
// Make newform 6010.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6010_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_6010_2_a_c();".
// 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_6010_2_a_c();".
// 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 := [-10, -41, 295, 244, -1344, -440, 2332, 164, -1986, 228, 872, -232, -178, 75, 9, -8, 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], [-2, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-17499053, -89999479, 462320623, 414738405, -1879892501, -438850745, 2649268970, -171959962, -1628277055, 422463844, 419917416, -180887926, -24822547, 22373644, -3039938, 25986], [-2133268, -24830756, 127539985, 145436552, -522240622, -225824922, 793919965, 59905022, -536624577, 79579696, 157537117, -48394711, -14421854, 7114944, -416320, -75052], [-20383548, 10010483, 233958171, -43939251, -763974869, 157843278, 988201962, -327354943, -560118954, 274101206, 118706510, -91910225, 2661981, 9497505, -2419781, 180376], [706474, -9102667, -67060795, -39287261, 331067185, 150259400, -442765003, -157995531, 279407436, 74146068, -100540960, -13794772, 20718600, -441924, -1819192, 284786], [-6358506, -61280117, 116648156, 201791930, -319570964, -81890064, 347372878, -176603303, -139100342, 182827079, -10924214, -57805768, 18928481, 4353048, -2892660, 362678], [-714592, 12448056, -34425448, -8887973, 178775499, -230548103, -280058413, 429564885, 129035112, -294055924, 16650207, 82865319, -22475282, -6474594, 3421318, -384044], [-43793, -556740, -4266261, 6564984, 29632797, -29093811, -46199287, 47347456, 23795599, -32730627, -971276, 9493309, -2133030, -785482, 366746, -39704], [-7297457, -48865179, 181790025, 323396742, -601061929, -731389267, 816274892, 709158898, -600603130, -323031488, 255794130, 60827688, -58445943, -244087, 5318060, -800612], [4252139, -44134004, -211092724, 392375534, 852144102, -1049225189, -1099582230, 1324407527, 503191029, -818923620, 3659483, 225954995, -54357008, -18013961, 8826787, -972224], [5288765, -25743241, -173968012, 363077413, 658088239, -1110390251, -757194522, 1382719669, 238488877, -804630339, 92532283, 206543323, -64724264, -14466751, 8878763, -1048336], [-1358606, -10778289, 112016908, -119306620, -392052900, 650190464, 424081214, -1000042516, -69804760, 645278405, -128681409, -173501774, 67320977, 11444848, -8847625, 1124047], [-7526881, 27985975, -170186194, 97238878, 958014404, -862892792, -1396057392, 1555714470, 654630562, -1108457254, 34111348, 323120564, -88167362, -25482926, 13952863, -1618763], [-10162035, -16102888, 214322353, -146521530, -739948253, 939591930, 858348612, -1527505049, -243563866, 1017753578, -160180219, -280173236, 100594394, 19404207, -13768724, 1717836]];
