
// Make newform 4235.2.a.bp in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4235_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_4235_2_a_bp();".
// 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_4235_2_a_bp();".
// 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 := [-176, -16, 3048, 3146, -9817, -10744, 14618, 14224, -12411, -9228, 6290, 3158, -1874, -580, 317, 54, -28, -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, 0], [0, 1, 0, 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, 0], [0, -6, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [168084952, -216452570, -2761902073, 351411224, 9357207550, -1022378476, -13389191277, 2576263422, 9309964908, -2433071454, -3339503298, 1019559172, 632416299, -210456368, -60082628, 21055270, 2255867, -816960], [-323928792, 411389508, 5031635418, -1056764613, -16449014832, 2734933636, 22779563346, -5105784911, -15476616276, 4312537074, 5471648118, -1724705334, -1026800452, 347707617, 96973182, -34314866, -3626264, 1319945], [298271312, -434813104, -4677606553, 1297181256, 15830872676, -3168374124, -22610713437, 5576925138, 15678706052, -4629724958, -5612196832, 1841735098, 1061364933, -370573528, -100751078, 36540544, 3780947, -1404970], [817140288, -1016134956, -13004973606, 2211117527, 43073936872, -5863959586, -60304065576, 12287191969, 41268166936, -10911924870, -14653546126, 4462271110, 2757128768, -910154271, -260823938, 90446516, 9764098, -3494855], [-881675160, 1156444596, 13693983540, -3238435689, -45018627968, 8249862870, 62570241838, -14731770147, -42593425200, 12210250374, 15074037090, -4844609962, -2830400454, 972933489, 267404218, -95815724, -10001908, 3681009], [487365336, -588831410, -7708927626, 1279478681, 25242671100, -3380420346, -35063523236, 7093748799, 23887170908, -6304189942, -8460573146, 2578526366, 1589551438, -525947833, -150234226, 52264236, 5620788, -2019375], [1262317704, -1621011268, -20135817862, 3323305787, 67202957572, -8744938822, -94914129924, 19126905513, 65367889636, -17292234094, -23305214606, 7125121230, 4396529884, -1458787503, -416660350, 145277224, 15618254, -5620963], [-876816348, 1074734650, 13732699798, -2767323217, -45188553084, 7382847570, 62927500708, -14066376001, -42917345898, 11969811462, 15209105388, -4802559124, -2858362608, 969897915, 270214862, -95822274, -10111498, 3688663], [2180404840, -2863714444, -33858344916, 7957130171, 111253613000, -20214537106, -154590680766, 36185979109, 105228170632, -30038423490, -37242277318, 11927306270, 6993458170, -2396335763, -660782758, 236054868, 24718536, -9070279], [-1411206752, 1751862786, 22066138029, -4435335212, -72788048540, 11704825162, 101585091069, -22470072148, -69371525690, 19218362010, 24601248878, -7731294326, -4625326767, 1563734178, 437355758, -154638528, -16368237, 5956650], [-2766807104, 3565229508, 43607448488, -9321097623, -144426557712, 24227665706, 202051896914, -45626346569, -138175741352, 38754588478, 49043404274, -15547648298, -9225906882, 3140449751, 872715794, -310321848, -32671464, 11947639], [1807951924, -2235461278, -28219452272, 5509444921, 92821313642, -14560176524, -129311316056, 28279515887, 88216405924, -24302139180, -31267758290, 