
// Make newform 539.2.q.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_539_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_539_q_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_539_2_q_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_539_2_q_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 := [625, 625, 375, 500, -300, -1375, 390, 240, 251, -156, 10, -29, 4, 16, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-81680866488250, -105743499205250, -162385826946000, 716191995079275, -161148237103450, -30916376621980, -128695406473289, 62191980906094, -4306261565400, 13772300331741, 1387553663334, -6934783544984, 2023744375705, 595438889614, -390646149564, 10747578565], [153447866875, 187325355750, -107615676875, 118432915400, -96741534250, -564367165, -45437262620, 37111437582, -7524269367, 1405593120, 496882252, -3522648197, 547473112, 202974090, -216782207, -11294463], [-11984339635250, -16183969928000, -23716493712625, 114842922010965, -21982568599150, -1437348812935, -15280985144283, 8449126663418, -1350437679285, 1507562081227, 137356700948, -847043836223, 277539827085, 57359873198, -50549404333, 4442741005], [-322941345212125, -381835489799000, 269200498832200, -228790206715425, 230037797816170, -33669067543990, 79980076478579, -76629971948588, 18065368943364, -2364013856961, -1497683300668, 4758645678158, -1669199906909, -149323399494, 350661785078, -45506971144], [-334386818347500, 167804604156125, 226617505715125, -283504981985225, 182426884437775, -41006490690815, 89320024758233, -69457776599693, 25735467004125, -6612074340177, -151223147723, 5145355461023, -2333144026885, -21593146433, 367332418783, -83632252305], [-20998151463750, -82629449657375, 622151968595825, -138675258120350, -139014480052425, -39013786059435, 65255565416770, 17312365591157, -7418645458657, 1645530675560, -7015585976493, 1771198003753, 1820794839302, -732120607075, -73661012057, 95874717082], [38634849837750, 55999400861125, 75235145215125, -139348684847645, 59956356442775, -14818453535215, 27509782939463, -22811269611173, 8826339491805, -1843743060247, -24649968003, 1643333019103, -767290363435, 4417433767, 119405641263, -30198383055], [-221616106153750, 14502483941500, -11405350455625, 404789439834025, -57339484155250, -93373444817095, -121627565899485, 23154598502726, 21076024732419, -1747566143665, 2032193985536, -5399279631671, -1104221745559, 919001199770, -220169951101, -154162898659], [288196456635125, -7878555726875, 27853396132625, -510650758411475, 82238742955775, 113501533227125, 137833352978403, -32955100495645, -24623976535863, -4606969356647, -2533300411795, 7017151182115, 737207740433, -1142785999873, 295656442235, 169638367088], [224933071900000, 156264822677125, 499015951885625, -81055574547325, -458338835044675, -94501899542125, 55780375629075, 101300812445949, -25673900926259, 5941055817805, -13824006840431, -2426784958929, 4731361480299, -1156604341685, -541590004299, 176090223129], [-236118590095250, -121564313236625, -582082324757025, 