    Rf_basisdens := [1, 1, 1, 3115183, 3115183, 3115183, 3115183, 3115183, 3115183, 163957, 3115183, 3115183, 3115183, 3115183, 3115183, 3115183];
    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_6010_a();"
function MakeCharacter_6010_a()
    N := 6010;
    order := 1;
    char_gens := [3607, 2411];
    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_6010_a_Hecke(Kf)
    return MakeCharacter_6010_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], [-1, 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, -1, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, 0], [-1, -1, 1, -1, 0, 0, -1, -1, 0, 1, 0, 0, 0, 0, -1, 0], [-1, -1, 1, 0, 0, 1, 0, -1, -1, 1, 0, 0, -1, 0, -1, 0], [0, -1, 0, 2, -2, -1, 0, -1, -1, 3, -2, 0, 1, -2, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, -1, 0, 1, 0, -1], [-1, 0, -1, 0, 0, 0, 1, 0, 1, -2, 1, 0, 1, 0, 1, 0], [-1, 0, -2, 0, 1, -1, 0, 1, 2, -1, 0, 1, 0, 0, 1, -1], [-2, 1, 0, 2, 0, -1, 2, -1, 0, -1, 0, 0, 1, 0, 1, 2], [-3, 2, -2, -2, 1, 1, 1, 3, 2, -3, 1, 0, 1, 2, 1, -1], [-3, 1, 1, 0, 0, -1, 0, 0, 0, -2, 3, -1, -1, 1, 0, 0], [-2, 0, 0, -1, 4, 0, 0, -1, 1, -1, -1, 2, 0, 3, -1, 0], [0, -1, 1, 1, 1, -1, 0, 0, 1, 1, 0, 0, -2, -2, 0, 0], [-2, 1, -2, 0, 1, -1, 0, 0, 2, -2, 0, -2, 1, -1, 2, 0], [0, 0, 0, 3, -3, -1, 1, 0, -1, 3, -2, 1, -1, -4, -3, 1], [-4, 0, 3, -3, 2, 2, -2, 0, -1, 0, 0, 1, -2, 1, -1, -1], [1, -1, 2, 0, -3, 1, -1, 1, -3, 0, 2, -3, 2, 1, 2, -1], [0, 1, -2, 4, -2, 1, 4, -2, 1, 2, -4, 0, 0, -3, 1, 2], [0, -1, 4, 0, -3, -3, -1, 0, -3, 5, 1, -2, 0, -3, -3, 1], [-1, -1, 1, 0, -3, 0, 0, 1, -2, 2, -2, -3, 0, -2, 0, -1], [-3, -2, 2, -4, -1, 4, -2, -1, -1, 1, 0, 3, -2, 3, -3, 0], [-2, 0, -2, 0, 1, 2, -2, -1, 0, 0, -1, 3, 0, 2, 1, 2], [-4, -3, 5, -3, 0, 1, 1, -2, -1, 2, 3, 1, -3, 2, -1, -3], [-3, 1, -2, 1, -1, 0, 0, -2, 0, 0, -1, -1, 1, -1, 2, 1], [0, 0, 2, 1, 2, 0, -1, -2, 1, 2, -3, 2, -1, -2, -2, 2], [2, -1, 0, 1, 1, -2, -3, 0, 0, 0, -2, -2, 1, -2, 1, 1], [-2, -1, 1, -2, 2, 3, -2, 1, -1, -4, 2, 1, 0, 2, 1, 0], [-3, -1, 2, -3, -1, 2, -2, 1, -1, 0, 3, 1, -2, 1, -4, -4], [-3, 4, -1, -5, 7, 4, -3, 4, 2, -9, 2, -2, 2, 7, 2, -5], [-1, -1, 3, 0, -1, -1, 1, -1, -1, 4, -2, 2, 1, -5, -4, -1], [-2, -5, 5, -3, 0, -1, 2, -2, 1, 5, 1, 6, -5, -4, -4, 2], [-4, 0, -1, -3, 4, 4, 1, 0, 1, -2, 1, 6, -4, 7, -1, 2], [-3, 2, -1, 0, 0, -3, 2, 2, 0, -4, 0, -3, 3, 0, 3, 0], [-5, 1, -2, 1, -2, 1, 4, 0, 1, 1, 0, 0, -1, 1, -1, -1], [-4, 1, 1, 3, 1, 0, 3, -2, 2, 1, -2, 1, -1, -2, 2, 2], [7, 0, -2, 6, -4, -1, 2, 0, 0, 1, -2, -1, 1, -3, 1, 1], [-3, -2, 2, -6, 5, 0, -3, 0, 1, 0, 1, 3, -4, 1, -1, -2], [-3, 6, -2, 1, 0, -2, 0, 4, 0, -3, -1, -5, 3, -2, 2, -2], [4, -3, 3, 3, -8, -1, -2, -1, -5, 6, 1, -4, 2, -4, -1, 1], [-2, 5, -4, 6, -4, -2, 4, 1, -1, -4, 0, -4, 2, 0, 7, 2], [1, 1, 2, 2, -6, -3, 1, 1, -3, 4, -3, 1, 