
// Make newform 6034.2.a.l in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6034_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_6034_2_a_l();".
// 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_6034_2_a_l();".
// 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 := [-21776, -109144, -112848, 205294, 391391, -101137, -477646, -47208, 312410, 74594, -124276, -36075, 31784, 9121, -5329, -1292, 573, 97, -36, -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, 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], [-4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1809016113384, 11129234398640, 17476023229232, -13461222095163, -47354517541381, -3408061802102, 49475485804836, 13552222306822, -28036276822386, -9140268418762, 9589231744201, 2948268091502, -2047106484441, -504766627077, 268014365098, 43709410711, -19663940787, -1462616462, 614360943, -3960343], [335027839864, 1778261525636, 2608653060254, -389081593375, -4498670133799, -3563958517610, 1855579562496, 4051274477878, 524324073778, -1972951737436, -641981540601, 530311897740, 208383070343, -87337002091, -32772811792, 9155450243, 2574973259, -577306280, -81044325, 16644951], [-2198305565808, -7872031684318, 423891695997, 22030650504305, 7824943444266, -26098069899884, -11605231024486, 17092287131486, 7567057791828, -6808718394193, -2697245206372, 1715817274811, 554472938489, -275173394620, -65128150081, 27150657839, 4034068656, -1492020713, -101591649, 34635180], [-434932067360, -1577973995415, 140670079897, 4863664147150, 2015862861242, -6173786067610, -3510755640666, 4292697618894, 2655993621793, -1833575792166, -1104063935083, 510837094381, 268421167748, -94984626483, -37991837591, 11460917862, 2893666957, -806665861, -91273626, 24749436], [-3456518219736, -13584964832756, -3701357233648, 34185117118823, 22694772257685, -36728041496050, -29044895255988, 22292286183426, 18400675058302, -8570078476466, -6735921243469, 2218857147522, 1488124645517, -395224629647, -195724039610, 46757370765, 14088317343, -3269423778, -425556827, 100169175], [-6827044858744, -23206861195928, 7430391971228, 74468175567565, 14323866889735, -98399132432526, -31448906153828, 70662011375414, 24434559246422, -30600553577726, -9986677320235, 8371072116058, 2340509122487, -1464415784497, -315571793034, 159530819887, 22739279353, -9871031970, -677141369, 264438933], [-1733210336848, -6630457002476, -1039701681626, 17888086708617, 10123784644597, -20850420947606, -14287583245004, 13891466884386, 9778625961422, -5893209464936, -3841472821561, 1669140863184, 905324889811, -317530122351, -126057411084, 38959352751, 9521282651, -2760722388, -299043217, 