163715215109050, 580928878355345, -16660884340455, -108255183265766, -110077052963765, 57062946239641, -5578051683996, 15682266957205, -314121333825, -6592373517306, 1993098800331, 508748668505, -354585769846], [-6717236603125, -88398103091375, -109773841167125, -167759616228500, 719416268648775, -146370316576575, -35107932262330, -131274825328889, 59494338686279, -2629639309260, 13664824546091, 1699233441719, -6977773859244, 1851783118665, 649176782439, -379898570999], [-228316588875, -199684397000, -4325378100, -23086558150, 337020939960, -141188731550, -21812543298, -69148932704, 39706180692, -1958283618, 3597332826, -434593236, -3725291302, 1010800728, 234222124, -245516587], [516245422374375, 408665383985875, 499909686111375, -56727128738700, -894721324555875, 138739310527935, 86069405698760, 190479026663079, -76777990078999, 8518785412090, -18708691847541, -2121604357809, 9563407738064, -2475405530945, -864985162329, 493169051379]];
    Rf_basisdens := [1, 1, 451236007807750, 262194077750, 90247201561550, 451236007807750, 451236007807750, 451236007807750, 90247201561550, 451236007807750, 451236007807750, 451236007807750, 451236007807750, 451236007807750, 262194077750, 451236007807750];
    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_539_q();"
function MakeCharacter_539_q()
    N := 539;
    order := 15;
    char_gens := [199, 442];
    v := [10, 6];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_539_q_Hecke();"
function MakeCharacter_539_q_Hecke(Kf)
    N := 539;
    order := 15;
    char_gens := [199, 442];
    char_values := [[-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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 2, -1, 0, 0, -2, 2, -1, 0, 2, 0, 0, 1, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 2, 0, -1, 0, 0, 0, 2, -1, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, -1, 0, 1, 2, -1, 0, -1], [2, -1, -2, 1, 0, 1, 0, -1, 1, 0, -1, 0, 1, -2, 1, -1], [3, 2, 0, 0, 0, 0, 3, -3, 0, 0, -3, 0, -3, -2, 0, 0], [0, 3, -3, -3, -3, 0, 3, 0, 0, 3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, -1, -1, 0, -2, -1], [0, -2, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0], [-5, 2, 5, -2, 0, 0, 0, 0, -2, 0, 1, 0, 0, 0, 0, 0], [5, 0, 0, 0, 0, 0, 6, 0, 0, 0, -6, 0, 0, 0, 0, 0], [-2, 4, 2, -4, 0, 4, 0, -4, -4, 0, -4, 0, -6, -1, -6, 4], [0, 0, 0, -3, 0, 2, 0, 0, 0, 0, 0, -4, -2, 0, -2, 3], [0, 0, 0, 1, 0, 4, 0, 3, 0, 1, 0, 1, 0, 1, 3, 0], [-2, -1, 3, 5, 1, 1, -3, 2, 5, 0, 2, 1, -1, -5, 0, -5], [0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, -1, 1, -1, -4, 1], [1, -3, -3, 3, 3, 0, 3, 0, 2, 2, -1, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 2, 0, 2, 0, 5, 0, 5, 0, 5, 2, 0], [0, 0, 3, -2, 1, 0, 0, 0, -2, 0, 0, 1, -3, 0, 0, 2], [1, 5, 1, -5, -5, 0, -1, 0, -3, -3, -1, 0, 0, 0, 0, 0], [5, -6, 0, 1, 1, 0, -2, 2, 1, 0, 2, 1, 2, 5, 7, -1], [7, 0, -1, 0, 0, 0, 7, 0, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 0, 4, 0, 0, 0, 0, 0, -4, -1, 0, -4, -3], [0, 