3, -7, -1, 0], [1, 7, -7, 9, -2, -5, 8, 2, 2, -7, -4, -4, 5, -1, 7, 3], [-3, 4, -5, -2, 5, -1, -2, 1, 3, -4, -3, 2, 3, 6, 4, 1], [5, 0, -3, 5, -5, -3, 4, 2, -2, 0, 0, -1, 0, -7, 3, 4], [-1, -3, 3, -4, -1, 4, -6, -3, -1, 9, 0, 3, -5, -4, -5, 0], [1, -2, 3, -3, 6, 4, -1, 0, 2, 1, 2, 2, -4, 0, -2, -2], [2, 1, 2, 8, -6, -5, 4, -3, -3, 4, 1, -4, -2, -5, -2, 2], [-2, 4, -2, 6, -2, -7, 3, 1, -1, -2, -4, -5, 1, -6, 3, 0], [-5, 5, -7, -2, 0, -1, 4, 2, 5, -6, 1, -1, 1, 5, 4, -1], [5, -6, -1, 4, -2, -2, 3, -3, 1, 5, -2, -1, 0, -5, 3, 1], [-5, 3, 3, 1, 4, 0, -2, 3, 3, -2, 3, -4, -2, 0, 1, -1], [0, 2, -2, 2, 3, -2, -1, 0, 2, 1, 2, 2, 0, 1, 0, 1], [-8, 4, 7, -2, -1, -1, -2, -1, -2, 2, 1, -2, 1, 3, -6, 0], [-1, -2, -2, -3, -1, 2, -1, 0, -2, -4, 2, 4, 3, 7, -1, -2], [3, -1, -5, 3, 0, 0, 2, -1, 0, 1, -5, 1, 7, -3, 3, 4], [-6, 2, 1, -2, -1, 1, 2, 0, -4, 2, 1, 0, 0, 0, -3, -1], [-2, 2, 6, 2, -5, -2, 1, -2, -2, 3, 4, -4, 1, -3, -1, 3], [-4, 7, -6, -2, 6, 2, -2, 3, 3, -4, -3, 0, 4, 3, 2, -4], [0, -3, 1, -6, 6, 2, -1, -3, -1, -1, 0, 1, -1, 6, 2, -5], [2, -8, 5, -6, 0, 5, -2, -1, 1, 8, 2, 5, -6, -4, -6, 1], [-3, 9, -5, 2, -1, -8, 2, 7, 2, -10, 1, -9, 7, 2, 6, -2], [-2, 8, -6, 2, -1, -4, 1, 6, 3, -6, 0, -5, 2, -2, 6, -2], [3, 0, 0, 3, -2, 2, -4, 2, -3, 4, 1, -1, 1, 3, -2, -2], [0, 1, -3, 8, -5, 4, -1, -7, -4, 4, -5, 3, 3, 1, 2, 6], [0, 0, 0, 2, -2, -2, -2, 0, -3, 0, 2, -4, 3, 0, 1, -3], [-4, 4, -7, -3, 7, 3, -3, 0, 2, -8, 0, 2, 2, 13, 3, 0], [-7, 6, -4, -3, 5, 2, -3, 2, 2, -5, -1, 3, 3, 10, -5, -3], [-3, -3, 1, -2, -10, 3, -1, 2, -5, 4, 3, -1, 1, -4, -1, -2], [-1, -5, -3, -4, 5, 3, -3, -4, -1, 3, -3, 9, 0, 1, -4, 0], [-1, -1, 1, -2, 3, -1, -5, -3, -3, 5, -3, -3, 1, -1, -3, 0], [4, -8, 2, 3, 1, 3, 1, -8, -5, 7, -3, 9, -3, -4, 1, 4], [1, 3, -4, 4, 2, -5, 4, -1, 2, -1, -6, 1, 0, -6, 3, 1], [9, 1, -7, 7, 0, -1, 3, 2, 4, -5, 0, 0, 3, -3, 6, 3], [0, 6, -7, -4, 9, -1, 0, 5, 8, -8, 3, 0, -2, 8, 8, -2], [-1, 8, -6, 5, -5, -4, 7, 2, 0, -6, 0, -2, 7, 0, 3, 4], [1, -6, 9, 2, -4, -2, -3, -3, -3, 13, -3, 2, 1, -11, -7, 1], [-5, 6, -1, -4, 3, 2, -2, 2, 4, -5, 3, 4, -3, 3, -5, 0], [1, 1, -5, 0, 0, 5, -4, 4, -3, 0, -5, -6, 2, 1, 1, -6], [-3, -7, 7, -15, 13, 4, -7, 0, 1, -1, 7, 5, -5, 11, -5, -5], [-15, 15, -11, -3, 8, 0, 3, 8, 8, -20, 5, 1, 2, 12, 6, 0], [-3, 7, -2, 3, -1, 0, -2, -3, -3, -1, 1, -4, 9, 4, -2, 2], [3, 0, -2, -2, 3, 7, -6, 0, 0, 1, 3, 1, -4, 9, 1, -3], [-11, -5, 5, -14, 13, 8, -8, -3, 0, -1, 1, 7, -3, 9, -2, -5], [4, -5, 12, -6, 1, -4, -5, -4, 1, 8, 6, -2, -2, -5, -5, 1], [-3, 0, 5, -6, 1, 6, -5, -1, -1, 3, 3, 3, -8, 3, -9, -3], [1, -1, 5, 14, -17, -4, 1, -11, -9, 13, -4, -6, 7, -13, -5, 9], [-9, 7, -5, 0, 3, 1, 7, 3, 0, -8, -1, 0, 1, 5, 3, 0], [1, -5, 5, -8, -1, 5, -1, -5, -5, 3, 6, 3, -7, 6, -3, -3], [1, 0, -3, 8, -11, -3, 1, -3, -7, 