
// Make newform 6223.2.a.k in Magma, downloaded from the LMFDB on 19 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6223_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_6223_2_a_k();".
// 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_6223_2_a_k();".
// 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 := [1, 1, -83, -280, 474, 1279, -1150, -1655, 1177, 957, -593, -275, 155, 38, -20, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-128712, 6759511, 34856907, -41913198, -162346281, 126910371, 205570931, -138531569, -115867538, 70696203, 32533154, -18385016, -4412026, 2347523, 228893, -116093], [299938, -4109094, -26089138, 32149637, 121751459, -79215634, -178289744, 91508371, 114412902, -52971862, -35484771, 15614719, 5170999, -2211727, -282272, 118542], [1167877, -2974546, -64193652, 40444858, 306694906, -167476866, -416685796, 213453449, 250361358, -121920963, -74008394, 34632721, 10433381, -4734663, -556793, 246643], [-1601912, 10770031, 112731547, -101394473, -519191331, 361459511, 679630526, -425687974, -397334578, 229254488, 115239599, -62183706, -16039026, 8205093, 848713, -416133], [-2488852, 15316331, 142341487, -127789533, -611916381, 431479801, 758894461, -486918174, -423867643, 252322388, 118505764, -66156781, -16040641, 8486798, 831558, -420868], [403609, -2391162, -27300111, 19521756, 123008780, -69683347, -159344793, 83323677, 92287029, -45669957, -26550396, 12597346, 3671187, -1686213, -193256, 86516], [-2398759, 14520502, 169828364, -104612936, -766742622, 412351657, 1000551217, -513565088, -583976571, 287448331, 169127043, -80263942, -23508057, 10825416, 1242406, -558146], [2939939, -18219637, -197861309, 142265856, 894448927, -514021097, -1145738667, 610664783, 655602681, -331094336, -186671853, 90323612, 25605932, -11975331, -1339991, 609731], [-2942448, 14029589, 195153183, -127489287, -878849284, 501713704, 1127337289, -608767076, -646723747, 332488247, 184614846, -90934424, -25378599, 12063962, 1330387, -614167], [-5229541, 12764513, 232294846, -121901624, -1013016703, 508299188, 1294215363, -632322762, -742060049, 351724054, 211857962, -97629743, -29129163, 13105079, 1527129, -673264], [5318248, -20676569, -308068303, 198561717, 1362704444, -797268019, -1714237034, 954511391, 966370827, -512234712, -271915646, 137780884, 36966909, -18024552, -1921932, 907437], [-1000302, 5138082, 59983859, -44412449, -267285935, 159092271, 341269962, -187405313, -195111036, 100969347, 55548928, -27413174, -7620756, 3621479, 398886, -183898], [5430944, -24599592, -296868004, 195849826, 1334031662, -721664127, -1730295122, 875314583, 1003060396, -483981381, -288743733, 134317747, 39940442, -18058381, -2103041, 929511]];
