 1, 0, 4, 2], [-8, 12, -2, 2, -4, 9, 4, -3, -1, 6, -5, 4, 1, 0], [-8, 5, 3, 9, -1, 4, 4, -2, 0, 7, -8, 4, -3, 2], [5, 2, -8, -4, -8, -2, -2, -1, 1, -2, -2, -2, 4, 3], [-6, 5, 1, 10, -1, 6, 13, -7, 1, 3, -4, 5, -1, -4], [3, -9, 4, 0, 9, 5, -5, -1, 1, -5, 4, -1, -3, 4], [1, -4, -3, -3, -8, 7, 1, -13, -2, 7, -3, 3, 2, 5], [-4, 3, -4, -1, -1, -5, 9, 4, -5, 2, 1, 0, 1, -6], [4, 8, -1, -6, 4, -1, -4, 5, -3, -1, 0, -8, -1, -2], [-11, 2, 6, 5, -4, 0, 6, -3, -2, 4, 2, 2, -7, -5], [-21, -9, -2, -8, 6, 1, -10, 6, 2, 2, -4, -4, -1, 5], [-9, 4, -3, -2, -10, 5, 3, -5, 0, 6, -7, 3, 3, 0], [-11, -3, 2, -4, 3, -1, -8, 3, 0, -6, 2, 2, 6, -4], [1, -3, 3, -7, -6, 4, 3, -2, -4, 2, 1, -4, 3, 0], [3, -6, -1, -3, 7, -3, -6, 3, -1, -5, 6, 6, 7, -4], [-38, 1, -4, -2, 0, 1, -4, -1, 0, 1, 1, -1, -1, 2], [9, -3, 3, 5, -2, -4, -1, 3, -2, -1, 8, -1, 4, -1], [-8, -4, -4, -8, -8, -4, -6, 8, -3, -1, -1, -5, 0, 4], [-21, 2, -3, 4, -7, -1, 6, -8, 3, 1, -3, 9, 1, -1], [6, 1, 2, 4, 7, -3, -4, 6, 0, 2, 0, 3, -9, -4], [10, -2, 0, -7, -6, -8, -5, 3, 1, -5, 2, 1, 9, 1], [6, -2, 11, 1, 5, 1, 3, 3, 1, -1, -2, 1, -3, -6], [-14, 8, -10, -6, -3, -6, -5, -4, 3, 0, 0, -3, 1, 0], [-13, 0, 0, -11, -5, 2, -2, 4, 0, 2, -2, 2, 4, 6], [7, 1, 1, 1, -3, 2, 4, -6, -2, 5, 2, -1, 2, -1], [-10, 10, -3, -2, 4, 3, -4, 9, -2, 0, -5, 1, -3, 0], [14, -7, 9, 4, 0, 0, -2, 10, -2, -9, 13, -9, -2, 0], [0, -5, 1, -1, 9, -6, -6, 2, -3, -2, 1, 5, -1, 2], [-14, -7, 4, 0, 0, 11, 2, -1, -1, -2, 4, -6, 1, 6], [16, -6, -6, 2, 3, -12, 2, -2, 3, 0, 3, 5, 2, -4], [14, 0, 3, 11, 5, -1, 7, -14, 5, 1, 1, 8, 1, -4], [6, 1, -2, 7, 2, -1, 9, -4, -2, 2, 5, 2, -1, -6], [10, 4, 1, -1, 4, -2, -7, 10, -6, 2, -1, 2, -6, 2], [-11, -7, 0, -7, -2, -2, -3, -2, 1, 2, -6, 4, 9, 8], [-25, -10, 3, -1, 2, 5, 0, -7, 6, -1, 0, -7, -1, 2], [-1, 2, -4, 4, 2, -6, 1, -3, 5, 2, 3, 1, 1, -3], [-16, -1, -7, -4, -11, -4, 6, -12, 1, 7, 4, 2, 0, -5], [-2, 2, -5, 5, -9, 1, 9, -14, 1, 3, 1, 0, -1, 4], [-1, -2, -7, -6, -5, -1, -6, -5, 8, 3, -10, 6, 8, 4], [2, 0, 1, -1, -3, -9, 0, 5, -1, 1, 2, 8, -2, -5], [14, 1, -3, 3, 2, -2, 