1, -2, -1, 5, -4, 3, 2], [-1, -4, -1, -11, 8, 7, -9, 1, 4, 2, 7, 2, -4, 5, -1, -2], [-3, 9, -4, -5, -1, 8, -2, 5, 2, -11, 10, 0, 0, 11, 3, -1], [2, 0, 1, 4, -6, -2, 7, -1, -1, -1, 1, -3, 3, 2, 8, 2], [-6, -5, 7, -11, 4, 5, -7, -1, 1, 3, 3, 2, -1, 12, -3, -1], [8, 2, -4, 14, -6, -4, 8, -1, 0, -3, -10, -7, 5, -13, 2, 3], [1, -4, 2, 2, -5, -6, 3, -2, 3, 2, 1, -4, 0, -6, 4, -4], [5, -4, 8, 2, -10, 1, -1, -3, -2, 16, -2, -4, -2, -11, -6, 2], [7, 8, -4, 5, 1, -7, 3, 0, 6, 0, -2, -4, 3, -8, -1, 5], [-8, 1, -4, -7, 10, -1, -8, 3, -3, -8, 0, 2, 4, 13, 5, -2], [11, 2, -3, 5, -12, -7, 4, 6, 2, 1, 3, -6, -1, -11, 4, 2], [-1, -1, 5, -1, -6, 3, -6, 1, -2, 6, 0, -3, 0, -8, -4, 2], [10, -18, 18, -5, -3, 3, -8, -5, -8, 19, 2, 4, -10, -5, -11, -3], [7, -6, 6, -1, 4, -2, -10, 0, 2, 1, 2, -4, -2, -7, -3, 0], [8, -4, 7, 4, -10, -4, -7, -5, -6, 7, -4, -5, 5, -9, 2, 7], [-1, -2, 2, 0, -3, -2, -4, 0, 0, 1, -2, 2, 0, -2, 0, 7], [2, -5, 7, 1, -5, 2, 2, -5, -2, 8, 2, 9, -9, -6, -7, 0], [3, -12, 8, 4, -8, -5, -1, -6, -6, 14, -2, -2, -1, -11, -3, 6], [-4, 6, -7, 7, -10, -6, 4, -2, 0, -3, -2, -4, 8, 5, 1, 3], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 8, -11, 8, -1, -3, 8, 0, 4, -9, -4, 3, 8, 7, 4, 8], [12, -9, -5, 2, -3, 6, -2, -1, -1, 5, -5, 5, 4, -5, -2, 1], [-4, -1, -6, -5, -1, 2, 3, 0, 1, 6, -5, 3, 0, 7, -5, -1], [-1, 0, -2, -8, -3, 4, -5, 2, -3, -7, 1, 0, 7, 9, -1, -3], [-3, 5, -5, 4, -8, -6, 8, 7, 5, -1, -6, -5, 8, -8, 1, 2], [8, -3, 0, 11, -5, -7, 7, -1, 2, -1, -8, 1, -1, -11, 1, 4], [10, -3, -3, 0, -2, 1, 3, 4, -2, -1, 4, 7, -4, 4, -1, -3], [-5, 2, 3, -3, 9, -9, -1, 0, 4, -5, 1, -8, 6, 7, 4, -4], [3, -5, 0, -6, 7, 4, -4, 3, 0, -1, 5, 7, -11, 9, -3, -6], [-2, 1, 0, 1, -8, 1, 2, 2, -1, -6, 11, -7, -1, 7, 3, 3], [7, -5, 1, 12, -11, -3, -1, -3, -4, 4, -5, -3, 3, -14, 4, 6], [5, -7, 7, 5, -3, 5, 6, -8, -2, 9, 0, 4, -6, 5, -4, -1], [3, 0, 2, 6, -5, -2, 6, 2, -1, 1, 5, -3, -5, -12, 3, 3], [1, -10, -4, -6, 6, 11, -2, -2, 0, -2, -6, 6, -1, 9, 4, -8], [1, 0, 0, -10, 13, 5, -10, -2, 5, 6, 2, 4, -2, 4, -6, 0], [0, -10, -2, -11, 9, 8, 0, 2, 1, -1, 0, 16, -8, 10, 0, -2], [8, -9, 7, 11, -5, -5, 3, -10, -4, 11, -6, -9, 2, -15, 1, 6], [-3, -1, 7, 1, -2, 4, 0, -3, -6, 13, -2, 6, -4, 3, -8, 1], [4, 5, -12, 0, 2, 7, 3, 3, 4, -13, 9, -4, 1, 9, 9, -5], [7, 3, -8, -9, 7, 2, -12, 10, -1, -3, -5, -1, 2, 5, 0, -2], [9, -9, 10, 2, 1, -6, 2, -1, 2, 6, 3, -1, -5, -10, 1, -4], [18, 4, -3, 6, -8, -3, 0, 4, -3, 1, 1, -11, 0, -5, 6, 2], [-2, -4, 5, 4, 1, 5, -1, -6, -4, 11, -11, -2, 5, 1, -2, -1], [-7, 8, -2, 0, 3, -6, 3, 4, 2, -7, 1, -8, 2, 9, 8, -1], [4, 6, -12, -2, 5, 5, -5, 9, 3, -14, -3, -2, 4, 2, 11, -7], [3, 1, -7, 14, -3, -4, 13, 2, 6, -1, -4, 2, -1, -11, -1, 2], [1, 2, 1, 2, 1, 2, 2, -3, -1, -6, 11, -4, 0, 11, 6, -2], [6, 2, -3, -2, -6, 0, 1, 7, -2, -4, 