    Rf_basisdens := [1, 1, 1, 239255, 239255, 239255, 239255, 239255, 47851, 239255, 239255, 239255, 239255, 239255, 47851, 239255];
    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_6223_a();"
function MakeCharacter_6223_a()
    N := 6223;
    order := 1;
    char_gens := [5589, 638];
    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_6223_a_Hecke( : Kf := Kf)
    return MakeCharacter_6223_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]];
    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 := [[0, -1, 0, 0, 0, 0, 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], [1, 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, 0], [-2, 1, -1, 1, 1, 0, -1, 0, 0, -1, 0, 1, 0, 1, 0, 0], [0, 0, -1, 1, 0, 0, 0, 0, 1, -1, 1, 1, 0, 1, 0, 0], [0, -1, 0, 1, 1, 0, 0, -1, 1, -1, 2, 2, 0, 0, 1, 0], [2, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, -2, 0], [-3, 1, -2, 3, 1, -1, -1, -1, 2, -1, 0, 2, 1, 2, 1, 1], [1, 0, 1, 0, 0, -1, 1, 0, -1, 0, 0, 0, 0, 0, -2, 0], [1, 1, 0, 4, 2, -1, 0, 1, 0, -1, 0, 2, 1, 2, 0, 1], [0, 1, 0, 1, 0, 0, 1, 2, 1, -1, -1, 0, 0, 1, 0, 0], [3, -1, -1, -2, -1, 1, 2, -1, 1, 1, -1, -2, 0, -1, 0, -1], [2, 0, 1, -1, -1, 0, -1, 0, -1, -1, 0, -2, -1, -1, -1, 0], [2, 0, -1, 2, 1, 1, 0, 0, 2, -1, 1, 2, 0, 0, 0, -1], [-2, 2, -3, 2, 0, 1, 1, 0, 2, 0, -2, -1, 2, 3, 2, 1], [5, 2, 3, 0, -1, -1, 0, 1, -1, 0, -1, 1, -1, -2, -3, -2], [-4, 1, -6, 4, 4, -1, -1, -1, 2, -3, 2, 2, 1, 3, 4, 1], [2, 0, 0, 2, 2, -1, 0, -1, -1, -1, 1, 1, 0, 0, -1, -1], [-6, 1, -2, 4, 3, -1, 1, 2, 1, -2, 1, 2, 1, 2, 0, 0], [1, 1, 1, 0, 0, -1, 1, 0, 0, 2, -3, -1, 0, 1, 0, 0], [-6, 0, -4, 3, 4, -2, -1, -1, 0, 0, -1, 0, 3, 1, 1, 2], [6, 0, 4, 2, 0, -1, -1, 3, -1, -1, 1, 2, 0, 1, -3, -1], [4, 3, 1, -5, -2, 0, 0, 0, -3, 1, -2, -2, -2, -1, -2, 1], [-2, 0, -3, 0, 0, -1, 0, -2, 3, 0, 0, 1, 0, 1, 1, 0], [2, 1, -2, 7, 3, -1, 2, 1, 1, -2, 2, 4, 3, 4, 1, -2], [1, 2, 1, 0, -1, -2, 0, 1, -1, 1, -3, -2, 2, -2, -3, 0], [-1, -3, 1, 0, -1, -1, -2, -1, 0, 1, 0, 1, -1, -1, -1, 2], [2, 0, -1, -1, -2, -1, 1, 0, 1, 1, 0, 1, -2, -1, -1, -2], [1, -2, 0, 2, 2, 0, 2, -1, 1, -1, 3, 3, 2, 1, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, -3, 2, 3, 0, -2, -1, 0, -1, 1, 4, 0, 3, 4, 0], [0, 5, -2, 3, 3, -2, 0, 0, 3, -2, 0, 3, -1, 1, 2, 2], [-2, 1, -2, 1, 0, -2, -4, -1, 0, -2, 1, 2, -1, 0, 1, 1], [-8, 3, -6, 6, 3, -4, -1, 1, 2, -1, -5, 0, 4, 5, 3, 3], [1, -1, 2, -3, -2, 3, 1, 0, -1, -1, -3, -2, 2, -2, -2, 0], [2, -2, 0, -8, -4, 2, 3, -1, -2, 4, -3, -9, -1, -6, -1, -2], [4, 1, -1, 2, 2, -4, -4, 0, -2, 1, 1, 1, -1, 2, 1, 0], [2, -1, -1, -1, 0, 1, 0, 1, -1, 0, -4, -5, 3, 1, 0, 3], [7, -3, 4, -4, 0, 1, 0, 0, -3, 3, 5, -1, -4, -4, -1, -4], [2, 2, -2, -3, -5, -1, -1, -1, -2, 1, -2, -7, 0, -1, 0, 2], [-4, 