84627259], [-2401691382064, -10464986875224, -6016009624925, 24561108636495, 25037390039884, -24546970398728, -31007101914510, 14086688762286, 19833998059986, -5373267279201, -7441733031364, 1489592807765, 1695729550403, -303351747064, -230447535695, 41775750129, 17119353404, -3327989547, -531994539, 112313458], [-5022246308728, -22319090928660, -13916717002464, 52250106029851, 57289050280281, -51199959091866, -71711913218188, 28224832586138, 46386347886198, -10240996838690, -17571019110121, 2766913680226, 4035613393209, -577091775883, -552284871818, 83884106449, 41300189627, -7059152210, -1291516455, 248833715], [-3367570379784, -14788998243232, -6866102538522, 41819809511415, 37945554762711, -50203746507230, -53852893948784, 34290680964482, 38096694582042, -15072335059492, -15440497361475, 4524429755200, 3730825904025, -933086775793, -529755791300, 125166030661, 40646537601, -9626405628, -1293022755, 315317385], [-4582432715876, -17766040350958, -3319275783001, 48367939070307, 29241168607046, -55931170695232, -41105211794078, 36210148063178, 27913124781788, -14657547984787, -10805281715184, 3937430053521, 2495980296331, -718699678124, -340072159095, 86433528077, 25168764824, -6117797523, -776958427, 189297804], [-20980523930824, -87948678687112, -41734671242616, 209047703145903, 186645175207745, -212352079365906, -230993811060164, 123486106092610, 146507264783378, -47030140681646, -54481031610765, 12707364215778, 12324747482549, -2478852923031, -1666268316042, 327212594709, 123370536655, -25262436066, -3826397139, 835590203], [19966247957256, 85005626154864, 41787055500476, -208738589070129, -193027604258035, 218562657331126, 247481464312132, -130670424420302, -162220478086054, 51071504508166, 62115207658047, -14170320998026, -14409203636595, 2844198628989, 1989813546586, -385986008075, -149948611701, 30511731298, 4718179533, -1027938729], [-6299303348056, -26573508701288, -10490555195360, 73339444737417, 60797505146367, -86809450322110, -85545372062572, 58785626890430, 60093801761878, -25658775696594, -24302200221347, 7642558381022, 5883644866955, -1562543194369, -839298144558, 208012032923, 64771735209, -15916321158, -2072767045, 519967205], [4753242395944, 19202507682118, 5630388544830, -51989409745755, -36936455863561, 60341382365326, 51180850872076, -39808268971858, -35051513158072, 16739364778226, 13806485341891, -4750930316416, -3260250462441, 921739033949, 454747815080, -117006906077, -34420744897, 8622426264, 1083920107, -274087579], [5764506969456, 23329830760960, 7613668131281, -60749876331836, -44579225210911, 68183513110990, 59478980973710, -43818194314372, -39653842084616, 