5, -1, -2, 6, -3, 1], [-6, -1, -1, -4, 7, 3, 6, 4, 3, -15, 13, 3, 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-3, 9, -11, -1, 0, -1, -4, 1, -1, -5, 4, 8, -6], [-10, -7, 1, -18, 21, 13, -13, 7, 7, -12, 12, 3, -3, 11, 1, -6], [2, -7, -7, 20, -22, -7, 14, -2, -10, 8, -16, -6, 6, -27, 11, 5], [4, 6, -4, -5, 13, -4, -1, -5, 12, -1, 6, 5, -3, 8, 10, 6], [8, 15, -13, -1, -7, 4, 0, 18, 2, -16, 4, -16, 3, -1, 13, -5], [7, 8, -20, 28, -20, -10, 19, 4, -1, 2, -19, -11, 8, -22, 16, 8], [-24, 17, 5, -13, 8, 12, -10, 7, -4, -13, 1, -4, 9, 21, -1, -10], [-14, 14, -7, -18, -1, 9, -9, 10, -1, -20, 22, -6, 4, 25, 4, -10], [-17, 2, -16, -10, 15, 6, 5, 0, 10, -21, 3, 10, 8, 26, 15, 5], [-14, 14, 0, 0, 8, -9, -1, 4, 7, -11, 6, 1, 0, 19, 2, 5], [11, -10, -14, 7, -1, 7, 0, 3, -5, 9, -10, 10, -5, -8, -1, 2], [10, 3, -11, 0, 10, 6, -4, 15, 0, -9, 2, -1, -6, -5, 0, -3], [-16, 19, -5, -3, -2, -1, 3, 7, 5, -10, 16, -8, -1, 11, 1, 3], [-19, 14, -12, -6, -5, 5, -5, 17, 7, -14, 9, -8, 5, 6, 12, -2], [-10, -8, 9, -1, 4, -4, 1, -1, -1, -6, 8, 14, -1, 2, 1, 2], [-30, -6, 4, -14, 27, -1, -3, -9, 3, -5, -5, 4, -5, 10, -1, -14], [5, 1, 5, 21, -28, -7, 10, -8, -15, 22, -17, -11, 8, -31, -10, 7], [13, -2, 1, -8, -8, 8, -9, 2, -5, 12, 1, -3, -3, 1, -1, -1], [-17, 8, 2, -1, 19, -8, 1, -1, 12, -11, 6, -5, 3, 7, -6, -11], [7, -7, 6, -6, 0, 11, -6, -1, 3, -1, 1, 6, 3, 9, -1, -7], [-17, 5, -14, -1, 4, 4, 4, 3, -4, -4, -4, 4, 4, 6, -8, -3], [-11, 2, 5, 8, -6, -15, 0, 4, -2, -12, 1, -18, 10, -8, 23, -1], [-11, 2, -3, -3, 6, 2, -5, 4, -2, 5, -2, 8, -6, -8, -4, -3], [-4, -3, 5, 16, -13, -6, 10, -6, -6, 15, -10, -1, 4, -8, -5, 0], [-13, 0, -3, -3, 10, -1, 0, 1, -3, -1, -12, 10, 8, 9, 8, 3], [-10, -19, 22, -20, 11, 7, -17, -3, 0, 9, 20, 12, -12, 16, -19, -16], [23, -21, 14, 6, -13, -2, -14, -17, -12, 39, -12, -4, -4, -17, -12, 2], [20, -14, 5, 5, 0, -12, -3, -8, 0, 19, -18, -2, 0, -19, -7, 4], [1, 10, -10, 6, 4, 8, 9, 3, 0, 2, -14, 14, -1, -2, -3, 9], [-2, -3, 5, 5, -3, -13, -6, -1, 10, 7, -1, 4, 0, -27, -9, 9], [-15, -8, 11, -3, 6, -6, -2, -3, 10, 8, -4, 13, -12, -4, -5, 6], [1, 11, 6, 6, -7, -5, -2, 7, -1, -2, -4, -12, -5, -13, 5, -9], [12, 14, -9, 10, -8, -11, 2, 5, 7, -13, 2, -14, 8, -10, 19, 8], [-13, -12, 8, -17, 10, 13, -10, 0, 7, -9, 4, -5, -1, 16, -1, -17], [-14, -5, 14, 2, -7, 1, -6, 2, -8, 13, 7, -5, -1, 8, -12, -2], [2, -9, 0, 0, -1, 5, -13, 2, -6, 11, -4, -2, -1, -10, -3, 4], [9, -18, 11, 1, 0, 8, -7, -14, -11, 11, -2, 5, -4, 8, -1, 2], [-4, 0, -1, -13, 21, -4, -3, -9, 8, 5, -15, 17, -4, 6, -9, 1], [1, 16, -1, 3, -11, 4, 7, 14, -9, -13, 8, 0, 1, 6, 1, -3], [0, -9, 8, 4, -9, 0, 4, 9, 0, 11, -2, 1, -17, -14, 4, 1], [-4, 7, -4, 10, -3, -4, -3, 2, 4, 5, -16, 6, 10, -14, 3, 15], [4, 12, -14, 3, -8, -13, -4, 0, 8, -11, -3, -10, 8, -5, 15, 