0, -3, 1, 5, 0, 1, -2, 3, -1, 0, 2, 1, 1, 1, 2], [0, 2, 3, -4, -4, 1, 2, 0, -1, 0, 1, 0, -1, -4, -3, -2], [1, 4, 0, -3, 1, 1, 1, 2, -1, 0, 1, -1, -1, -1, -1, 1], [0, -5, 4, -5, -1, 3, 1, 0, -1, 2, -2, 0, 0, -4, -1, 0], [0, 1, 3, -2, -1, 2, 0, 2, 0, 0, -1, -1, -2, 1, 0, -2], [1, -1, 3, 0, -1, 4, 1, 0, 1, 0, 2, 1, -2, -1, -2, -1], [-4, 0, -5, 8, 5, -2, 0, 2, 2, -4, 2, 3, 2, 7, 3, 1], [0, 3, -1, 5, 4, -3, -1, 3, -1, -1, 0, 2, 0, 5, 1, 3], [-9, 1, -6, 6, 0, 0, -3, -1, 4, 0, -2, 3, 4, 5, 4, 3], [3, 2, 0, 0, 0, 3, 3, 1, 0, 0, -2, -3, 4, 0, 1, 0], [-4, -2, 4, -6, -3, 1, -1, 2, -3, 1, -3, -3, -1, -2, -1, -1], [-4, -1, -4, 7, 6, -1, 1, -2, 3, -4, 5, 8, 1, 1, 2, -4], [11, -1, 4, -8, -5, 1, 1, 2, -4, 2, -6, -6, -3, -3, -2, -1], [-6, 2, -6, 2, 0, 0, -1, -2, 2, 0, -3, -1, 5, 0, 3, 4], [-2, -3, 3, -2, -1, 2, 1, -2, -2, 3, 4, 0, -1, -3, 0, -2], [-1, 6, -5, 4, 0, -2, 0, 2, 2, 0, -2, -2, 2, 3, 2, 3], [2, 0, 3, 0, -6, 0, 0, 3, 1, -2, -3, 1, 0, 0, -3, 0], [6, -2, 6, 1, 0, -1, -2, 1, -4, -1, 0, 2, 1, -1, -6, -2], [0, 6, -2, -1, 2, 0, 3, 0, 1, -1, 4, 1, -2, 3, 3, -1], [-4, 3, -3, -6, -3, 4, 0, -3, 1, 0, -5, -2, -4, -2, 1, -1], [-1, 0, -2, 10, 3, -4, 0, 1, 2, -2, 2, 8, 3, 5, 0, 2], [7, -2, 8, -6, -5, 3, 2, 4, 0, 3, -1, -1, -1, -4, -5, -2], [3, 2, -8, 5, 0, -1, 4, -1, 4, 0, -2, -3, 7, 3, 4, 2], [3, -1, 1, 2, -1, 4, 3, 1, 2, -3, -2, 0, 2, 0, -1, 0], [3, 1, 2, 1, 2, -2, -4, 1, 0, 1, 0, 4, -1, 2, 0, -3], [1, -4, 0, 2, 1, 1, 0, 2, 0, -1, 1, 2, -1, 2, 0, 0], [1, -3, 3, -3, 0, 0, -3, 0, -2, 5, 0, 0, -3, -2, -1, -4], [-3, 3, -4, -2, 1, -1, -2, -1, 1, 2, -2, -6, 2, 0, 4, 2], [2, -2, 1, -10, -4, 5, -1, -2, -2, 2, 1, -6, -2, -1, 1, 1], [4, -2, 0, 2, 0, 1, 3, 2, 1, 1, -4, -1, -1, 0, -2, -3], [-11, -1, 1, 5, 3, 0, -2, 3, 4, -1, 2, 7, -2, 2, 2, 0], [-10, 9, -7, -2, -3, -1, -4, 1, -1, 3, -8, -11, 4, 1, 3, 7], [10, -10, 4, -6, -6, 6, 6, 2, -2, 6, -2, -4, -1, -1, -1, -4], [-1, 6, -1, 4, -3, -1, 1, 3, 1, 0, -4, 2, 2, 2, -4, 0], [4, 0, 4, -2, 1, -2, -7, 3, -7, 0, 0, 1, -2, 0, -6, 1], [9, -8, 6, -3, 0, 3, 5, -2, -4, 1, 9, 2, -2, -3, -1, -4], [5, -3, 2, -1, 3, -1, 2, -2, -2, 2, 1, -3, -4, -2, -1, -1], [1, 3, -1, -5, -3, 0, 3, -1, 0, 1, -4, -8, -4, -1, 0, 2], [3, 3, 2, 0, 2, -3, -4, -2, -1, -2, 3, 0, -1, -3, -3, -2], [9, 3, 6, -8, -7, 2, -4, -1, -3, 2, -1, -5, -2, -5, -6, 0], [-5, 2, -1, 12, 10, -2, -2, 3, 1, -3, 2, 8, 5, 7, 2, 0], [4, 2, 12, -14, -10, 0, -3, 3, -10, 5, -6, -9, -4, -5, -10, 2], [3, 1, 7, 8, 0, -3, -2, 0, -2, -4, 8, 11, 0, 4, -4, 0], [-1, -5, 4, 1, 1, 1, 0, -1, 0, 1, 1, 0, 1, -5, 0, 1], [11, -2, 10, -2, 0, 1, 3, 4, -2, 0, 2, 5, -5, -1, -3, -6], [11, -1, 11, -3, -3, 1, -4, 1, -1, 3, 5, 4, -3, -1, -6, -3], [10, 1, 5, 2, 0, 1, 5, 2, 1, -3, -1, 1, 0, 0, -3, -3], [10, -2, 2, -9, -3, 0, -1, 1, -4, 