3, 6, -1, -3, -4, 2, -4, -3], [15, -10, -2, -2, -1, -1, -1, -1, 3, -3, 3, -1, 3, 6], [-5, -2, -2, -1, -5, -10, -6, 3, 0, 1, 5, 3, 1, -1], [2, -3, -5, 1, -1, 0, -7, -10, -2, -5, 8, 1, 6, 4], [13, -1, 3, 2, 4, -6, 0, 0, 1, -4, 7, 3, 3, 0], [-7, 0, -6, -7, -2, -8, 2, 8, -4, 1, 3, -8, -6, 0], [-4, 8, -1, -5, 3, -1, -6, 3, -2, 6, -8, 6, 0, 2], [-8, 5, -1, -2, -6, 0, -6, 1, -4, 2, 1, 7, 7, 1], [1, -3, -5, 0, -5, -7, -1, 7, -1, 4, -1, 9, -3, 2], [-3, -1, -2, -7, -7, 1, 1, -3, 0, -1, -3, -4, 6, 6], [-9, 9, -1, -4, -4, -4, 7, 0, 0, 3, 0, -3, -2, -4], [1, 4, 1, 4, 5, -1, 0, 9, -5, -6, 2, -3, -4, -6], [18, -6, -3, 1, 6, -9, -6, 4, 2, 3, 2, -1, 0, -2], [1, 9, -12, 0, 1, -10, -3, 0, -1, 4, 3, 3, -2, -2], [6, -6, -5, -3, 1, 3, -11, -1, -6, 0, 2, 0, -3, 4], [1, -1, 1, -9, 0, 1, -9, 15, 2, -9, 1, 1, 4, 0], [-14, 3, -7, 6, 2, 0, 7, -8, 5, 6, -5, 16, 3, -6], [-7, 5, 5, 12, 2, 8, -1, 2, 1, 2, 2, 2, -4, -1], [-15, 2, 0, -2, -7, -10, 0, 2, -1, -8, 3, 0, 7, -6], [4, -2, -1, 1, 4, 7, -4, 11, -2, -1, -4, -1, -2, 7], [-19, 0, 6, -5, -4, 8, 2, -5, 0, 2, -3, -6, -3, 5], [18, 11, 3, 4, 5, -6, 0, 7, -10, -3, -1, 3, -5, -3], [-18, 3, -1, 4, -8, 10, 8, -10, 2, 3, 3, -3, 0, 1], [7, 2, -4, -7, -8, -6, 2, 2, -3, 4, 2, -7, -2, 3], [13, -4, 3, 5, 7, -2, -3, 2, -5, 4, 0, 2, -5, 0], [21, -4, 3, -4, 6, 3, -5, -1, -3, -3, 0, 1, 1, 0], [22, -7, -4, -2, 6, 0, 2, 0, 1, -5, 5, -3, 2, -2], [-13, -1, 3, 0, 0, 0, -4, 3, -3, 0, 4, -6, 3, 6], [1, -5, 1, 4, 0, -2, 3, -2, 0, -1, 2, -1, -2, 0], [9, -1, -2, -10, -10, -1, -4, 5, 1, 0, 0, 4, 6, 0], [25, 2, 7, 2, 6, 5, -4, 4, 1, -4, -4, -7, -1, 1], [-1, 7, 0, -5, 1, -1, -1, 1, -1, 0, -2, 1, 2, -2], [18, 14, -3, 8, -2, -1, 4, 0, -9, -2, 7, -3, -10, -1], [1, 3, 1, 4, 4, -4, 0, 8, 0, -7, 2, 1, 1, -5], [3, 2, 2, -4, -4, -8, 7, 2, 2, -4, -8, -3, 2, -2], [24, -4, 0, 5, 3, 8, 11, -14, 11, 5, -1, 1, 0, -4], [29, -11, 4, -4, 2, 0, -11, 2, -1, -11, 8, -10, 4, 0], [17, 14, -5, 3, 0, -5, 9, 2, 0, 5, -6, 6, -2, -8], [-5, 2, 6, 10, 10, 2, -2, -1, -5, -1, 2, 4, -5, -4], [-31, -2, 9, -3, -3, 13, -6, 1, -1, -1, -3, -5, 3, 8], [12, 