12], [14, -16, 16, 12, -18, -3, -11, 2, -18, 32, -13, -1, 0, -23, -15, -11], [10, -2, -4, 24, -13, -12, 6, -5, -4, 12, -18, -2, 18, -9, 0, 10], [-43, 6, 6, -17, 18, 1, -1, -4, 14, 4, -2, 10, -8, 4, -11, -2], [7, 10, -11, -4, -9, 4, -2, -3, 4, 8, -3, 10, -2, -8, -13, -1], [0, 22, 2, -5, 0, -5, 0, 15, 2, -1, -1, -6, 0, 9, 1, -2], [0, 4, 6, 5, -8, -21, -2, 5, -1, 1, 3, -7, 7, -12, 7, 0], [-3, 0, -7, -11, 0, 12, -12, 6, -3, -5, 0, -5, 12, 14, 1, 0], [-26, 14, -8, 3, 14, 0, 11, 3, 15, -14, -2, 7, 0, 15, 0, -4], [-6, 24, -14, 18, 3, -7, 6, 4, 5, -19, -8, -20, 20, 1, 29, -4], [5, -6, -9, -8, 10, 7, -11, -12, 9, -3, -5, 8, 1, 13, 7, -1], [12, -1, -5, 12, 5, -2, -7, 8, 0, 4, -3, 4, -6, -3, 5, -2], [23, -12, -6, 24, -16, -6, 7, -1, -8, 11, -13, -12, 12, -21, 17, 9], [-14, 2, -2, -1, 0, -16, -1, 2, 8, -2, -3, -5, 2, -1, 6, 1], [-19, 16, -5, -10, 14, 8, 13, 4, 9, -17, 6, 5, -6, 11, 3, -15], [-9, -3, 5, -3, -4, -8, 3, 6, -3, 6, 3, 6, -5, 4, -7, -11], [10, -14, 10, 21, -23, -8, 14, -19, -11, 20, -3, -3, -4, -21, -8, 14], [12, -28, 28, -1, -21, -10, -10, -13, -19, 29, 4, -16, -6, -14, -8, 2], [-3, 1, -8, -4, 0, 1, 14, 8, 2, -17, 13, 1, 2, 10, 12, -1], [12, 0, -5, 17, -9, -18, 2, 3, 1, 4, -7, -11, 23, -13, 8, 2], [6, -1, -12, 16, -17, -7, 19, 10, -2, -14, -2, -6, 8, -9, 14, 0], [-12, 11, -23, 1, 28, -10, -2, 0, 15, -16, -4, 5, 9, 17, 14, 3], [-23, -4, 14, -9, -8, -7, -2, -9, -4, 8, 13, -16, 0, 3, -2, -13], [-12, -2, -7, -12, 1, 1, -9, 6, -4, -1, -9, 8, 3, -3, -8, -5], [-10, 9, 4, 16, -10, -14, 6, -2, 2, 6, -12, -8, 2, -26, 2, 4], [-25, 18, 3, -10, 11, 1, 9, 2, 2, -22, 7, -6, 8, 22, 6, -12], [18, -5, 14, 14, -13, -14, -5, -8, -8, 28, -16, -19, 9, -32, -9, 3], [12, 2, -5, 7, -5, -11, 10, 11, 10, -12, 8, -7, 5, 1, 1, 6], [0, 5, 4, 21, -12, 2, 9, -11, -15, 23, -8, 6, -9, -18, -14, 10], [-5, -11, 7, 1, -8, -1, 1, 10, 5, 7, 4, -9, 0, -6, 2, -8], [-4, 19, -16, -2, 26, 6, -7, 10, 11, -22, -5, 3, -3, 21, 13, -6], [-6, -8, -2, -15, 7, 13, -1, 5, 10, -13, 9, 13, -8, 9, 5, -3], [-2, -7, 12, 9, -11, 6, 4, -22, 0, 22, 10, 6, -9, -7, -15, 9], [3, -24, 0, -13, 6, 6, -5, -4, -6, 7, 8, 19, -13, 5, -11, -3], [11, 6, -9, -5, 17, 4, -5, 4, 9, -20, 6, 10, 2, 11, 8, 6], [3, 12, -20, -1, 7, -17, 0, 7, 9, -10, 4, -10, 5, -9, 7, 2], [-19, 9, -20, -2, 0, 11, 20, 9, 13, -15, 6, 8, 0, 14, 8, -7], [-22, 14, 3, -15, 13, 17, -3, 4, -3, -12, 5, 5, 9, 32, -2, -1], [28, -17, 16, 2, -9, -11, -19, -12, -10, 35, 2, 5, -8, -15, -12, 10], [-10, 3, 6, 13, 10, -10, 24, -5, 9, -10, -2, -7, -10, -6, 16, 3], [-14, 14, -7, -3, -8, -12, 6, 14, 10, -10, 4, -6, 7, 4, 2, -8], [21, -4, 5, 8, -2, 2, -14, -6, -5, 6, -8, 3, 3, -5, -3, -2], [35, 12, -15, 7, -4, -7, 5, 12, 8, -13, 5, -17, 6, -10, 17, 3], [-11, 29, -40, 8, 16, -3, 25, 17, 25, -51, 0, 2, 11, 23, 25, 2], [-11, 25, -10, 3, 