2, -3, -7, -5, -1, 0, 2], [-9, 4, -5, 8, 5, -3, -1, -2, 4, 0, -2, 4, 1, 2, 4, 1], [2, -1, -2, 6, -3, 1, -2, -1, 2, -1, 4, 3, 4, 1, 0, 1], [14, 0, 1, 1, 2, -5, -4, -3, -1, 1, 0, 0, -1, -3, 0, 2], [-2, -3, 2, 2, -4, 1, 2, 2, 3, -1, 0, 0, 2, 0, -5, 2], [-1, -6, 2, -8, -2, 3, 1, -1, -7, 2, -2, -7, 0, -3, -1, -3], [-7, -2, -5, 11, 6, 2, -4, 3, 4, -2, 0, 4, 5, 7, 9, 3], [14, -4, 2, -16, -5, 4, -3, -2, -6, 4, -1, -12, -7, -8, 0, 2], [-1, 0, -5, 10, 4, 2, 1, -1, 3, -6, 3, 6, 0, 5, 0, 2], [-3, -3, -8, 0, 5, -1, 2, -4, 5, -1, -2, -5, 6, -2, 6, 2], [6, -8, 7, 0, -4, -1, -2, 3, -3, -1, 1, -2, -2, -1, -4, -2], [-4, 2, 0, 7, 0, -3, -8, -1, 2, -5, 4, 10, 4, 4, -1, 6], [8, 5, 7, -8, -4, 4, -3, 2, -4, -1, -4, -4, -6, -4, -6, -2], [-1, 0, -1, -8, -6, 6, 1, -3, 3, 0, -1, -2, -2, -7, 0, -2], [-1, -4, 2, 4, 0, 1, 4, 1, 5, 1, 1, 5, 1, -5, -3, -5], [11, -3, 14, -3, -4, 2, 0, 2, -2, 4, -4, 3, -4, -9, -6, -5], [-3, -4, -7, 5, 8, -2, -3, -1, 2, -5, 1, 7, -1, 8, 7, 1], [7, -2, 4, 1, 3, 1, -2, -3, 0, -2, 10, 9, -4, 0, -2, -6], [16, -6, 15, -9, -1, 3, 1, 0, -8, 3, 2, 2, -3, -9, -8, -7], [8, -2, 5, -1, 2, -1, 4, 2, 0, -1, 8, 8, -6, 1, -1, -7], [-15, 3, -10, 4, 6, -4, 2, 1, 1, 5, -10, -3, 6, 3, 3, 2], [-7, 1, -1, -3, -3, 3, 4, -2, 1, 2, -1, -3, 1, -5, -1, 3], [7, -2, 5, -13, -4, -1, 3, 1, -4, 7, -2, -11, -7, -7, -2, -3], [1, 2, -2, 4, 6, -9, -5, -3, -5, 2, 0, 2, 2, 5, 2, 5], [13, -1, 4, 0, -2, -4, -1, -3, 1, 3, 4, 0, 0, -6, -3, 4], [-1, 5, 1, -7, -4, 3, 4, -5, 3, 0, -1, -1, -2, -4, -2, -1], [-6, -2, 1, 6, 3, 6, -2, -4, 2, -2, 2, 9, 0, 3, 1, -3], [-1, -1, -4, 0, 1, -3, -5, -3, 5, -4, 3, -1, -2, -5, 2, 2], [9, -1, 3, -6, -2, 6, 2, 5, -6, 3, 1, -5, 0, -1, -1, -4], [7, 6, 6, 3, -3, -4, -1, 3, -1, -1, 7, 0, -1, 2, -2, 3], [10, -5, 3, -8, -4, 5, 5, -2, -1, 3, -2, 1, -6, -4, -3, -7], [3, -5, 2, -12, 2, 2, -5, -3, 0, 3, -1, 0, -6, -7, -1, -5], [12, -1, 16, -12, -10, 2, -7, 4, -8, 2, -1, -3, -7, -10, -12, 1], [-8, 3, -6, -5, -1, 2, -6, -5, 5, 1, -3, -2, -1, -1, 6, 6], [-5, -4, -7, 0, 0, -2, 2, -7, 5, -1, 2, 1, 4, 2, 0, 2], [3, -1, 3, 1, 1, -3, -2, 2, 0, 3, 3, 6, 0, -1, -3, 0], [2, -7, 1, 9, 4, -6, -3, 2, -1, 4, 1, 2, 3, 5, 0, 5], [11, -2, 10, -7, -7, 2, -5, -1, -3, 1, 8, 6, -5, -4, -4, -3], [-8, -5, -9, 4, 3, -2, 2, -8, 3, -2, 0, 1, 6, 4, 8, 5], [-2, 12, -2, 13, 1, -3, 1, 3, 5, -7, 1, 7, 7, 5, 3, 3], [1, -1, 4, -4, -5, 3, 0, 1, 3, 3, 2, -3, -5, -5, 8, -4], [-10, -4, -7, 5, 8, -4, 4, -5, 1, 3, -2, -3, 9, 0, 5, 4], [-11, 5, 0, 8, 0, -4, -5, 0, 0, -3, 3, 5, 6, 3, -2, 5], [3, 9, -2, 0, 4, -3, 1, 0, -2, -4, -1, -1, 1, 1, 3, 3], [0, 6, 3, 2, -3, -1, 0, 1, 0, -7, 5, 3, 2, -2, -7, 3], [19, -4, 9, -10, -4, -2, 3, 2, -4, 5, 2, -8, -5, -9, -6, -9], [-7, 2, 1, 12, 9, -3, 2, 1, 6, -6, 11, 17, 2, 6, 0, -3], [7, -4, -1, -6, -7, 