18105460574843, 15306357628247, -5090068253835, -3560419069126, 983119021817, 491110264125, -124471421788, -36865983801, 9150086029, 1153750064, -290115955], [5699083771564, 23834003996814, 10163029426582, -60574697135363, -51620949387853, 65906318041966, 67960867918620, -41023320358782, -45314356507148, 16601593774334, 17575111887427, -4686813636124, -4117650204609, 935317640093, 572895705936, -124377259165, -43406255205, 9610163980, 1370868643, -317683195]];
    Rf_basisdens := [1, 1, 1, 3040618192, 1520309096, 760154548, 380077274, 1520309096, 3040618192, 760154548, 760154548, 1520309096, 1520309096, 760154548, 3040618192, 3040618192, 1520309096, 760154548, 760154548, 760154548];
    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_6034_a();"
function MakeCharacter_6034_a()
    N := 6034;
    order := 1;
    char_gens := [1725, 869];
    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_6034_a_Hecke(Kf)
    return MakeCharacter_6034_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, 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], [-1, 0, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -2, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1], [1, 1, 0, 0, 1, 0, -1, 0, 0, 1, 1, -1, 0, 0, -1, -1, -1, 0, -1, 0], [1, 1, 1, 1, -1, -1, -1, -1, 0, 0, 1, 1, 0, 1, 0, 1, 0, -1, 1, -1], [-1, 1, 0, 0, 0, 0, -1, 0, 0, 1, -1, 1, -1, -1, 0, 0, 1, -1, 1, 0], [-2, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 0, 1, 0, 1], [-2, 0, 0, 0, 1, 0, -1, -1, 1, 1, -2, -1, 1, 0, 0, -1, -1, 2, 0, -1], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0], [-2, 0, -1, -2, 1, 0, 1, 1, 1, 1, 1, -3, 1, -1, -1, -2, 0, 2, -3, 1], [0, 1, 1, 3, -2, 0, 1, -1, -1, -1, -1, 1, -1, 2, 1, 1, 1, -2, 2, -1], [-3, 1, 3, 1, 0, 0, -2, 1, -1, 0, -1, 1, 0, 0, 0, 0, -1, -2, 2, -1], [1, -1, -1, -1, 0, 1, 1, 0, -1, -1, 2, -1, -1, 0, -1, 1, 0, 0, -1, 1], [-1, -1, -1, -3, 1, 1, 2, -2, 1, 0, 0, -1, 2, 0, 0, 1, 0, 2, -1, -1], [-2, 1, 3, 2, -2, -1, 0, 0, 0, -3, -2, 2, 1, 1, 2, 0, 0, -1, 3, 0], [-2, -1, -2, 0, -2, 0, 3, -3, 1, -2, -2, 1, -1, 1, 3, 0, 2, -1, 1, 1], [0, 1, -1, 0, 0, 1, 1, -1, 1, 1, -2, 0, 0, 0, 1, 0, 0, 1, 0, 1], [0, 3, 1, 3, -2, -2, 1, 1, -1, -3, 1, 2, 0, 1, 1, 2, 0, -2, 0, 0], [-5, 1, 0, -1, 0, 1, 1, 1, 0, 1, 0, 0, -2, -2, -1, -2, 1, -3, -1, 2], [-3, 0, -1, -1, 1, 2, 0, -1, 1, 1, -3, 1, -1, -1, 0, 0, 0, 1, -1, 0], [-2, -1, 1, 1, -1, 1, 0, -1, -1, 0, 0, 1, 0, 0, 1, 2, 0, -2, 0, -1], [0, 1, 3, 2, 0, 0, -2, 0, -2, 0, 0, 1, 0, 0, 1, 1, -1, -1, 1, -1], [0, 2, -1, 0, 0, -1, -1, -1, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-5, 0, -1, -2, 2, 0, 0, 0, 1, 2, 1, 1, -2, -1, -1, 0, 1, 0, -1, 2], [1, -1, 0, 0, 3, 1, 