
// Make newform 4010.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4010_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_4010_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_4010_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 := [-2, -8, 136, 67, -1839, 485, 6456, -2573, -6415, 2544, 2854, -1075, -638, 228, 70, -24, -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, 1, 0, 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, 0], [-113062883, -2951997384, 17753806708, 15957834852, -107935060143, -12946666199, 157200326808, -16507498270, -88325683716, 17099745144, 22855792597, -5467625787, -2742036456, 741135014, 120153563, -36703983, 517292], [-1191999696, -5708177003, 35278488729, -4830190853, -129782414532, 117093675538, 76557067027, -108891365197, -8919225821, 39522423700, -3832677742, -6920964417, 1249910253, 583174545, -132711821, -18840353, 4837565], [30057660, -3427129740, 9021163238, 13581871588, -62713131246, -80263510040, 135261243188, 70835731577, -96979101208, -25754222788, 31725390068, 4595130637, -5245102879, -396189248, 426684545, 13049122, -13533000], [-133252285, 2530180189, -20618036438, -23538978861, 140382802694, 76514749485, -180587342672, -97602977359, 110121572033, 52998995529, -37253291795, -13595084080, 7106995572, 1640652448, -712942292, -75243059, 29164450], [2506119991, -2388994893, -57121635689, -5721251743, 289410490306, 88683444929, -422865996766, -86805681473, 258464542138, 34708011918, -78482109680, -6839928369, 12518001396, 653567956, -1004560849, -24013167, 31959553], [734658957, -9595715223, -15556414844, 99969508807, 61669212078, -288436995914, -25061014027, 293829879846, -28457584155, -137168820096, 24429820825, 32010911487, -7069307355, -3638281524, 898763550, 160512035, -42578708], [-455720149, -1005069177, 31186774911, 34335215026, -203405448987, -156468744478, 315848585860, 168438708936, -207100673241, -78753218317, 68978681863, 18190214717, -12352098451, -2036841696, 1139626837, 88292181, -42697450], [-171215480, -10017945931, -276287293, 158965030720, -57832557032, -530055382254, 227835002030, 518021909831, -214756568868, -228720423838, 88901872108, 50928586221, -18628834335, -5575538085, 1945071424, 238642459, -80578876], [1635727447, 6224528889, -38782026641, -149037962658, 239307987195, 605792160984, -493500671988, -621335089574, 384895427290, 280528180900, -144528189191, -63236140377, 28404799275, 6970241055, -2829478710, -299359609, 113055721], [2741181689, 