11, -6, 0, -4, 2, 4, -6, 0, 0, -6, -4, 0, -4, -6, 0], [1, 1, -1, -1, 0, 1, 0, -1, -1, 0, -1, 0, 0, 5, 0, 1], [0, 0, 0, -3, 0, 0, 0, 4, 0, 3, 0, -2, 4, 3, 8, 2], [3, 2, -3, -2, 0, 0, 0, 0, -2, -4, 13, 0, 0, 0, 0, 0], [2, -3, 8, 3, 3, 0, -8, 0, -2, -2, -2, 0, 0, 0, 0, 0], [6, -2, 0, 0, 0, 0, 1, -1, 0, 0, -1, 0, -1, 2, 5, 0], [0, 0, 0, 2, 0, 8, 0, -14, 0, -2, 0, 0, -8, -2, 0, -2], [0, -1, -3, 1, -4, 0, -4, 0, 0, 4, -3, 0, 0, 0, 0, 0], [2, 0, 7, -3, 0, 9, -7, -2, -3, 0, -2, 0, -9, 3, 0, 3], [0, 0, 0, 0, 0, 7, 0, -7, 0, -3, 0, 0, -11, -3, -11, 0], [0, 0, 0, 0, 0, -2, 0, 2, 0, -2, 0, 0, -5, -2, -5, -5], [-3, 0, 6, -2, -3, 3, -3, 0, -2, 0, 0, -3, -6, 0, -3, 2], [0, 0, 10, 0, 6, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 4, 0, 0, 11, 0, -11, 0, 0, -11, 0, -15, -4, -15, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 2, 6, 6], [-4, -2, 1, 2, 2, 0, -1, 0, -4, -4, 4, 0, 0, 0, 0, 0], [0, -5, -2, 0, -1, -1, 3, -2, 0, 0, -2, -1, 0, -1, -2, 0], [0, 0, 0, 7, 0, -3, 0, 5, 0, -4, 0, -3, 3, -4, 0, -4], [1, 6, 0, -6, -1, 0, 3, 0, -1, 5, -3, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 2, 0, 2, 0, -1, 2, 2, 18, 1], [-4, 0, -1, 0, 1, 0, -4, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 5, 0, 0, 0, 0, 5, 6, 3, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, 9, 0, -12, 0, -3, 0, -3, 0, -3, -12, 0], [5, 0, -2, -3, -4, -5, 5, 0, -3, 0, 0, -4, 2, 0, 5, 3], [2, 0, -9, -1, -2, -2, 2, 0, -1, 0, 0, -2, 9, 0, 2, 1], [1, -7, -1, 7, 0, 0, 0, 0, 7, -3, -3, 0, 0, 0, 0, 0], [0, 0, 0, 4, 0, 0, 0, 4, 0, -4, 0, 4, 4, -4, 12, -4], [-6, 0, -5, 0, -2, 0, -6, 0, -11, 0, 0, 0, 0, 0, 0, 0], [0, -5, 9, 0, 6, -16, 7, 9, 0, 0, 9, 6, 0, 6, 9, 0], [0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, -10, 0, -2, 0], [0, 0, 0, -5, 0, 13, 0, -14, 0, 1, 0, 4, -13, 1, 0, 1], [0, 8, -8, 0, -6, 0, 8, -8, 0, 0, -8, -6, 0, -6, -8, 0], [0, 0, 0, 6, 0, -12, 0, 0, 0, 0, 0, 5, 13, 0, 12, -6], [-1, 2, 11, -2, -2, 0, -11, 0, -1, -1, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, -3, 0, -2, 0, 0, 5, -2, 5, 4], [0, 1, -1, -1, -3, 0, 4, 0, 0, 3, -1, 0, 0, 0, 0, 0], [12, -1, 0, 1, 5, 0, 14, 0, 5, 4, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 5, 0, -5, 0, 0, -9, -5, -9, 3], [-7, -6, 14, 8, 6, 7, -14, 7, 8, 0, 7, 6, -7, -8, 0, -8], [-4, 0, -10, -7, 3, 4, -4, 0, -7, 0, 0, 3, 10, 0, -4, 7], [3, -17, 0, 9, 9, 0, -4, 4, 9, 0, 4, 9, 4, 8, 7, -9], [0, 0, -2, 0, 4, 0, 0, 0, 0, -4, -2, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 1, 0, 16, 0, 5, 0, 0, -1, 5, 0, 5], [3, -1, -3, 1, 0, 0, 0, 0, 1, 2, -24, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 1, -6, 0, -18, -1], [0, 2, 7, 0, 0, 0, -7, 7, 0, 0, 7, 0, 0, 0, 7, 0], [0, 0, 0, 8, 0, 10, 0, -4, 0, -15, 0, -15, 0, -15, -4, 0], [15, -1, 0, -2, -2, 0, 8, -8, -2, 0, -8, -2, -8, 3, 7, 2], [-6, -5, 3, 5, 5, 0, -3, 0, 4, 4, 6, 0, 0, 0, 0, 0], [-12, 0, -7, 2, 1, 12, -12, 0, 2, 0, 0, 1, 7, 0, -12, -2], [0, 0, 0, -13, 0, -12, 0, 22, 0, 10, 0, 10, 0, 10, 22, 0], [13, -4, 0, 3, 3, 0, 10, -10, 3, 0, -10, 