1, 1, 0, 0, -1, -1, 1, 0, -1, -1, 0, 2, -1, 0], [2, 3, 5, 2, -1, -2, -2, 3, -2, 0, 1, 0, -1, -1, -2, 0, 0, -1, 1, 0], [0, 1, 2, 1, 0, -1, -1, 0, -1, 0, 0, 0, -1, 2, 0, 1, 0, 0, 1, -3], [-4, -2, -3, 0, -2, -1, 3, -1, 2, -2, -2, 1, 1, 2, 3, -1, 2, 2, 1, 1], [-4, -1, -2, -1, 1, -2, 1, 2, 1, 0, 1, 0, 1, -1, 2, 2, 1, 3, -1, 1], [-2, -1, 1, 1, 3, 2, -1, 0, -1, 2, -3, -3, 2, -2, -1, -3, -4, 0, 0, -1], [1, -2, -1, 0, -1, 1, 0, -1, 0, -1, 0, 1, -2, 1, 0, -2, 2, -1, 1, 0], [-3, 1, 1, 1, 1, -2, -5, -1, 3, 2, -1, 1, 0, 0, 0, 0, -2, 0, 1, -1], [4, 2, 5, 5, -3, 0, -4, -1, -2, -2, -1, 4, -2, 1, 1, 2, -1, -6, 6, -2], [-1, 1, 5, 2, 0, 1, 0, 4, -3, 1, 1, -1, 0, -1, -2, 1, 0, -3, 1, -1], [0, 0, 0, -2, 3, -1, 1, 0, 2, 3, 1, -1, -1, -2, -1, -1, 0, 2, -3, 1], [-1, -3, -2, 0, -1, 0, 1, 0, -1, -1, 2, 2, -1, 1, 0, 1, 2, -2, 2, 0], [-2, -1, 2, 1, -3, -1, -1, -1, 0, -1, 1, -1, 2, 2, 1, 1, 0, -1, 3, -2], [1, -4, -5, -6, 0, 3, 2, -1, 3, 1, 1, 1, 0, 0, -1, 0, 4, 3, -2, 0], [-2, 1, -1, 0, -2, -1, -2, -2, 0, 3, -1, 0, -3, -2, -1, -1, 0, -3, 2, 1], [4, -2, -1, -2, 0, 2, 0, 1, -1, 3, 5, -3, -2, -2, -3, 0, -1, 0, -4, 0], [0, -4, -6, -5, 3, 2, 5, -1, 1, 1, 1, -4, 1, -2, -1, -1, 1, 4, -7, 1], [1, -3, -1, -1, -2, -1, 1, -1, -1, -1, 3, -1, 1, 2, 0, 1, 2, 1, 0, -1], [3, 1, 3, 2, 0, 0, -4, -1, -2, -1, 2, -1, -1, 1, -3, -1, -4, -4, 2, -2], [3, 1, 0, 5, -5, 0, 0, -4, -1, -6, -2, 7, 0, 3, 2, 4, 2, -6, 5, -2], [-8, -1, 1, -1, 4, 1, -2, 2, 1, 3, -1, 0, 1, -3, -1, -1, -1, 3, -3, 1], [2, 2, 1, 0, 0, -3, -2, -1, 3, -1, 1, 0, 0, 3, 0, 1, 0, 0, 3, 0], [-4, -3, -1, -2, 4, -1, 1, 2, 1, -3, 3, -1, 2, 1, -1, 0, 0, 2, -1, 0], [1, -1, 0, 2, -1, 2, -2, -1, -1, 2, 0, 0, -4, -1, -1, -2, 1, -5, 1, 1], [-5, -7, -6, -1, -3, 4, 3, -1, 0, 0, 0, 0, -1, 1, 3, -1, 2, 1, 0, 1], [3, 3, 3, 2, 1, -3, 0, 2, -2, -3, 4, -2, 2, 3, -3, 1, -3, -2, 1, 0], [1, 1, -1, 0, 3, -3, -1, 1, 2, 1, -1, -3, 4, 1, 0, 0, -2, 4, -2, 2], [4, -4, -6, -3, 0, 0, 2, -3, 1, 1, 6, -3, -2, -2, -2, -1, -1, 0, -5, 2], [-1, -2, 2, -3, 1, 2, 1, 4, -2, 0, 2, -4, -1, -1, -3, -2, 0, 2, -3, 0], [2, -3, -2, 1, -1, 0, 1, -1, -1, -3, 1, 0, 3, 1, 2, 2, 0, 4, -1, -1], [-4, -2, -3, -1, 0, -2, 3, -5, 4, -3, -2, 0, 3, 1, 3, 0, 0, 1, 2, 1], [6, -2, 0, -2, 1, 1, -4, 2, -2, 5, 5, -7, -2, 0, -5, -3, -2, 3, -4, -3], [-9, -4, -4, -2, -1, 1, 1, 0, 1, 1, -2, 3, 0, -1, 2, -2, 0, 0, 1, 1], [0, 1, 1, 1, 1, 1, -2, 2, 0, 1, -1, -4, 0, 1, -2, -1, -2, 3, -1, -2], [1, -3, -3, -4, 3, 2, 0, -2, 2, 0, -3, -3, 3, 1, -2, -2, -2, 3, -2, 2], [-3, -1, -3, -3, -2, -3, 3, -1, 3, -4, 1, -1, 3, 4, 1, 1, 2, 2, -1, 1], [-2, 3, 4, 3, -1, 1, -1, 2, -4, -2, -1, 5, -3, -2, 0, 2, 0, -7, 2, 0], [-3, 4, 2, 3, -1, -1, -6, 1, -1, 2, -2, 3, -2, -3, 1, 