3, 4, -3, -1, 4, 3, 2, -5, -4, -4, -2], [14, -2, 12, -11, -10, 5, -4, 4, -6, 6, -11, -5, -4, -6, -5, -2], [-3, 16, -10, 6, 3, -5, -2, -1, 5, -4, -2, 2, 3, 9, 6, 5], [-3, 4, 3, -4, -2, 3, -3, -8, 4, 2, -1, -2, -1, -4, 1, 3], [-15, 15, -7, 13, 4, -7, 1, 6, 5, -4, -6, 6, 9, 16, 6, 7], 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0, 4, -5, 11, 9, 5, 2, 1, 4], [-8, 6, 4, 5, -6, 5, 8, -5, 13, -3, -5, 3, 4, 3, 2, -5], [-2, -6, 3, 9, 0, -2, 2, 4, 0, -3, 0, -4, 3, -1, -8, 9], [-8, 2, 4, -11, -13, 0, 0, 2, -5, 10, -2, -13, 4, -8, -4, -1], [0, 0, 12, 17, 10, -3, 10, 9, 1, -9, 9, 29, -2, 7, -8, -9], [14, -3, 23, -18, -8, 7, 8, 5, -4, 1, 13, 5, -14, -16, -10, -14], [24, -6, 16, -18, -12, 6, 8, 2, -6, 7, 5, -5, -5, -15, -4, -6], [0, -12, -1, -9, -1, 4, 2, -8, -3, 8, -11, -18, -1, -6, 2, -2], [30, 2, 15, -11, -10, -6, -6, 8, -12, 2, -1, -9, -10, -12, -15, -3], [-9, 10, -3, -4, 5, -4, 3, -2, -3, -3, 5, -5, 0, -3, -3, -2], [-14, -2, -23, 16, 16, 2, 9, -5, 13, -10, 7, 10, 13, 8, 12, 4], [-4, 24, -18, 8, 6, -14, -18, 2, -5, 0, -6, -6, 0, 17, 8, 11], [35, -14, 12, -29, -17, 12, 7, -1, -14, 8, 3, -17, -5, -15, -17, -8], [12, -12, -9, 11, 1, -7, -4, 1, 7, -10, 6, 3, 1, 9, 1, 2], [-5, -8, 1, -13, -4, 2, -8, -3, -2, 0, 1, -6, -5, -9, 7, 7], [16, -2, 5, -19, -16, 0, -9, -1, -15, 4, 5, -12, -5, -9, -11, 2], [16, 4, 10, 16, 17, -4, 6, 3, -4, -10, 14, 24, -1, 3, -5, -14], [-5, 5, -11, 3, 5, 3, -9, -9, 5, -7, 0, 5, -2, 9, 4, 1], [-1, 0, 11, -6, 0, 0, 4, 6, -9, 10, -2, -6, -8, -10, -2, -4], [-14, 8, -22, -2, 1, 3, 14, -5, 12, 1, -8, -11, 4, 11, 14, 4], [-11, 3, -16, 15, 13, 1, 5, -1, 8, 0, 10, 5, 2, 9, 21, -11], [26, -3, 16, -14, -10, 2, -11, 4, -7, 3, -4, -3, 0, -6, -10, 0], [-31, 20, -14, 4, 1, -1, -2, -4, 10, -1, -5, -4, 7, 8, 14, 14], [2, 7, 1, -12, -11, 4, -4, 0, -1, -4, -7, -7, -7, 3, 0, 11], [21, 1, 7, 7, 7, -2, 10, 6, 1, 0, 4, -2, -2, -3, 1, -11], [15, 8, 3, -3, 0, -2, -7, -6, -3, 1, 16, 0, -6, -12, 2, -4], [-14, 20, -7, 20, 8, -12, -6, 6, 6, -7, -11, 12, 10, 17, 3, 3], [12, 7, 2, -14, -14, 4, -20, 1, -1, -2, -3, -3, -6, -3, -2, 6], [3, -1, -7, -7, -3, 7, 9, 5, -1, -4, 9, -3, -6, 9, 0, -5], [7, -5, 24, -18, -9, 10, -2, 5, -14, 0, 6, -2, -6, -6, -7, -7], [7, 9, 9, 5, -5, 2, -14, 5, -7, -4, 4, 9, -5, 4, -3, 9], [-1, 6, 6, 9, 3, 2, 1, -1, 8, -11, 8, 10, -3, 1, 2, -2], [2, 11, -8, 22, 15, -5, -9, 4, 7, -13, 3, 18, -2, 21, 7, -4], [4, -4, 5, -10, -4, 6, -3, 3, -10, 8, -3, -12, -8, -8, -1, -9], [8, -14, 16, -12, -7, 5, -3, -6, -7, 8, 13, 0, -10, -13, -3, -7], [1, -6, -15, -10, 2, 6, 11, -14, 6, -2, 3, -1, -1, -1, 5, -7], [-41, 6, -27, 29, 21, -9, 5, 3, 15, -5, -4, 7, 16, 23, 28, 10], [12, 4, 10, 0, -14, 2, -3, 7, 1, 9, -6, -12, -5, -3, 2, 1], [10, 11, 23, -3, -7, 3, 2, 8, -3, 