5, -4, 0, -8, 1, 4, 1, 2, 3, 1, -1, 3, 4], [5, -16, -4, -2, -2, 2, 7, -10, 9, -1, -1, -3, 4, -4], [31, 4, 2, 8, 11, 0, 6, 2, 4, -1, 9, 2, -1, -7], [4, 3, 6, 2, 6, -3, 0, 5, 3, -6, 1, 2, -1, -9], [25, -3, 1, 12, 0, 0, 10, -7, -2, 7, 6, 5, -1, -6], [6, 5, -2, 4, -8, -5, 4, 7, -2, 3, -8, 3, -3, 2], [-8, 9, -2, 2, 7, -5, 2, 0, -1, -1, 6, 4, 5, -6], [-11, -3, -1, 6, -6, 0, -1, -2, 1, 0, -6, -1, 2, 2], [-8, 1, -6, -10, -5, 1, -5, -3, 1, 5, -5, -4, 2, -2], [11, -6, 3, 0, -4, -7, -9, 6, 2, -2, 7, -1, 5, 0], [24, 7, -1, 1, 6, 4, -4, 6, 0, 6, -7, 9, -1, 4], [-11, 6, -7, 1, -9, -1, 1, -9, 1, 5, -6, -5, 2, 8], [5, -5, 1, 7, 6, -2, 10, 3, -4, -3, 10, 4, 1, -7], [-2, -3, -2, 4, -3, -5, -1, 2, -8, -1, 7, -4, 1, -1], [6, -6, -10, 1, -2, -5, 0, -5, 3, -1, 6, -1, 4, -2], [-15, 4, -4, 5, 0, 8, 0, -4, -2, 7, -3, 3, -6, 6], [11, 4, 4, 4, -4, -8, 5, 10, -5, 1, 2, 5, -2, -6], [0, -8, 3, 5, 6, 1, 0, 3, -1, 0, 5, 1, 0, 2], [20, 3, -5, 6, -2, -4, 5, -6, -3, -2, 0, 1, 3, 0], [5, 6, 4, 5, 16, -1, -2, 11, 0, 0, -5, 8, -3, -4], [3, 10, -8, -3, -10, -9, 6, 1, -4, 4, -6, 3, 1, -4], [-17, 6, -3, -6, -10, -2, 8, -2, 5, 6, -4, 1, 5, -4], [-22, -8, -1, -10, -9, -3, -13, -2, 0, -2, -4, -2, 7, 7], [11, 5, 5, 5, 5, 8, 5, -9, 1, 5, -7, 7, 0, -3], [-14, 0, 6, 0, -8, -4, 2, -2, -2, -2, -1, 0, 0, 0], [-8, 2, -5, -5, -9, -1, 4, -1, -1, 7, -4, 1, -6, -1], [-14, 7, -2, -7, -2, 5, -1, -2, 2, 4, -9, 5, 5, -2], [9, 2, 3, 4, 4, -5, -2, -6, -2, -6, 4, 8, 2, -6], [6, -8, -5, 1, -2, -11, -5, 11, -10, -1, 4, 4, 3, 0], [-9, 0, 1, 4, -5, 5, 4, -9, 1, 4, 4, 0, -6, 0], [-9, 12, -1, 0, 9, 3, 0, 5, 1, 0, -5, 4, -2, 0], [9, -1, 5, 2, 11, -1, 1, 5, 5, -10, 1, 1, 5, -2], [21, -8, 2, -5, 7, -6, -1, 4, 1, -5, 1, 2, 9, 0], [9, 17, 7, 6, 10, 2, -2, 16, -8, 3, -2, 6, -7, -4], [24, -1, 3, 0, 1, -3, 6, -5, 0, -5, -7, 4, 3, -4], [-8, 8, -4, 1, 1, 9, -3, -1, 0, -1, 4, -8, -3, 4], [-13, -2, 8, 3, 1, 16, 6, -8, 10, 0, -2, -8, 4, 6], [39, -8, 7, 2, 3, -3, -1, 3, -7, -2, 9, 0, -3, -4], [10, -11, 4, 3, 10, -3, -2, 5, -4, -4, 7, -4, 0, -6], [-20, 5, -9, -3, -5, -6, 4, -8, 6, -3, -2, 