0, -2, -3, 3, 3], [-1, -1, -1, 2, -2, 1, 2, -4, 0, -3, -7, 5, 2, 4, 5, 1, 0, -2, 4, -1], [4, -1, 2, 0, 2, 1, -5, 1, 0, 3, 1, -2, 1, -4, -1, -1, -2, 1, -2, -2], [-4, -2, -1, -1, 0, -3, -3, 1, 4, 1, 4, 2, -3, 0, -1, 1, 2, 1, 0, 0], [5, -1, -3, -3, 2, -3, -3, -2, 3, 3, 4, -1, -1, -1, -1, 1, 0, 3, -5, 1], [-9, 3, 3, 0, -1, 1, 0, 8, -1, 1, -2, 2, -2, -4, 1, 0, 3, -5, 3, 2], [1, -2, -5, -5, 5, -1, 2, -5, 4, 1, -3, -3, 5, 0, 1, -2, -4, 8, -6, 1], [1, -3, -3, -2, -1, 0, 4, -3, 1, -4, -3, 1, 4, 1, 2, 2, 0, 3, 1, 0], [2, -2, -3, 1, -4, -2, 3, -4, 0, -5, 0, 2, 0, 1, 4, 3, 2, 0, 0, 1], [-3, -2, -4, -2, -2, 0, 1, -1, 2, 1, -2, -1, -1, 1, 3, 1, 4, 2, 1, -3], [-2, -5, -3, 0, -9, 3, 6, 1, -1, -4, 0, 2, -3, 1, 2, 1, 6, -3, 4, 0], [2, 0, -1, 0, -2, 2, 2, -5, -1, -1, -1, 3, 1, 0, 0, 3, 1, -1, -1, -2], [-4, 5, 3, 1, 2, -1, 0, 5, -3, -1, 1, 0, 2, -3, 1, 1, 1, -1, -1, 0], [0, 2, 5, 4, 4, 0, -5, -2, 0, 4, -4, 3, 1, 0, 0, 0, -2, -2, 5, -3], [-2, -1, 0, 5, 0, -1, -3, 0, -3, 2, 2, -3, 0, 4, 1, 0, -2, 0, 0, -4], [-1, -3, -1, 0, 1, 0, -2, -3, 2, 4, -2, -2, 0, -2, -1, -2, -4, 0, -4, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 2, -1, 2, 1, -3, 4, -2, 5, 1, -3, 1, -1, -4, -1, -1, 5, -3, -3], [6, 4, 0, 6, -5, -6, 1, -4, 2, -7, -2, 3, 1, 4, 5, 3, 1, 0, 6, 0], [6, 1, 3, 5, -4, 1, -3, 5, -7, 1, 1, 1, -3, 0, -3, 1, 0, -4, 2, -4], [-2, -8, 0, -6, 4, 3, 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-2, 4, -8, -9, 8], [5, 3, 1, -1, 0, 4, -6, 14, -3, 11, -2, -7, -8, -12, -8, -7, -1, -4, -9, 2], [-3, 4, 17, 10, -7, -6, -2, 10, -10, -10, -1, -3, 8, 9, 4, 4, -1, -3, 15, -6], [0, 2, -9, -10, 12, -4, 6, 1, 1, 4, 12, -7, 0, 4, -7, -2, -1, 12, -9, 1], [-12, 7, 13, 4, 8, -6, 7, 16, -7, -9, -7, -4, 17, 6, 5, 6, -2, 11, 5, -7], [6, -8, 3, 5, -9, 3, 3, -7, 1, -9, -3, 5, 7, 8, 2, 6, -1, -4, 5, -8], [-16, 10, 10, 9, -2, 0, 1, 2, -6, -3, -3, 10, -6, 1, -1, 6, 3, -4, 6, -7], [3, -12, -11, -11, 1, -1, 5, -5, -1, 10, 7, -8, -5, -3, 1, 3, 4, 15, -10, -2], [0, 6, 2, 8, -5, -7, 2, -3, -5, -7, 1, 11, -4, 8, -1, 12, 8, -10, 10, -4], [27, 8, 3, -2, 15, -3, -4, 4, 0, 4, 6, -16, 8, 2, -13, 1, -10, 14, -16, -3], [-2, -6, 5, 4, 3, 9, -4, 4, -2, 3, -12, -1, -1, -4, 1, -11, 1, 1, 6, 1], [-3, 6, 18, 11, 1, 5, -2, 9, -11, 1, -11, 6, 1, -3, -2, -2, -3, -11, 10, -2], [-11, -7, 1, -5, 17, 1, 2, 8, 7, 4, 1, -9, 4, -6, -6, -12, -7, 13, -14, 5], [4, 4, -16, 0, -5, -1, 5, -13, 10, 0, -8, -7, 1, 1, 10, -10, -2, 7, -6, 3], [3, -8, 3, -3, 10, 1, -5, 11, 5, 10, -1, -9, 10, -1, -6, -6, -7, 16, -6, -2], [-14, -17, -7, -3, -4, 1, -1, 1, 2, 5, -3, -7, 3, -4, -1, -14, -1, 5, -3, 1], [-14, -9, -3, 2, -3, 3, 9, -1, 2, -2, 2, 6, -8, 5, 1, 0, 11, -5, 4, 2], [-7, 2, -10, 1, 1, 