5, -1, 5, -6, -6, -11, -1], [-10, 13, -13, -2, -2, -5, -9, -5, 9, -3, -10, -5, 4, -4, 5, 9], [9, -12, 11, -2, 0, 3, 7, 6, -2, 10, 6, 5, -6, 1, -3, -11], [3, -6, -6, 2, -1, -7, 0, 0, 6, 2, 1, -9, -1, 6, 5, 0], [-4, -5, -20, -7, 2, -2, 2, -11, 4, 6, -3, -24, 6, -1, 14, 11], [-20, -6, -6, 20, 5, -6, 0, 4, 2, -6, 15, 6, 10, 8, 7, 12], [14, 3, 11, -21, -13, -3, 6, -2, -6, 18, -5, -23, -5, -14, -4, 2], [7, 0, -2, 21, -2, 0, 2, 6, 9, -6, -10, 7, 7, 15, -2, 3], [-20, -6, 7, -18, -5, 4, 2, -1, 1, 6, 1, -20, -2, -6, 7, -2], [8, -2, -3, 1, -6, 0, 4, -3, 8, -2, -3, -3, -1, -5, -5, -5], [3, 6, 7, -2, -10, 5, 4, 8, 4, -9, 2, 1, -14, -2, 2, 1], [14, -2, 13, 1, 1, -5, 1, 3, -4, -1, 3, 9, -4, -2, -9, -11], [16, -3, 9, -10, -2, -2, -1, 4, -10, 13, -2, -19, -1, -7, -2, -2], [25, -15, 19, -38, -14, 11, 1, 4, -7, 13, -2, -9, -11, -19, -4, -9], [4, 7, -15, 6, 3, 8, 12, 5, 7, -2, -2, -6, 7, 11, 6, -3], [8, 3, -3, 14, 10, -3, 4, 2, -1, 1, -4, 4, 0, 5, -1, -9], [12, 0, 6, -12, 4, 5, 1, 1, -10, 4, 11, -5, -8, -4, -2, 0], [-5, -1, -3, -12, -5, -1, -2, -10, 5, 2, 4, -6, -5, -6, 15, 9], [-45, 9, -11, 20, 8, -4, -6, -1, 17, -7, -9, 3, 10, 12, 10, 7], [-7, -20, 5, -3, 6, 6, 11, -7, 8, 4, 2, 3, 8, -1, 8, 1], [36, -9, 9, -14, -8, 5, -1, -5, -3, 2, -2, -7, 4, -10, -7, -3], [-11, -6, 3, -2, -6, 16, -5, -1, 4, -4, -11, 1, 3, -3, 0, 3], [-15, 5, -29, 16, 6, -7, -2, -5, 18, -4, -19, -2, 15, 11, 17, 10], [-31, 3, -19, 18, 5, -6, -2, -3, 6, -1, -9, 2, 5, 14, 11, 10], [-16, 0, -14, 18, 12, -2, 0, 8, 13, -4, -7, -1, 12, 17, 19, 5], [14, 5, 4, 2, -6, -4, -8, -1, -3, -6, -6, 5, 3, -2, -7, 2], [8, 4, 1, 0, -1, 3, -12, 2, -2, -2, -7, 1, -1, 1, -2, -1], [-28, -5, -15, 18, 11, 1, 11, -3, 12, -6, 0, 21, 5, 12, 10, -2], [-26, 12, 3, -19, -6, 2, -6, 1, -12, 11, -15, -25, -2, -10, -2, 5], [19, -6, 7, -6, -1, -1, 2, 4, -11, 8, -12, 0, -4, -1, -9, 0], [13, 7, 4, -15, -16, 0, -3, 5, -8, 2, -2, -16, 1, -9, -15, 7], [-21, 16, -18, 20, 14, -10, 3, -5, 9, -8, -5, 9, 9, 7, 6, 5], [-21, 6, -17, 25, 16, -10, 3, 4, 8, 0, 4, 1, 6, 17, 18, 5], [-34, 13, -18, 7, 5, -7, -8, -1, 9, -7, -10, -7, -1, 10, 16, 10], [-8, -9, -3, 1, 9, 0, 3, -9, 8, -9, 9, 13, 2, 5, -4, -6], [9, 10, 10, -12, -9, 4, 9, 6, -1, 11, -2, 3, -5, -3, -4, -7], [19, -13, 4, 12, -3, -4, -6, 2, 1, -4, 3, 12, -9, -2, -9, -6], [-5, 4, 2, 6, 4, -7, -7, -1, -2, -5, 3, 4, 1, 5, 13, 14], [15, -11, 10, -27, -8, 7, 12, -4, -1, 14, -8, -15, -9, -20, -13, -8], [21, 9, 4, 5, 7, -15, -2, 7, -6, 3, -6, -1, -2, 2, -10, 1], [2, -15, -2, -7, 2, 2, 5, 11, -9, 5, -7, -20, 7, 0, 0, 10], [-11, 24, 0, 13, 8, -15, -6, 3, 1, -6, 0, 3, 2, 11, 1, 11], [22, -6, 24, -8, -14, 2, -2, 3, -16, 11, 6, -3, -5, -9, -12, -8], [20, -7, 7, -13, -11, 10, 9, 1, -3, 8, -1, -7, -12, -10, -8, -15], [17, -4, 