10, -1, 17, 3, 13, -17, 1, 7, 7, 15, 5, 8], [5, 11, -4, 13, -15, -8, 6, 8, -4, -2, -9, -13, 12, -4, -1, -1], [-4, 12, -8, 15, -14, 1, 4, 1, -6, 2, -14, -1, 0, -18, 2, 12], [8, -13, 8, -9, -4, -5, -12, -1, 0, 20, -1, 11, -1, -16, -12, 6], [6, -17, 19, 1, 1, 12, 5, -21, -2, 30, -4, 31, -32, -15, -19, 13], [-3, 7, 0, 9, -16, 7, 9, -3, -6, 11, 1, 1, -6, -14, -8, 12], [4, 23, -19, 12, -5, -8, 6, 8, 8, -24, -3, 1, 17, 5, 19, 8], [-3, -5, -10, -26, 27, 27, -14, 4, 6, -22, 9, 12, -5, 35, 4, -7], [-4, -11, 1, -1, -6, 11, 11, -4, -7, 10, -8, 1, -6, 13, -8, 1], [13, -11, -22, 7, -8, 8, 3, -5, 4, 6, -10, 11, 12, -6, 10, 9], [-16, 10, -10, 16, -3, 4, 24, -7, 10, -7, -10, 9, -3, -14, 8, 13], [23, -16, 36, 1, -15, 6, -4, -13, -17, 31, -9, 14, -19, -19, -20, 13], [12, 6, 2, 5, 0, 14, 7, -9, -1, 7, -4, 13, -8, -2, -4, 12], [8, -21, 18, -1, 3, 4, 7, -4, -6, 15, 2, 15, -14, -10, -16, -4], [12, 25, -34, 33, -10, -4, 22, -1, 0, -14, -21, -2, 21, -2, 9, 11], [-9, 5, -8, -19, 6, 15, -12, 17, 5, -14, 5, 14, 0, 12, -11, -17], [3, -6, -1, 1, 1, 5, -7, 2, 2, 6, 1, 14, 0, -10, -1, 8], [-14, -2, 12, -14, 12, 7, 2, 6, -2, -22, 22, -14, -2, 16, 5, -10], [-1, 16, -18, 4, 5, -1, 5, 12, 2, -5, -24, 11, 14, 4, 6, -5], [15, 6, -9, 27, -32, -11, 11, 16, -21, 1, -4, -28, 14, -25, 13, 4], [-22, 1, 4, -25, 7, 4, -13, 0, 0, -8, 14, -4, 7, 26, 5, -7], [2, -15, -1, -17, 12, 7, -14, 6, 0, -7, 12, 2, -7, 18, -11, -14], [0, 5, -24, -8, 8, 10, -1, 16, 5, -35, -7, -10, 19, 20, 14, -19], [-8, -6, -2, -40, 46, 23, -11, 13, 12, -36, 16, 29, -11, 38, 1, -15], [7, -11, 1, -11, 13, 10, -6, -4, 3, 5, 1, 11, -5, 6, 5, -3], [-37, 7, -2, -17, 20, 4, -10, 22, 8, -32, 15, -15, -1, 24, 14, -7], [24, 12, -11, 19, -15, -16, 1, 15, 0, -1, -16, -30, 7, -26, 13, 0], [13, -33, 24, -11, 2, 3, -17, -3, -5, 24, 9, 6, -11, 3, -13, -13], [10, -5, 6, 9, -11, -4, 2, -9, -6, 18, -10, 6, 0, -2, -20, 3], [23, -11, 7, 26, -35, -7, 6, -14, -14, 31, -11, -3, 2, -15, -10, 8], [10, -10, -6, 17, -10, -10, 7, 0, 0, 11, -7, -21, 8, -10, 10, 2], [8, -3, -26, 15, 2, -3, 13, 6, 6, -3, -14, -1, -2, -12, 19, 6], [16, -11, 5, -23, 8, 18, -13, 1, 7, 13, 2, 14, -15, -2, -11, -7], [13, -6, 5, 9, -6, -1, 13, 1, 7, 18, -7, -3, -5, -29, 0, 5], [-15, -3, -1, 5, 2, 5, 23, 6, 3, -12, 10, 4, -15, 5, 2, -1], [14, -9, 9, -3, 1, -4, 2, -3, -3, -13, 8, -9, 7, 14, 4, -4], [-4, -9, 3, -40, 33, 27, -24, 20, 3, -30, 26, 5, -11, 35, -3, -26], [13, -9, -10, -4, -1, -5, 4, 3, 1, -1, -5, 3, 6, 3, 3, -9], [-16, 7, 7, 14, -13, 1, 3, -2, -3, 3, -3, 1, 11, -9, -10, 4], [-2, -17, 18, 8, -4, -1, -12, -11, -9, 15, -2, -4, -1, -10, 5, 5], [7, 9, -5, 6, 7, 1, 6, -16, 12, -21, 0, -5, 5, 12, 16, 3], [-21, 1, -4, 13, 1, -9, 16, -9, -1, -8, -2, 2, -2, 8, 10, 10], [-3, 7, -11, 11, 12, -17, 10, -9, 13, -24, 1, -7, 2, 13, 15, 5], [1, -6, -2, 