3861941654, -61108503632, -127655594119, 316758674053, 579787781632, -549342878384, -637341894902, 408725147379, 302367912296, -153317794321, -70628441588, 30622076627, 7992434926, -3118269472, -350251889, 127487096], [1300063361, 12492996538, -29094172573, -211409732500, 202932340288, 797125646421, -495886805261, -797062040583, 412232176906, 353717142183, -160079093236, -78734188184, 32151835641, 8598825523, -3252935367, -366833857, 131491739], [-820821813, 14665985027, 34234490641, -272833266410, -49144481099, 1022958505739, -277759642068, -1005248527892, 337091453304, 438713481887, -150418300729, -96340887561, 32519190575, 10415391985, -3444899317, -441098466, 143660013], [-456913120, -9885630770, 27086625562, 243290631232, -244022585342, -979513317111, 614763282936, 1003050087593, -517697609720, -452195502624, 203184397219, 101813111297, -41202594024, -11215905374, 4204915171, 481721894, -171291908], [2910598480, 26265785258, -64166588488, -519085786871, 489226814098, 2069300872805, -1218459358638, -2161251749155, 1047036217357, 992835992437, -421019731065, -226952470094, 87385432777, 25302804816, -9106434527, -1097265728, 377743308]];
    Rf_basisdens := [1, 1, 1, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047, 309515047];
    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_4010_a();"
function MakeCharacter_4010_a()
    N := 4010;
    order := 1;
    char_gens := [2407, 3211];
    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_4010_a_Hecke(Kf)
    return MakeCharacter_4010_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, 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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, -1, 0, 1, -1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0], [0, 0, -1, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 1, -1, 0, 0, 0, 1, -1, -1, -1, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0], [1, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1, -1, 0, 0, 0, 0, -1], [2, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 1, 1, 0, 0, 0], [-2, 1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1], [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1], [1, 0, 1, 0, -2, 0, -1, 1, 1, 1, -1, 0, 0, 1, -1, -2, 0], [0, -1, 0, -1, -2, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, -1, -1], [-1, 2, 1, 0, 1, -1, 1, -1, 1, 0, 0, 2, 1, 0, 0, 0, 0], [0, 1, 0, 0, -1, 1, -1, 2, 1, 0, 0, -2, -2, 0, 0, 1, 1], [4, -1, -1, 1, -2, -1, 1, -1, 0, 0, 0, -1, 0, 0, 1, -2, -1], [2, 0, -1, -1, -1, -1, -1, -1, -3, 