13, -8, 3, 2, -4, 1, -10, 0, -4, 10, -3, -6, -4, -3], [1, -6, 8, 10, -1, -7, 2, 0, 1, 1, 10, 3, 2, -6, -12, -3], [-8, 5, -26, 7, 4, 7, 3, -7, 20, -13, -2, 1, 2, 11, 7, -3], [3, 17, 12, -11, -1, -3, -5, 3, -12, 4, -12, 1, -7, 0, -8, 3], [17, 4, -1, 6, 2, 3, 16, 4, -6, -4, -1, -2, 8, 8, -11, 1], [19, 7, -6, -24, -2, -2, -2, -7, -7, 7, -6, -15, -7, -8, 0, -6], [-12, -8, -1, 0, -6, 1, 1, -3, 1, 4, -23, -4, 8, -6, -9, -4], [6, 2, 9, 14, -2, -3, 1, 8, -5, 1, -12, 12, 17, 3, -12, 4], [-9, 4, 8, -17, 0, 1, -7, 1, -8, 1, -2, 3, -5, -4, -5, -2], [-26, 6, 11, -16, 2, 0, -5, 1, -13, -3, 2, 0, -9, -2, -8, 0], [-19, 20, -15, 9, -1, -15, -13, -8, 4, -3, -9, -7, 6, 6, 1, 15], [-22, 1, 3, -8, -8, -1, -11, -7, 2, 0, 0, 0, 1, 0, 1, 12], [23, 1, 15, -3, 0, -8, -3, 12, -17, 5, -12, -13, -6, 2, -13, 3], [21, 3, -8, 2, -1, -1, -1, -1, -6, 8, -15, -18, 8, -6, 7, 7], [-3, -12, 12, -23, -18, 6, 0, -2, -10, 9, -13, -24, -3, -10, -5, 0], [-19, -7, -1, 2, 4, 3, 7, 1, 10, -5, 8, 14, -5, -8, 2, -4], [-24, -16, -16, 16, 16, 3, 1, -10, 13, -8, 18, 17, 9, 8, 19, -1], [-3, -13, 4, -14, -8, 6, 12, -4, 0, 16, -12, -11, 1, -11, 1, -1], [17, 11, 9, 6, -3, 2, -7, 6, 1, -2, 2, -3, -1, 0, -5, 7], [6, 6, -11, 27, 18, -6, 0, 5, 3, -8, 4, 10, 5, 12, 0, -6], [-7, -5, 9, -10, -11, -7, -2, 6, -4, 13, -16, -4, -9, -3, -9, -2], [1, 14, -7, -19, 3, 2, -7, -4, -2, 3, -10, -8, -12, -3, -3, -3], [-5, 23, -14, -7, 1, -6, -12, 3, -3, 3, -10, -16, 3, 9, 14, 22], [-26, 1, -6, -6, 7, 4, -4, -8, 0, -3, 9, 10, -5, -1, 0, 1], [-4, -8, -5, 8, 13, -6, 1, 7, -7, -13, 11, 11, -2, 3, 2, 0], [0, 1, -10, 22, 0, 0, 0, -5, 16, -11, 15, 24, 10, 10, 11, -1], [-1, 5, 2, -5, -1, 1, -14, -2, 1, -4, -5, -3, 1, -4, 3, 14], [44, -21, 7, -3, -5, 4, 1, 3, -7, -7, 16, 3, -5, -5, -7, -9], [-2, 1, -18, 9, 10, 6, 1, 1, 5, -2, -2, 2, 4, 11, 11, -2], [6, 20, -7, 11, 5, -1, -4, -1, 7, -7, 14, 17, -4, 8, -2, -4], [23, -16, 27, -45, -28, 9, -2, 5, -13, 15, 4, -19, -16, -26, -17, -12], [-9, -4, -1, 5, 15, 0, -4, 2, -5, -12, 14, 20, 4, 11, 4, 4], [7, -2, 4, 2, 9, -4, 9, 4, -6, 0, 8, 3, 9, 4, -1, 0], [1, -5, -6, -4, -6, 12, 9, -7, 3, 6, -3, -5, 3, -1, 9, -2], [-57, 2, -12, -2, -2, 5, 4, -2, 8, -3, 0, 1, 2, 1, 11, 4], [-11, -4, -9, 10, 20, 2, 0, 4, 7, 1, 4, 15, -2, 8, 11, -6], [-3, -15, -9, 2, 12, -1, -2, -8, 5, -2, 16, -2, 0, -5, 19, 1], [16, -10, 9, -25, -22, 3, 11, -2, -3, 11, -8, -27, -4, -25, -7, -12], [-15, 1, 10, -5, -16, 5, -2, 7, -8, 10, -3, -10, -1, -7, -15, -9], [-42, 17, -28, 13, 0, -7, -9, -4, 13, -5, -24, -8, 7, 15, 8, 14], [-2, -18, 10, -21, -18, 10, 8, -9, -2, 11, 9, -8, -5, -20, -1, -10], [3, -10, 8, -20, -9, 7, -3, 3, -7, 4, -4, -17, 3, -2, 0, 7], [2, -5, 3, -9, -16, 1, -6, -12, 5, 4, -12, -3, 2, -4, -6, 7], [-12, 2, -30, 12, 18, -6, 0, -7, 