3, -10, 1, 3, -3], [-9, 2, 5, -2, -2, -4, -5, 9, -2, 0, 9, -2, 4, 2, 0, 2], [0, 0, 0, -1, 0, -13, 0, 0, 0, 0, 0, 2, -12, 0, 13, 1], [14, -15, 0, 15, 12, 0, -1, 0, 12, -3, 1, 0, 0, 0, 0, 0], [-2, 9, 2, -9, 0, 0, 0, 0, -9, -3, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 8, 0, 0, 0, 0, 0, 3, 16, 0, -8, 4], [0, 18, -2, -18, -8, 0, 10, 0, 0, 8, -2, 0, 0, 0, 0, 0], [-9, 7, 0, -6, -6, 0, -18, 18, -6, 0, 18, -6, 18, -1, 9, 6], [5, -1, -5, 1, 0, -14, 0, 14, 1, 0, 14, 0, 19, 5, 19, -1], [0, 0, 30, 0, 4, 0, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0], [11, 12, -5, 4, -12, 6, 5, -11, 4, 0, -11, -12, -6, -4, 0, -4], [-10, -12, 0, 12, 2, 0, -5, 0, 2, -10, 5, 0, 0, 0, 0, 0], [5, -7, 0, 7, 10, 0, -3, 0, 10, 3, 3, 0, 0, 0, 0, 0], [9, -2, 9, 8, 2, 18, -9, -9, 8, 0, -9, 2, -18, -8, 0, -8], [0, 0, 0, 4, 0, -11, 0, -1, 0, 2, 0, -6, 11, 2, 0, 2], [0, 0, 0, 0, 0, 7, 0, -7, 0, -4, 0, 0, -2, -4, -2, -13], [0, -13, -3, 13, 12, 0, -8, 0, 0, -12, -3, 0, 0, 0, 0, 0], [0, 11, -11, 0, -2, 2, 9, -11, 0, 0, -11, -2, 0, -2, -11, 0], [1, -2, -1, 2, 0, 0, 0, 0, 2, -5, 18, 0, 0, 0, 0, 0], [13, -15, -13, 15, 0, 1, 0, -1, 15, 0, -1, 0, 12, -1, 12, -15], [0, 0, 0, -4, 0, 20, 0, 3, 0, -4, 0, -4, 0, -4, 3, 0], [-3, 9, -8, -9, -9, -11, 8, 3, -9, 0, 3, -9, 11, 9, 0, 9], [-4, 0, 11, 7, 8, 4, -4, 0, 7, 0, 0, 8, -11, 0, -4, -7], [-12, -6, 12, 6, 0, 0, 0, 0, 6, -1, 24, 0, 0, 0, 0, 0], [0, 0, 0, -10, 0, 3, 0, -5, 0, -4, 0, -4, 0, -4, -5, 0], [0, 0, 0, -8, 0, 8, 0, 0, 0, 0, 0, 8, -1, 0, -8, 8], [-19, 6, 19, -6, 0, -17, 0, 17, -6, 0, 17, 0, -2, 1, -2, 6], [0, 0, 0, 9, 0, 0, 0, -11, 0, -9, 0, 2, -11, -9, -3, -2], [0, 0, 0, 10, 0, -4, 0, -1, 0, -1, 0, -9, 4, -1, 0, -1], [0, 0, -5, 0, -1, 0, -2, 0, 0, 1, -5, 0, 0, 0, 0, 0], [13, -11, 0, 10, 10, 0, -5, 5, 10, 0, 5, 10, 5, 1, 18, -10], [0, 0, 0, 8, 0, -2, 0, 21, 0, 4, 0, -12, 2, 4, 0, 4], [0, 11, -5, -11, -4, 0, 32, 0, 0, 4, -5, 0, 0, 0, 0, 0], [6, 0, 17, -9, 13, -6, 6, 0, -9, 0, 0, 13, -17, 0, 6, 9], [0, 0, 0, 0, 0, 3, 0, -3, 0, 0, 0, 0, 6, 0, 6, -10], [-21, 3, 0, -3, -10, 0, -4, 0, -10, -7, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 3, 0, 13, 0, 0, 14, 13, 14, -10], [0, 4, 7, -4, 7, 0, 11, 0, 0, -7, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -14, 0, 14, 0, -4, 0, 0, 12, -4, 12, 1], [-1, -4, 2, 4, 4, 0, -2, 0, 9, 9, 1, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -16, 0, 0, 0, 0, 0, 7, 11, 0, 16, -2], [0, 8, -3, 0, -8, -2, 5, -3, 0, 0, -3, -8, 0, -8, -3, 0], [-7, -7, 0, 7, -5, 0, -12, 0, -5, -12, 12, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 0, -5, 0, 6, 0, -16, -5, 6, -13, 16], [-6, 0, -3, 0, -10, 0, -6, 0, 4, 0, 0, 0, 0, 0, 0, 0], [-27, -1, 12, -7, 1, -15, -12, 27, -7, 0, 27, 1, 15, 7, 0, 7], [0, 0, 0, -10, 0, 0, 0, -7, 0, 10, 0, -11, -7, 10, -15, 11], [-22, 0, 10, 0, -1, 22, -22, 0, 0, 0, 0, -1, -10, 0, -22, 0], [33, 2, -33, -2, 0, 0, 0, 0, -2, 2, -17, 0, 0, 0, 0, 0], [17, 1, -20, -11, -1, -3, 20, -17, -11, 0, -17, -1, 3, 11, 0, 11], [0, -11, 8, 0, 7, 20, -28, 8, 0, 0, 8, 7, 0, 7, 8, 