1, 0, 1, 2, -1, 0, -2, -2], [1, 1, 1, 2, 2, 1, 1, 0, 0, -2, -1, 0, -1, -1, -1, 2, 2], [1, 3, 0, -4, 3, -2, 1, -3, -2, 0, 0, 3, 3, 0, -1, 0, 0], [1, 1, 0, 0, 1, 1, -2, 0, 0, -2, 1, -1, -3, -1, 0, 1, 1], [-1, 1, -2, 1, -1, 1, 0, -1, 1, 0, 1, -2, -1, 0, 1, 0, 0], [3, 2, 1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, -1, 0, -1, 2], [-2, 1, -1, 0, 1, 0, 1, 0, 0, -1, 1, -1, 0, 0, 0, 2, 0], [4, -1, -1, -2, -3, -1, -1, 0, 1, 2, 0, 0, 1, 0, 0, -3, -1], [3, 1, 1, 2, 2, 0, 1, 0, 0, -1, -1, 0, -1, 0, -1, 0, 0], [2, 2, -2, -3, -1, 1, 0, -1, -1, 1, 1, 3, 1, 0, 0, -1, -1], [-2, 0, 1, 3, 0, 1, 0, 3, 0, -1, 0, -1, -3, 1, 1, 4, 2], [4, 0, 0, -2, 2, -1, 1, -1, -1, 2, 2, 4, 3, 0, 0, -1, -1], [-1, 2, 2, -2, 3, 0, 1, 0, -2, -2, -2, 0, 0, 0, -2, 2, 4], [1, 0, -2, 2, -2, 0, 0, 0, 2, 1, 1, -2, 1, 0, 1, -1, -1], [2, 1, -1, -1, -3, 0, 1, -1, 1, 1, 3, 1, -2, -1, 2, -1, -1], [0, 1, 0, -2, 1, -1, -2, 0, 0, -2, 2, 1, -1, -1, -1, 1, 1], [0, 0, -1, 3, 2, 2, 0, -1, 1, -2, 0, -1, -1, 1, -1, 2, 0], [6, 1, 1, -1, -1, -2, -1, 1, 0, -1, -3, -1, 0, 0, -2, -1, 1], [5, -2, -1, 1, -1, 0, 2, -1, 1, 0, 2, 0, -2, 1, 0, 0, 0], [0, 1, 0, 2, 0, 4, -1, 2, 2, -1, 3, -1, -3, 0, 0, 1, 1], [0, -2, 2, 2, 0, -2, 0, 1, 0, -1, -1, -3, -3, 0, 0, 2, 1], [1, 2, 1, 0, -1, 2, -2, 1, 1, -1, 1, 0, -1, -1, -1, -1, 1], [2, 0, 0, -1, -1, 0, 2, -1, 0, 0, 1, 1, 1, -2, 0, -1, 1], [2, -2, -1, 1, 0, -1, 0, -1, 0, 3, -2, 1, 3, 1, 1, -2, -4], [3, -2, 2, -3, 0, -2, 2, 1, 0, 2, -2, 4, 1, 2, 0, -1, -1], [7, 2, -2, -3, 1, -2, 0, -3, -2, -1, -2, 3, 1, 0, 0, -1, -3], [1, 2, -1, 0, 2, 0, 0, -1, -4, -1, -2, 1, 1, -1, 1, 1, 0], [2, -4, 1, -1, -2, -1, -1, 0, 1, 0, -2, 1, 0, 0, 1, -1, -1], [3, 1, 1, 0, 4, 2, -2, 0, 2, -3, 0, -2, -3, 0, -4, 3, 3], [6, 0, -2, -1, -2, 2, 0, 0, 1, -1, 5, -3, -3, 0, 1, 0, -2], [2, -2, 1, 2, -2, 3, -4, 3, 1, 0, 0, -4, -3, 0, -2, 0, 0], [-3, -3, 0, 1, -1, 1, 1, 0, 1, 1, 3, -2, -1, 1, -1, 2, 1], [-5, -2, 4, 2, -2, 1, 2, 3, 1, 2, -4, -1, 1, 4, -1, 1, 1], [6, 3, -3, -3, 3, -3, -1, -4, -2, -2, 2, 4, 1, -3, 1, 0, 0], [1, -4, 1, 4, 0, 1, -3, 1, 2, -2, 0, -3, -3, 1, 1, 2, 0], [5, 2, 0, 0, -1, 0, -4, 1, 1, 1, -2, 2, 1, -1, 0, -1, -1], [4, -1, -3, -1, -2, 1, -3, 1, -1, 0, 3, -2, -2, -2, 1, 0, -1], [-2, -4, -1, 0, -6, 2, -2, 2, 3, 3, 5, -3, -1, 1, 2, -2, -2], [4, -1, 2, -3, -3, 0, -1, 3, 1, 1, 2, -2, -2, -1, 0, 0, 4], [2, 2, -1, 4, 1, -1, 1, 0, 0, -3, -2, -1, -2, 1, 1, 5, 3], [1, -2, 0, 0, -3, -1, 1, -2, 1, 1, 2, 0, 2, 0, 2, -2, -2], [3, -3, 0, 4, 0, 2, -1, 1, 1, -2, -2, -2, -1, 0, -1, 2, 2], [0, 2, -1, 