0, 3, -17, 11, -1, -18, 2, 4, 6, 14, -4, -2, 7, 6, -1], [-12, -13, -8, -5, -3, -5, 7, 5, 2, -2, 5, -3, -3, 5, 4, 5, 8, 16, 0, -2], [3, 13, 14, 6, 15, 0, -14, 12, -5, 12, 0, -9, -3, -9, -8, -5, -9, -2, -6, -2], [5, 1, 2, 2, 1, 5, -1, 2, -2, 2, -4, -6, 0, 7, -4, -4, 4, 3, 2, -5], [-11, -9, -2, 0, -9, 2, 4, -2, 7, -13, -8, 3, 9, 7, 10, 3, 4, -1, 13, -1], [4, 11, 1, 7, -2, -7, -3, -11, 0, -7, 2, 5, 1, 8, -3, 2, -1, -5, 2, -4], [2, -2, 1, 12, -17, 1, -13, -8, -1, -5, -6, 11, -9, 5, 10, 8, 8, -11, 24, -8], [1, -1, 10, 13, -2, -5, -14, -6, -6, -1, -1, 0, -8, -2, -1, -4, 0, -12, 13, 0], [-19, 16, 6, 5, 6, -9, -2, -2, 9, -3, -22, 9, 2, -2, 11, -2, 0, 8, 8, 3], [15, 5, 5, 1, -3, -8, -3, -7, 4, -11, 9, -5, 10, 10, 6, 5, -6, 5, 4, -9], [1, 0, -2, 5, 0, 3, 1, -3, 0, -9, -9, 0, 11, 5, 5, 3, -6, 8, -2, -6], [14, 11, 1, -1, 15, -3, -3, 0, 1, 7, 6, -7, 2, -1, -8, 4, -11, 13, -16, -2], [-1, 7, 5, 2, 1, 4, -5, -7, 0, 8, -3, 1, -8, -4, -5, -3, -1, -5, 1, 0], [30, 4, 4, -1, -5, 0, 7, -5, -3, -12, 10, -6, 9, 8, -2, 9, 1, -3, -3, -6], [-19, -15, -11, -2, -5, 0, 10, -2, 6, -10, 6, -3, 2, 4, 7, -5, 4, 3, -8, 6], [-20, -12, -12, -7, -4, 11, 9, -3, -1, -1, -6, -1, 2, -6, 3, -5, 2, 3, -1, 7], [1, -1, 7, 5, -3, -4, -10, 6, -8, 0, 11, 6, 2, 2, -7, 5, -4, 0, 2, -8], [6, -10, -7, -7, 4, 16, 5, 2, 1, 9, -3, -7, 1, -6, -2, -6, 1, -2, -6, -1], [4, 0, -9, -2, 1, 7, 3, 0, 3, 12, 1, -10, -8, -3, 0, 0, 5, 6, -2, -1], [-12, -7, -15, -1, -1, 7, 2, -14, 5, 4, -14, 2, 2, -1, 4, -2, 2, 3, 0, -1], [10, 2, 10, 13, -8, -11, -2, -12, -6, -19, -1, 2, 12, 19, 8, 9, -5, 0, 10, -13], [5, 10, 4, 7, 1, 0, 2, -1, -4, -1, 2, 9, -9, 3, 2, 7, -1, -9, 4, 5], [-15, 3, -7, -7, 5, 0, 6, 0, 6, 1, -2, 3, -3, 2, 3, -4, 6, 2, -2, 3], [-17, -4, -5, -4, -7, -3, -3, -6, 12, -3, -10, 6, 4, 3, 12, -4, -2, -2, 11, 3], [-8, -5, -16, -12, 2, 6, 6, -9, -1, 1, 12, -14, -1, 0, -1, -5, -3, -2, -12, 4], [-13, 5, -9, -15, 1, 4, 8, -4, 10, 11, 0, -10, -3, -5, -1, -12, 4, 15, -13, 3], [15, -2, 10, 10, 1, 4, -3, 12, -14, 7, 10, 1, -4, -8, -5, 5, -6, -10, -4, -6], [21, 5, -5, -9, -4, -1, -6, -13, 9, -2, 2, 8, -8, 1, 1, 5, 4, -7, 3, 2], [8, -1, -9, -5, -2, -3, -3, -15, 2, -7, 1, 5, 4, 1, 9, 5, -1, 2, 1, -1], [11, 4, -3, 4, -3, 0, -2, -7, -5, -9, 1, 4, 6, 5, 1, 12, 7, 7, 4, -9], [5, -5, 0, -9, 0, 14, 8, -1, -8, 7, 3, -4, -5, -8, -9, -7, -5, -8, -12, 5], [24, -9, 2, 8, -2, -3, 1, -5, 0, 1, 8, -2, 6, 5, 3, 1, -3, -1, -3, -2], [-10, 1, -4, -11, 1, 6, 11, 4, 1, -3, 9, -8, 3, -2, 0, 4, 3, 7, -14, 1], [1, -8, -1, 3, -2, -3, 1, 3, 2, -2, 11, -8, 8, 11, 2, 4, -5, 8, -6, -5], [-6, -1, -15, -3, 7, 4, 0, -13, 5, 4, -3, -2, 5, 1, 4, -4, -6, 7, -17, 5], [14, 0, 1, -7, 10, 7, -3, 7, -6, 12, 