10, -17, -2, -5, -1, -3, 18, -1, 4, -1, -10, -8, -7], [-5, -3, 12, 3, 4, -18, 12, -1, 4, 12, -7, 5, -6, -26, -5, 8], [-8, 8, 0, 9, 2, -5, 7, -5, 7, 4, -1, 3, -15, 5, -5, 11], [21, -21, 21, 11, 1, -7, 2, -4, -10, 11, -4, -10, -13, -17, -9, -1], [15, -18, 1, 3, -5, -3, 0, 9, -4, 1, 6, -2, -4, -5, 11, 0], [-18, -4, -5, -9, 1, -2, -10, 2, 0, -12, 6, -14, 8, 18, 15, 0], [-5, 0, 2, 14, -8, -20, 14, -2, -2, -7, 2, -8, -1, -4, 11, 6], [-20, 14, -8, -17, 17, 2, -17, 10, 11, -11, 11, 15, 7, 22, -3, -8], [5, -22, 9, 3, -5, 1, 2, -12, -6, 32, -1, 9, -17, -10, -29, 2], [-9, 13, 1, -5, 13, 4, 5, 10, 10, -8, 10, 4, -12, 16, -13, -4], [-16, 2, 12, -3, 27, 1, 12, 0, 11, -9, 4, 0, -20, 7, -2, -8], [11, -29, 5, -4, 9, 9, 6, -2, 1, 5, -7, 10, -14, -5, -13, -4], [-21, 16, -2, -7, 22, 1, 5, 12, 8, -30, 14, 5, -5, 20, 7, -4], [-6, 0, -2, -16, 29, 15, 6, 1, 7, -20, 20, 19, -15, 30, 3, -7], [20, -24, 10, 0, -9, -3, 7, -8, -6, 17, 0, 9, -19, -15, -10, -5], [-30, -14, 18, -36, 25, 9, -10, 7, 4, -4, 20, 4, -16, 20, -14, -16], [6, -10, 9, 17, -12, -2, 2, -2, -10, 18, -12, 7, 2, -17, -13, 4], [1, 10, -4, 21, -14, -4, 19, -14, -5, 11, -3, 0, 12, 2, -6, 10], [-35, 3, -4, -10, 9, 14, 2, -8, 15, 0, 2, 1, -4, 9, -3, -3], [23, -14, 19, 24, -33, -9, -2, -3, -20, 33, -3, -18, -1, -44, -22, 10], [-11, 3, 1, 13, 10, -8, 5, -12, 5, 0, -10, 3, 12, -2, -6, 4], [10, -10, 3, 5, 8, -15, 1, -1, 2, -8, -6, -10, 4, -13, 4, 0], [-7, 25, -28, 11, 9, 0, 15, 1, 17, -25, -9, 7, 14, 6, 8, 17], [-20, 6, -7, -5, -5, 13, 1, 10, 4, -4, 14, -1, 1, 17, -4, 1], [-6, -17, 10, -9, 4, -14, -17, -6, 0, 21, 0, 14, -12, -11, -13, 1], [-24, 29, -29, 6, -5, 1, 23, 3, 0, -21, -4, 9, 17, 8, 9, 9], [26, -6, 6, 12, -16, 2, -8, -4, -13, 33, -12, 9, -8, -21, -17, 15], [5, -25, 8, -4, -1, 7, -5, -17, -8, 18, 10, 10, -12, 5, -18, 7], [6, -3, -2, -6, 13, 18, -11, -12, -3, -7, 7, 2, 3, 33, 4, 4], [-4, -3, 19, -3, -19, -2, 5, 3, -11, -8, 12, -10, 3, 0, -7, -9], [8, 24, -5, 18, -10, -9, 11, -4, 5, 3, -12, -5, 23, -11, -3, 10], [6, 6, -7, 12, -7, -15, 15, 8, -4, 5, 1, 0, 3, 2, 7, 0], [16, 4, 4, -12, -2, 9, -12, 13, -12, -4, 14, -13, 3, 10, 6, -6], [-15, 8, -22, 6, 6, -10, 13, 18, 17, -33, 2, -8, 7, 8, 32, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6010_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6010_2_a_c();".
// 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_6010_2_a_c(:prec:=16)
    chi := MakeCharacter_6010_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_6010_2_a_c();".
// 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_6010_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6010_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-19, 38, 268, -370, -1132, 979, 2183, -864, -2142, 71, 1015, 206, -204, -79, 9, 8, 1]>],Snew);
    return Vf;
end function;