0, 2, 3, -1, -1, 3, -2, 3, -1, -5, -2, 0, 4, 4], [2, -2, 0, 3, 1, 2, 0, 3, 2, -2, 3, -5, -1, 0, 2, 5, 3], [-1, -4, 1, 0, -2, -2, 1, 1, 1, 3, 0, 3, -1, 1, 1, 0, -2], [2, -2, 3, 0, 0, -1, 2, 3, 1, 2, 0, -2, 1, 2, -2, 0, 2], [6, 3, -4, -8, 3, -2, 2, -5, -2, 0, -1, 6, 2, 0, 1, -2, -2], [10, -2, 1, 0, -1, 0, 2, -2, 1, -1, 1, 0, -2, -1, 0, -3, -1], [0, 0, 1, 1, 0, 1, 2, 0, 2, 0, -3, 0, 0, 3, 1, 3, 1], [2, 0, 2, -1, -5, 0, 1, 2, 1, 4, 1, 1, -1, 3, 0, -1, 1], [-3, -7, 2, 3, 0, -1, 1, 2, 2, 2, 0, 2, 0, 1, 2, 0, -2], [-1, -1, 0, -1, -7, 3, 0, 2, 4, 3, 3, -2, 1, 1, 1, -5, -1], [0, 3, 2, 0, 3, -1, -2, -2, -2, -2, -1, 1, -2, 1, -4, 1, 1], [6, 0, -1, 0, -1, -1, 0, 0, 1, -1, 1, 2, -2, -2, -1, -1, -1], [6, -7, -4, 4, -5, 0, 1, 1, 4, 2, 4, -4, 0, 0, 2, -1, -3], [3, -4, 3, 1, -3, 0, 4, 1, 4, 0, 1, -2, -3, 0, 1, -3, 0], [3, -7, 0, 0, -4, -1, -4, 2, -1, 0, -3, -3, 1, 0, 0, -2, -2], [-7, -2, 1, 3, -2, 0, -1, 2, 5, -3, 2, -5, -1, 2, -2, 1, 3], [8, -4, -4, 2, -3, -1, 1, 1, 1, 2, 4, 0, 1, 0, 2, 0, -4], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 1, -4, 1, 1, 3, 1, -2, 3, -4, -3, 0, 3, 2, -2], [-10, -5, 2, 0, 1, -3, 7, 0, 1, 1, -1, 2, 1, 4, 0, 1, 1], [5, -4, -3, 4, -3, 2, -2, 2, -1, -2, 2, -4, -4, -1, 0, -1, -3], [2, -1, -2, 0, -1, 1, 1, -1, -3, 1, -1, 0, 0, -1, -1, 1, 0], [1, -1, 2, -3, 0, -1, 2, 0, -2, 3, -1, 3, 3, -1, 2, -1, -1], [-1, -2, -4, 6, -2, 2, -1, 3, 4, 1, 2, -6, -4, 1, 1, 1, 1], [5, -3, -5, 3, -1, 3, 0, -2, 0, 1, -1, -1, 2, 1, 0, 0, 0], [-2, -6, 3, 2, -2, -4, 1, 0, 1, -2, -6, 0, 2, -1, 0, -5, -3], [14, 3, 0, -8, -3, -1, -2, -3, -2, 5, -1, 8, 6, 1, -1, -8, -4], [1, 6, 2, -4, 8, 0, 2, -6, -3, -2, -4, 4, 1, 2, -5, 2, 2], [8, -6, 2, -3, -2, -3, 3, -1, -2, 3, 0, 5, 5, 3, 1, -5, -5], [9, -4, 0, -1, -4, -2, 1, 0, 2, 4, -2, 0, 5, 4, 0, -5, -3], [-4, -4, 3, 2, 4, -2, 3, 0, -1, -2, 0, 0, 2, 5, -2, 1, -1], [-3, -2, 5, 0, 3, 1, 6, 2, 2, -1, -1, 1, 1, 3, 4, 5, 5], [2, 1, 4, 0, 5, -1, 0, 1, -2, 1, -9, 4, 4, -1, -3, 2, 2], [3, 3, 0, -2, 3, 0, 3, -3, -4, -2, -2, 4, 3, 0, 0, -4, 0], [1, 3, -1, -3, 0, 0, -4, 0, -1, -1, 4, -2, -5, -1, -1, 0, 0], [-8, 2, 2, 1, 6, 0, 0, 3, -2, -2, -2, 0, 0, 3, -1, 7, 3], [-1, 8, -2, -3, -1, 1, -3, 2, -4, 2, -2, -3, -2, 1, -1, 0, 2], [5, -4, 0, -1, 0, -1, 1, -1, 0, 1, -2, 4, 3, 0, 0, 0, -1], [6, -3, -1, -2, 0, -3, 1, 0, -2, 4, -2, 0, 5, -1, 4, -4, -2], [7, 0, -7, -5, -5, 0, -3, -2, -3, 3, 5, -2, 6, -1, -1, -3, -1], [2, 4, -2, 2, 0, -1, -1, -1, 0, 2, 2, 1, -1, -3, -1, 2, -2], [-1, 0, 4, 2, 5, -2, 4, -3, -1, -1, -2, 4, 1, 1, 0, 3, 3], [-6, 1, 