2, -14, -1, -8, -9, -4, 0, 8, -8, 2], [7, -4, 7, 6, -10, 2, 0, 2, -9, -10, 10, 9, -7, -4, -2, 13, 1, -22, 7, -2], [19, -8, -5, -14, 4, 3, 4, 13, -2, 6, 15, -10, 8, 0, -6, -4, 1, 16, -19, -2], [5, -15, -5, -11, 5, 4, 7, 8, -9, 1, 12, -23, 12, 0, -6, -9, -1, 25, -17, -4], [-31, 6, -4, -20, 8, -1, 15, 3, 15, -2, -7, -4, 8, -4, 0, -7, 2, 16, -12, 10], [3, 16, 8, 10, -9, -3, -7, -1, -2, 2, 3, 10, -12, -3, -8, 4, 8, -21, 14, 0], [18, -6, -4, -2, -6, 1, 12, -10, 3, -12, 1, -9, 13, 8, 5, 10, -3, 1, -1, -7], [-11, 0, -2, -1, -4, 2, -3, 4, -10, -2, 1, 5, -1, -2, -5, 0, 1, -8, 0, 1], [-27, 0, -3, -5, 1, -4, 3, 13, 0, -10, 4, 5, 1, 3, 0, 1, 8, -1, 7, 2], [-2, -5, -6, 10, -10, 5, 0, -20, 5, 1, -10, 1, -8, 1, 12, -2, 0, -13, 10, 0], [-36, -6, -9, -15, 7, 2, 5, 14, -1, 3, 10, -8, 5, -13, -2, -8, 1, 7, -18, 15], [9, 0, -5, 1, 7, 1, -7, 3, -10, 8, 12, -4, -9, 0, -8, 1, -4, 6, -8, 0], [3, -1, 2, 0, 8, 3, -11, -7, 7, 7, -5, 2, -2, -6, -5, -6, 0, 4, 7, 1], [-17, -1, -1, -17, 5, -2, 2, 5, 6, 6, 8, -4, -1, -7, -2, -4, -4, 4, -9, 14], [14, -6, -12, 2, -2, 6, -7, -14, -5, 8, -5, -4, -7, -3, -1, -2, -5, 3, -2, -3], [0, -12, -16, -9, -4, -1, 6, -6, 19, 1, -6, 2, 4, -2, 6, 0, 4, 7, -7, 1], [6, -10, -19, -5, 1, 4, 6, -2, 0, 7, 6, -14, -4, -4, -5, -8, -2, 4, -15, 3], [-11, 19, 9, 6, 4, -3, -2, 4, 2, -4, -16, 20, -6, -7, 3, 1, 3, -13, 10, 10], [-27, 4, -5, -11, -5, -3, 12, 8, 6, -13, 1, 7, -3, -1, 7, 2, 11, 5, -3, 10], [-29, -8, -12, -13, 10, 4, -5, 6, 9, 11, 8, -11, -2, -5, -10, -2, 0, 8, -4, 4], [-13, -5, -7, -10, 8, -7, 1, 8, 10, 2, 4, -11, 7, -6, 2, -11, -8, 13, -6, 11], [-25, 1, 0, -12, 7, 4, -4, 7, 2, 6, -6, -6, 4, -7, 0, 1, -2, 8, -6, 4], [-16, -2, 10, 3, 5, 10, -7, 5, -9, 6, 1, 7, -12, -8, -3, -1, 4, -13, 4, 7], [1, -4, -14, -12, 2, 8, 0, -3, 0, 13, 8, -13, 1, 3, -9, -9, 2, 13, -17, 1], [4, -6, -13, -13, 6, 1, 2, -7, 4, 5, 2, -5, -2, -5, -2, -3, 3, 16, -17, -4], [14, 7, 6, 5, -7, -5, -15, -3, -1, 10, 10, -4, -7, 5, -6, 4, -7, -1, 9, -7], [-22, 9, -7, -10, 3, -2, 6, 4, 3, 3, 1, 1, -2, -1, 3, 0, 2, 10, -8, 8], [-14, -4, -3, 6, 3, 3, -4, 0, 1, 10, -5, 3, -1, -6, 1, -2, -5, 1, -1, 5], [-3, -7, -18, -12, 2, 4, 6, -11, 9, 5, -2, -8, 2, 1, 4, 0, 3, 12, -8, -4], [-11, 8, 5, -4, 12, -2, -5, 9, 4, 6, -6, 0, 1, -6, -7, -3, -3, -5, -8, 6], [-29, 4, -7, -11, 7, 3, 2, 5, 4, 6, -5, 1, 2, 0, -6, -9, 2, 4, -2, 4], [5, -2, 1, 0, 7, 5, 6, 4, -7, 1, -6, 0, 2, 1, 0, -3, 3, 3, 7, 2], [3, -2, 3, -4, 1, 0, -8, 1, -8, 7, 0, -9, 6, 4, -8, -10, -4, 14, -1, -4], [-11, 2, 6, -1, 11, -2, 6, 8, 0, 0, 1, -12, 15, 0, -8, -10, -9, 12, -14, 0], [-26, -1, 5, -8, 9, -3, 0, 14, 2, 7, 14, -5, 3, -8, -6, 4, -5, 1, -13, 7], [-10, -8, -16, -18, 0, 9, 8, 5, 5, 0, 11, -9, -6, -6, -7, -1, 