6, -6, 13, 8, 0, 14, 21, 2], [0, -1, -2, 1, -8, 7, 10, -5, 4, 6, 5, 1, 5, -7, -3, -9], [8, -27, 11, -18, 3, 10, -2, -11, -1, -1, 5, 13, -9, -11, -5, -11], [-3, -4, 13, 10, 9, -1, -11, 12, -16, -6, 3, 13, 1, 6, -16, -1], [-5, 6, -13, -17, 4, 10, -3, -1, -4, -2, -5, -11, -5, -4, 1, 1], [32, 4, 15, -11, -15, 1, 3, -2, 7, 7, -4, 0, -5, -18, -11, -12], [14, -9, 9, 20, 17, -3, 1, 7, -7, 0, 4, 19, 2, 8, -6, -9], [-14, 10, -32, 41, 26, -7, -3, 1, 6, -21, 1, 8, 17, 30, 17, 16], [-4, -7, 13, -11, -3, 2, 9, 1, 2, 4, 6, 5, -4, -7, -8, -12], [-14, -1, 0, 26, 8, 3, -5, 5, 10, -16, 7, 25, 11, 11, -7, -7], [-3, 6, 0, 4, 15, -8, -4, -4, -2, 1, 19, 2, 1, 3, -1, 6], [21, 6, -10, 3, 9, -12, -1, 5, 0, -3, -10, -1, 7, 19, 17, 16], [9, 7, -12, -17, 1, -1, 5, -9, -7, 3, -3, -25, -4, -6, 7, 2], [1, -13, 19, -13, -8, 5, 0, 2, -13, 11, 2, -2, 0, -6, -13, 1], [3, 15, 19, -23, -14, 8, -9, -2, -6, 0, -23, -11, -14, -7, -9, -4], [6, 10, 9, 21, 11, -2, -3, 10, 3, -2, 3, 28, -1, 11, -1, -2], [-19, 7, -20, 16, 13, 0, 8, 6, 12, -15, -10, -3, 8, 14, 7, 9], [5, -5, 17, -18, -12, -2, 16, -2, -11, 9, -2, -19, 1, -21, -14, -5], [-9, 1, -1, 5, 3, 0, 6, 7, -7, 1, 5, -6, -3, -4, -10, -5], [16, -10, 3, 12, 11, 0, 11, -7, 1, -5, 18, 19, -11, 2, -1, -16], [19, 1, 30, -2, -2, 2, -2, 13, -11, 2, -3, 6, -4, 3, -14, 0], [2, -3, -3, -14, -1, 1, 0, -2, -6, -1, 7, -22, -8, -6, 3, 4], [-4, -5, -6, -9, 3, -1, 9, -1, 2, -2, -7, 0, 8, 1, 3, 1], [-12, -7, -11, 8, 5, 3, 11, -4, 5, -5, 6, 10, 5, 11, 1, -3], [13, 5, 10, -16, -12, -3, 7, 5, 1, 1, -10, 1, -9, -8, -8, -4], [36, 12, 19, -25, -15, 8, 7, 5, -8, 10, 1, -13, -10, -15, -6, -5], [8, -9, 7, -16, -19, 4, 2, -2, 5, 9, 14, 0, -11, -13, 2, -3], [3, -1, 3, 16, 7, 6, -8, -2, 7, -9, 15, 26, -5, 1, 7, -7], [-9, 5, -1, -6, -8, 10, 10, -2, 8, 0, -14, -4, -2, -5, -2, -2], [-11, -14, 20, -11, -4, 8, -3, -1, -13, 3, 4, -1, -7, -2, -7, -5], [10, 4, -22, 9, 3, 5, 1, 0, 1, -4, -8, -9, 7, 5, 8, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6223_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6223_2_a_k();".
// 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_6223_2_a_k(:prec:=16)
    chi := MakeCharacter_6223_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_6223_2_a_k();".
// 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_6223_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6223_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -1, -83, 280, 474, -1279, -1150, 1655, 1177, -957, -593, 275, 155, -38, -20, 2, 1]>,<3,R![-16, 368, 2392, -5098, -4525, 10815, 2617, -9547, -32, 4206, -486, -950, 175, 102, -23, -4, 1]>,<5,R![16, -184, 92, 3406, -4663, -13233, 18066, 12824, -21504, 223, 7523, -2135, -532, 273, -7, -9, 1]>],Snew);
    return Vf;
end function;