1, -3, -2, 2, -2, 1, -1, -1, 0, -1, -6, 1, -1, 1, 1], [10, -1, 1, -2, 1, -5, 2, -2, 0, 2, -1, 4, 5, 1, 1, -4, -2], [4, 6, 0, -1, 1, 0, -2, 1, -4, 4, 1, -1, 1, 0, 2, 1, -3], [3, 2, -1, 0, 5, -5, 8, -6, 0, -1, -2, 6, 6, -1, 4, -3, -3], [0, 3, -6, 3, -1, 4, -4, -1, -1, -5, 2, -6, -2, -2, -4, -1, 0], [-2, -4, -4, -1, -3, -1, 0, -4, -3, 2, 4, 0, 4, 0, 4, -4, -6], [1, 4, -5, -3, 3, 2, 1, -7, -1, -5, 6, -1, 3, -1, -2, 0, 0], [-9, -7, 1, 8, -4, 2, 4, 4, 4, -1, 2, -5, -5, -2, 3, 7, 4], [0, 8, 5, -4, 5, -2, 2, -1, 2, -3, -2, 7, -3, 2, -1, 4, 8], [-1, -1, 2, 1, -2, 6, -1, 1, 1, 1, 3, -3, -6, -1, 2, 2, 0], [-2, 5, -1, 0, -4, -2, 3, -3, 1, 2, -1, 5, 0, -3, 0, -4, 1], [5, 6, 1, -4, 4, 3, -2, -2, -1, 1, 3, 7, 1, 2, -2, 1, 3], [2, 0, -1, -3, -4, 2, 3, -4, 2, 1, 4, 5, 0, 0, 3, -1, -3], [5, -4, -1, -1, -6, -2, 1, 3, -2, 0, 4, -2, -1, 0, 4, 0, 0], [-7, 7, 3, 2, 7, 0, 2, 1, 2, -6, -5, -3, -4, 0, 0, 5, 7], [-1, -1, 1, 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5, 3], [-15, -5, 0, 7, 4, 1, 2, 2, 2, -1, 1, 1, -1, 1, 2, 5, 1], [-2, 0, 4, 1, -8, 2, 0, 1, 7, 5, 5, -2, -1, 3, 2, -5, 1], [1, 4, -1, 4, -4, 3, -7, 0, 0, -6, -1, -8, -7, -3, -5, 2, 0], [-9, 5, 1, -2, 5, 3, 3, 0, -3, 0, 0, 4, -2, -2, 2, 6, 2], [-4, 1, -5, 0, 0, -2, 0, -6, -6, 2, 1, 3, 3, 0, -1, -6, -8], [9, 6, -4, -9, 3, 1, -1, -6, -2, 0, 7, 4, -3, -3, 3, 2, 4], [-3, -7, 4, 6, -4, 0, -1, 4, 0, 1, -5, 1, -2, 1, 3, -1, -5], [2, 7, -6, -2, -3, 2, -1, -5, 5, 0, 2, 1, -2, 1, -3, -2, -2], [6, 1, 5, -3, 1, 1, -1, -3, 0, -2, -1, 9, 0, -3, -2, -1, -3], [-3, -4, 4, -2, -1, -3, -1, 1, -3, 4, -4, 8, 1, -1, -2, -4, 1], [1, -1, 3, -7, 2, -6, 0, -4, -2, 3, -6, 12, 4, 0, -2, -7, -3], [-5, 0, 2, -2, -7, 0, 0, 2, -1, 4, -4, -2, 5, -1, 2, -6, 0], [-2, 0, 3, -4, 6, -2, 3, -4, -3, 0, 4, 2, 6, 1, -2, 3, -1], [-7, -1, -5, 7, 0, 1, -6, 9, 2, -2, -1, -10, -4, -1, 1, 6, 0], [6, -5, -6, -4, -7, 2, -3, 1, -1, 4, 4, -5, 2, -1, 1, -10, -2], [-7, -5, 4, 4, 0, 2, 2, 6, -1, 7, -3, -2, 7, 3, 1, 1, 1], [-4, 4, 0, -4, 4, 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-2, 1, -3, -4, 0, -8, 0, 0], [-11, 7, -4, -3, 2, -2, 1, -1, 0, 1, 1, 5, 0, -6, 2, 2, 2], [-9, -2, 1, 10, 4, 3, 1, 2, -1, -7, -3, -2, -8, 2, 1, 11, -1], [-1, 6, 5, -2, -5, 1, 1, -1, 1, 4, -2, 0, 3, 0, -2, -4, 0], [-7, 6, 1, 1, 5, 3, 3, -3, -7, 0, -2, 1, 2, 2, -3, 1, 1], [4, -6, -1, 1, -8, 5, -7, 6, -3, 2, 4, -3, -4, -4, -1, 3, -1], [1, -8, -1, 4, -9, 0, -6, 10, 3, 0, -3, -10, -7, 2, -2, -5, 1], [-12, 11, 2, 0, 6, 5, -2, 0, -1, 0, 4, -2, -3, 0, -4, 7, 5], [-9, -6, 2, 2, -16, 5, -6, 