10, 2, -9, 8], [-24, -2, 5, 11, -4, -6, -4, 1, -2, 3, -2, 1, 3, 3, 9, 2, -4, 1, 4, -5], [10, -7, -7, 5, -4, -7, 10, -2, 5, -15, -1, 2, 6, 12, 10, 4, 14, 14, 3, -8], [4, -19, -7, -12, 6, 17, 0, -1, -7, 3, 6, -11, 6, -3, -4, -6, 1, 2, -8, -1], [7, 24, 11, 13, -4, -1, -16, 1, -12, 4, -4, 12, -17, -4, 0, -1, 1, -19, 14, 1], [15, 5, 4, 0, 0, -1, -10, 5, -7, 7, 22, -2, -19, -14, -15, -1, -3, -16, -1, 8], [-15, 5, 16, 2, 0, -5, 3, 7, -6, -11, -5, 2, 1, 5, 3, -2, 2, -1, 7, -3], [20, 15, 10, 6, 5, -7, -1, -1, -3, -6, -11, 8, 5, -1, 10, 13, -5, 4, 1, -5], [-16, -4, 9, -11, 6, 1, -1, 9, -4, 2, 13, -10, 10, -6, -12, -1, 0, 14, -5, -3], [-9, 0, 3, -8, 7, 0, -1, 1, 3, 0, -2, 4, -1, -9, 2, 1, 1, -6, 1, 3], [-14, -17, -17, -20, 14, 13, 12, 9, 3, 8, 1, -8, 2, -12, -6, -9, 6, 17, -13, 8], [-8, -8, -3, -16, 10, 9, 2, 15, 2, 8, 1, -4, -5, -13, -6, -12, 6, -7, -7, 8], [1, 8, 0, -5, -6, 11, 0, 4, -7, 4, 7, 2, -6, -12, -6, -2, 2, -15, -7, 6], [-6, -13, -8, -1, 4, 6, 5, -2, 5, -6, -7, -4, 2, -1, 7, -1, -3, 0, -2, 2], [-9, -1, 13, 6, -8, -3, 6, 6, -10, -21, -2, 13, -5, 2, 5, 8, 7, -14, 17, -5], [-3, 10, -8, 4, -2, -5, 1, -2, 2, -9, 2, -6, 10, 5, 5, 0, -12, 10, -9, 3], [-2, -7, -2, -15, 3, 0, -4, -4, 0, 4, 8, -5, 7, -3, -7, -4, 0, 15, -5, 0], [-4, -5, -3, 5, 5, -3, 0, -15, 9, -5, -13, -2, 15, 4, 11, 1, -5, 12, 0, -7], [6, 2, 10, 6, -1, 7, -16, -3, -5, 17, -3, -3, -6, -5, -10, -5, -7, -10, 6, -8], [-1, 1, -12, 2, 5, 2, 1, -15, 0, -2, -8, -15, 9, 4, 6, -3, -13, 14, -10, -3], [18, -11, -5, -14, 2, 3, -1, -1, 1, 8, 19, -10, 0, 0, -2, 0, -1, 2, -14, 1], [1, 2, -3, 0, 2, 5, 6, 8, -4, -11, 3, -3, 0, 3, -8, -2, 2, -3, -1, 4], [-15, -7, 0, -9, 10, 5, 2, 8, -2, 0, -10, -3, 7, -11, -5, -3, -5, 11, -3, 3], [7, 4, 3, 6, 3, 10, 3, -10, -6, 10, -1, 7, -14, -10, -1, 3, -2, -16, -3, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6034_2_a_l();".
// 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_6034_2_a_l(:prec:=20)
    chi := MakeCharacter_6034_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_6034_2_a_l();".
// 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_6034_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-21776, 109144, -112848, -205294, 391391, 101137, -477646, 47208, 312410, -74594, -124276, 36075, 31784, -9121, -5329, 1292, 573, -97, -36, 3, 1]>,<5,R![-128, 592, 4033, -9234, -47095, 11547, 138208, 6847, -178167, -24962, 118348, 23433, -42076, -10953, 7633, 2587, -566, -277, 0, 10, 1]>,<11,R![-345720, -6544172, 17033271, 9319078, -45473368, 7269758, 41492768, -12894744, -18994881, 5614107, 5322686, -1010760, -947182, 50528, 97570, 6516, -4654, -774, 42, 17, 1]>],Snew);
    return Vf;
end function;