6, -1, -2], [16, 3, 10, 3, 15, 3, -7, 17, -3, -6, -1, 3, -1, -2], [5, -5, 4, 6, -8, -3, 5, -7, 0, -7, 7, -5, -1, 6], [34, -6, 4, -5, 2, 2, -8, 0, -1, -2, -1, -1, 3, 6], [16, 5, 1, 9, -2, -4, 10, -5, -6, 3, 6, -4, -8, 1], [-6, 15, 1, -2, -8, 0, 6, -6, 1, 2, -2, -8, -4, -2], [2, -11, 1, -4, -4, 0, -7, -4, -4, 3, 2, -2, 5, 8], [-8, 4, -9, -9, 2, -1, 6, 1, 1, 10, -8, -2, -2, -3], [23, 10, -3, 3, 2, -7, -1, 12, -4, -2, -1, 6, 3, -5], [-11, -9, 0, -7, 3, -3, -2, -1, 6, -3, -6, -7, 1, -1], [32, -8, 7, 5, 5, -9, -5, -2, 0, -8, 8, -1, 5, -8], [5, -12, -7, -11, -2, -3, 2, 0, 3, 3, 2, 2, 8, 1], [8, 11, -7, -1, 1, -6, -4, 2, -2, 1, 4, -2, 1, 4], [-1, -6, -5, -4, 3, -4, -6, 14, 2, 0, -9, 6, -3, -4], [-18, -4, -9, -11, -14, -9, -7, -1, -1, 4, 2, 1, 3, 6], [26, 8, 5, 0, 7, 11, 6, 7, -3, 4, -4, -8, -5, -4], [19, 1, -7, 4, -1, -10, 6, 0, -1, 5, 1, 3, -5, -4], [4, 5, 0, 1, -5, 2, -1, -4, 7, 1, 0, -4, -6, 2], [-33, 2, -7, -3, -10, -3, 1, 4, -2, 12, -7, 8, -1, -3], [1, 3, -5, -2, 14, 0, -5, -2, 7, 0, -7, 8, -4, -2], [-12, -1, 2, 0, -5, 6, 9, 0, 1, 0, 0, -8, -3, 4], [9, 0, 1, 3, 3, 11, 9, -2, -1, 2, 0, 4, 1, -3], [-15, -4, -6, -9, -8, -2, -3, 0, 0, -1, -2, 1, 9, 4], [18, 5, 6, 5, 0, 3, 1, -6, 2, -5, 6, 0, 2, -1], [-10, -5, -6, -4, -8, -3, -13, 1, 0, 1, -2, -1, 0, 7], [0, 3, 8, -5, 4, 1, -5, -3, -1, -3, -8, 9, 1, -2], [0, 9, 4, 9, -5, 13, 7, -5, 0, -1, -1, 3, 1, 2], [14, -8, 2, 8, 3, 0, 9, -5, 2, -2, 11, -1, 6, -1], [8, 15, -9, -1, 2, 4, -5, 4, -3, 3, 1, -2, -4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_ba();".
// 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_8034_2_a_ba(:prec:=14)
    chi := MakeCharacter_8034_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_8034_2_a_ba();".
// 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_8034_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1492, 506, 10504, -2997, -22759, 4126, 18441, -2268, -5438, 451, 722, -36, -44, 1, 1]>,<7,R![-6624, -35664, 13691, 91773, -14049, -76528, 11179, 23945, -3857, -3377, 596, 216, -41, -5, 1]>],Snew);
    return Vf;
end function;