9, -1, 4, 6, -7, -3, -2, 5, -8, 1], [3, -7, 3, 7, -6, 3, -6, 6, 6, 1, 0, -10, -5, -1, 2, 0, 2], [-4, -8, 0, 0, 3, 1, -1, 5, -3, 2, -5, 2, 5, -3, -1, -2, 1], [-7, -1, 2, 9, 5, 5, 4, 5, 7, -6, 3, -4, -9, 4, 1, 8, 4], [-6, -2, 0, -7, -1, -2, -4, -1, -5, 4, 2, -3, 2, -2, 2, -5, -3], [-8, 1, -3, 3, 2, -3, 7, -2, -3, 1, -2, -3, 3, 2, 3, -3, 0], [4, -12, 3, 7, -1, -2, 4, 2, -2, 1, -8, 2, 5, 6, 0, -5, -5], [-6, -9, 1, -2, 9, -2, 0, -4, -7, -2, -7, 4, 7, 1, 0, -2, -2], [1, 6, 4, -3, 1, -2, 4, -4, -2, -4, 5, 2, -5, 0, 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6, 1, -3], [-4, -5, 4, 1, -4, 5, 2, 4, 7, -1, 6, 1, -7, -4, 3, -1, 1], [-1, 4, -5, -9, 6, -1, 4, -12, 2, -2, 4, 10, 2, -3, 2, 1, -1], [13, 11, -5, -6, 8, 1, 1, -9, -6, -2, 0, 9, 6, -5, -4, -3, 1], [-3, -6, -1, 1, 4, -3, 1, -4, 2, 0, -1, 3, 4, 5, 2, 3, -5], [2, 7, 3, -1, 10, -8, 3, -2, -6, 0, -8, 3, 9, -2, -1, 2, 0], [-3, -8, 6, 3, -6, 2, 1, 7, 2, 2, -3, -3, -3, -2, -2, 1, 3], [3, -1, 3, -1, -1, -6, 4, -2, 6, -4, -4, -1, -1, -2, 1, 1, -2], [11, 6, -6, -10, 12, 0, -6, -12, -12, -8, -3, 7, 4, -10, -4, 3, -1], [9, 9, -2, -6, 7, -5, 0, -5, -4, -1, -6, 6, 2, -3, 0, -2, -4], [15, 1, 3, -3, -1, -2, -2, -5, -2, -1, -1, 6, 4, -4, -5, -7, -3], [-7, 0, 11, 8, 6, 0, 9, -4, 3, -3, -1, 2, -1, 5, -1, 2, 8], [-1, 7, 4, 1, 3, 4, 3, 4, 4, 3, 2, 3, -2, 4, -5, 0, 4], [-13, 8, -3, -2, 4, 10, -5, 1, 0, 1, 8, -1, -5, 2, -3, 6, 6], [12, 0, 0, 0, -2, -7, 6, 3, 3, 8, -5, 1, 9, 2, -2, -7, 1], [1, 7, 0, 2, 10, -2, 1, -3, -1, -2, 1, 3, 2, 1, 0, 7, -1], [-1, 12, -1, 0, 5, 3, 2, 1, -4, 1, 1, -3, 3, 5, -1, 8, 4], [7, 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-9, -1], [-3, -7, -5, 8, -10, 7, -1, 6, 10, -2, 10, -9, -8, -7, 11, 6, 6], [22, 8, -11, -10, 11, -2, -2, -7, -1, 3, -10, 11, 17, -1, -5, -8, -10], [-10, 3, -1, -3, -4, 4, 2, -2, 1, 8, 3, 6, 3, 1, 5, 2, 2], [-11, 21, -4, 2, -1, -1, -3, -1, -5, 1, -2, -4, -2, -7, 2, 7, 4], [8, 7, 3, 0, -4, 4, -8, 10, 4, 1, -3, -7, -3, -6, 2, 6, 8], [-4, 6, 9, -3, 7, -2, -2, 3, -6, 0, -18, 10, -1, 8, -4, -2, 2], [-16, -6, 3, 2, 5, 8, 4, 6, 5, -6, 1, -2, -8, 5, 3, 14, 7], [-16, -14, 2, 14, -4, 2, 11, 13, -3, -1, -4, -4, -9, 3, 12, 9, 1], [17, 6, -3, -12, -5, 4, -6, -6, -3, 2, 7, 10, 3, -3, -5, -7, -3], [16, 4, 3, -13, 14, -2, 4, -4, -9, -5, -5, 14, -2, -5, 3, -1, 5], [-3, 18, 4, -9, -4, -6, -1, 0, -1, 5, -1, 7, 6, -2, 0, -3, -2], [12, -6, 16, -3, 4, -8, 8, 1, -1, 14, -9, 17, 9, 5, -2, -8, -4], [5, -5, 7, -3, -1, -4, 9, 1, 6, 11, 3, 6, 2, 9, 9, 0, -7], [5, 12, 4, -7, 14, -9, 5, -4, -9, -2, -10, 