0], [12, -4, 0, 4, 0, 0, -7, 0, 0, -4, 7, 0, 0, 0, 0, 0], [0, 1, 13, 0, 0, -14, 1, 13, 0, 0, 13, 0, 0, 0, 13, 0], [0, 0, 0, 4, 0, -14, 0, 0, 0, 0, 0, 3, -2, 0, 14, -4], [-5, -3, 19, 3, 3, 0, -19, 0, 8, 8, 5, 0, 0, 0, 0, 0], [-14, 9, 14, -9, 0, -7, 0, 7, -9, 0, 7, 0, -7, -10, -7, 9], [-9, 0, -23, 0, -11, 0, -9, 0, 7, 0, 0, 0, 0, 0, 0, 0], [10, 0, -17, -5, 6, -10, 10, 0, -5, 0, 0, 6, 17, 0, 10, 5], [5, 7, 0, -7, 11, 0, -4, 0, 11, 18, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, 0, -11, 0, -5, 0, 0, -4, -5, -4, -3], [0, 0, 0, 0, 0, 1, 0, 28, 0, 1, 0, -1, -1, 1, 0, 1], [10, -3, 8, 3, 3, 0, -8, 0, -12, -12, -10, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, -31, 0, 18, 0, 2, 0, 1, 31, 2, 0, 2], [-2, 7, 2, -7, 0, 0, 0, 0, -7, 9, 18, 0, 0, 0, 0, 0], [0, -22, -12, 0, 8, 27, -15, -12, 0, 0, -12, 8, 0, 8, -12, 0], [0, 0, 0, -1, 0, -20, 0, 0, 0, 0, 0, -2, 14, 0, 20, 1], [4, 0, 11, 0, 3, 0, 4, 0, 10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -8, 0, -18, 0, -10, 0, 0, 0, 0, 0, 0, -10, 0], [-4, 11, 4, -11, 0, 0, 0, 0, -11, 11, -7, 0, 0, 0, 0, 0], [-2, 9, -4, -9, -9, 0, 4, 0, -5, -5, 2, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 9, 0, 9, 0, 7, 9, 9, 3, -7], [16, 2, 0, 0, 0, 0, 34, -34, 0, 0, -34, 0, -34, -2, -18, 0], [-24, 3, 23, -7, -3, -1, -23, 24, -7, 0, 24, -3, 1, 7, 0, 7], [-7, -13, 7, 13, 0, -28, 0, 28, 13, 0, 28, 0, 21, 1, 21, -13], [-17, 11, 0, -11, -1, 0, 4, 0, -1, 10, -4, 0, 0, 0, 0, 0], [-5, 6, 5, -6, 0, 0, 0, 0, -6, 19, -23, 0, 0, 0, 0, 0], [0, -3, 5, 0, -2, 2, -7, 5, 0, 0, 5, -2, 0, -2, 5, 0], [0, 0, 0, -21, 0, 9, 0, 18, 0, 4, 0, 17, -9, 4, 0, 4], [16, 0, -9, 0, 6, 0, 16, 0, -6, 0, 0, 0, 0, 0, 0, 0], [4, -9, 9, -1, 9, 13, -9, -4, -1, 0, -4, 9, -13, 1, 0, 1], [-16, 15, 12, -9, -15, -4, -12, 16, -9, 0, 16, -15, 4, 9, 0, 9], [-18, 0, -5, 0, -3, 0, -18, 0, 10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, -2, 0, 17, 0, 18, 0, -15, 2, 18, 0, 18], [-9, 12, 0, -16, -16, 0, -15, 15, -16, 0, 15, -16, 15, 4, 6, 16], [0, -22, -9, 22, 4, 0, -4, 0, 0, -4, -9, 0, 0, 0, 0, 0], [-15, 13, 0, -13, -4, 0, 10, 0, -4, 9, -10, 0, 0, 0, 0, 0], [-7, 5, 7, -5, 0, -20, 0, 20, -5, 0, 20, 0, 13, 0, 13, 5], [0, 0, 0, 2, 0, 9, 0, 0, 0, 0, 0, 8, -10, 0, -9, -2], [0, 0, 0, -9, 0, -2, 0, 1, 0, 20, 0, 20, 0, 20, 1, 0], [0, 0, 0, 6, 0, 0, 0, -21, 0, -6, 0, -18, -21, -6, -16, 18], [0, 0, 0, -15, 0, -30, 0, 35, 0, 14, 0, 14, 0, 14, 35, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_539_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_539_2_q_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_539_2_q_c(:prec:=16)
    chi := MakeCharacter_539_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_539_2_q_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_539_2_q_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_539_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![625, 0, -875, -1250, 825, 850, 115, 445, 221, -101, 80, 56, -16, 1, 0, -1, 1]>,<3,R![1, -3, 5, -8, 9, -2, 0, -2, 1]>],Snew);
    return Vf;
end function;