10, 6, 2, -8, 4, -4], [-6, 1, -7, 11, -13, 5, -6, 6, 7, -9, 5, -13, -17, -10, 4, 4, 0], [-4, -9, 9, -1, 7, -7, 7, 0, 0, 1, -11, 13, 4, -5, 2, -8, -6], [-5, -12, 0, 11, -1, 2, 2, 11, 10, -4, -6, -17, -5, -2, 1, -7, 9], [-10, -17, -8, -3, -4, -1, -4, 2, -2, 7, -2, 5, 13, -1, 1, -17, -11], [2, 7, -7, -6, 7, -1, 0, -3, -4, 5, 10, 9, 6, 3, -7, 3, -5], [-11, 21, -3, 6, 2, 1, -5, 0, -4, -9, -7, -16, -8, -1, -9, 9, 9], [3, 20, -3, -5, 14, 5, -3, -11, -4, -7, 8, 6, 3, -13, -4, 12, 4], [18, 1, 0, -7, -10, 0, -5, 4, 2, 14, 1, -2, 5, 6, -1, -7, -5], [7, -5, -3, 1, -10, 1, -8, 6, 10, -5, 5, -13, -8, 5, -2, -4, -4], [-9, -16, -2, 17, -9, 2, -4, 10, -2, 3, -15, -14, 6, 0, -1, 2, -8], [2, 5, 1, 2, 8, -2, -2, -3, 4, 0, -7, 5, 4, -1, -2, -16, -9], [-5, 4, 4, 1, 9, 8, 2, -4, 2, -9, -5, -2, -6, 2, -1, 12, 12], [-28, -5, 9, 8, 1, -2, 6, -6, 1, -1, -9, 10, 3, -5, 1, -4, -4], [32, 11, -14, -11, 0, -10, 3, -22, -4, -6, -1, 10, 9, -4, -1, -9, -7], [-20, 2, -9, 12, 4, 8, -3, 3, -4, -7, 1, -17, -3, 2, -8, 18, 0], [3, -1, 4, 0, -3, 3, -1, -7, 5, 1, 11, -3, 2, -2, 0, -6, 0], [-12, -12, 0, 12, -5, 11, -9, 12, 3, -3, 3, -18, -12, 2, 3, 9, 0], [11, 8, 0, 3, 5, 8, -8, 1, -1, -2, 6, -7, -7, -5, -5, 11, 7], [-7, 4, -2, 6, 5, -1, 6, -3, -4, -6, 0, 6, -7, -7, 0, 9, 5], [24, -4, -4, 7, -4, -4, -8, -4, 3, -1, 2, -1, 2, -7, -5, 4, -10], [13, -20, 3, 9, -6, 1, 1, -3, 0, 3, 0, 0, 5, 1, 4, -2, -12], [7, -12, -8, 8, -6, 7, -9, 0, 2, 1, 4, -15, 4, -5, -3, -4, -8], [-5, 6, 4, -6, -4, 7, 5, -2, 12, -1, 24, -3, -8, 5, 5, 5, 8], [-4, 5, -9, -6, 7, -1, 3, -8, -1, -9, 6, -1, 5, 0, -10, 4, 2], [10, 11, -3, -4, 10, 2, 2, -8, -2, -17, 6, -1, -12, 2, 1, 12, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4010_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4010_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_4010_2_a_l(:prec:=17)
    chi := MakeCharacter_4010_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_4010_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_4010_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4010_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, -8, 136, 67, -1839, 485, 6456, -2573, -6415, 2544, 2854, -1075, -638, 228, 70, -24, -3, 1]>,<7,R![5296, -137208, 876612, -228471, -1609892, 576768, 1109863, -376262, -373043, 114701, 65552, -18517, -5951, 1582, 254, -65, -4, 1]>,<11,R![241764, -418428, -1184718, 1862395, 2205990, -2704627, -2044566, 1413876, 1008018, -233683, -211304, 4981, 18356, 1185, -652, -68, 8, 1]>],Snew);
    return Vf;
end function;