9795396730, 5877133890, -1983049725, -555649790, 196202226, 20794276, -7559885], [-4151506792, 5224692996, 65121956186, -14049552159, -215289093124, 37003951374, 300793219888, -68988249961, -205524724216, 58220639342, 72910025298, -23281031570, -13711285756, 4694030479, 1296748662, -463327892, -48539622, 17825543], [5546657016, -7215473472, -86511099796, 19687976769, 284382610852, -50457333038, -395473480546, 91650814599, 269397508516, -76561491406, -95394232990, 30484202262, 17919217902, -6133289141, -1693449166, 604662944, 63356008, -23245789]];
    Rf_basisdens := [1, 1, 1, 1, 2227868, 2227868, 2227868, 4455736, 4455736, 2227868, 4455736, 2227868, 4455736, 2227868, 4455736, 2227868, 4455736, 4455736];
    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_4235_a();"
function MakeCharacter_4235_a()
    N := 4235;
    order := 1;
    char_gens := [2542, 1816, 2906];
    v := [1, 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_4235_a_Hecke(Kf)
    return MakeCharacter_4235_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 := [[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, 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], [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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, -1, 0, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [0, -1, 1, -1, 1, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 1, 0, -1, 1, 0, 0, 0, 1, -1, -1, 0], [0, 1, -1, 0, 0, -1, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0], [3, 1, -2, 1, -1, 0, 0, -1, 1, 1, 0, 0, -2, 0, -1, 1, 0, 1], [2, -1, 0, -1, 0, 0, 0, 1, 0, -1, -1, 0, 0, 0, -1, 0, 0, -1], [1, 2, 1, -1, 1, 0, 1, 0, 2, -1, 0, 1, 2, -1, 2, -1, -1, -1], [0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, 0, -1, -1, 1, 0, 0, 1], [-1, -2, 1, 0, 0, 1, 0, 2, 0, 0, 0, -1, 0, 1, 0, -1, 1, 0], [0, 0, 1, -1, 1, 1, 0, 1, 0, -1, 0, -1, 1, 0, -1, 0, 0, -1], [1, -2, 0, -1, -1, 2, 0, 0, -2, 0, 1, -1, -1, 1, -1, -1, 1, 0], [0, 0, 2, 0, 0, -1, 0, 0, -1, -1, 0, 0, 0, 1, 0, 0, -1, 0], [2, 1, -1, 0, 1, -1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, -1, 0], [-1, -2, 0, -1, -1, -2, -1, 0, -2, 0, 0, -1, 0, 0, -1, 0, 0, 0], [0, 0, 1, -1, 0, -2, -1, 0, -1, -1, 0, -1, -1, 3, -2, 0, 0, -1], [0, -3, 0, 0, 0, 3, 1, 0, -1, 1, -1, -1, 0, 1, -1, 0, 0, -1], [0, 2, 0, 0, 1, 0, 2, 0, -2, 0, -1, 0, 2, -2, 0, 1, 0, -2], [3, 0, 0, 1, 0, -1, -1, -1, 0, 0, 1, -1, -1, 2, -2, 1, 1, 0], [3, 1, 0, -1, 0, 1, 1, -2, 2, -1, 1, 1, -1, -2, 1, -1, 0, 0], [-1, 2, 0, 0, 1, -3, 0, -1, 2, 0, 0, 2, 0, -2, 1, 1, -2, 0], [3, -3, 1, 1, -1, 3, 0, 1, 2, 1, 1, 0, 1, 0, 0, -1, 1, 0], [-2, 1, 0, -1, 0, -2, -2, -1, 0, -1, 1, 0, -2, 0, -1, 0, 0, 1], [0, -1, 0, -1, -1, -3, -1, 1, 1, -1, 0, -1, -1, 3, -1, 0, 0, 0], [2, 0, -1, -1, 2, 0, 1, 2, 2, 1, -2, 1, 1, -1, 0, 0, 0, -1], [-1, 0, 0, 0, 0, -1, -1, -1, -1, 0, 1, 0, 0, 1, 1, 1, 0, 2], [3, 1, -1, -1, 1, -3, 1, -1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0], [1, -5, 0, -2, 0, 0, 1, 0, -3, 0, 0, 1, 1, 1, 1, -1, 0, -1], [5, -3, -2, 1, -1, -1, 0, -1, 0, 2, 0, 0, 0, -1, 1, 1, 1, 1], [-2, 1, 1, -1, 2, 2, 2, 3, 1, 0, -2, 0, 3, 1, 0, -1, 0, -2], [3, -1, 0, 0, 2, -1, 2, 0, 4, 1, -2, 1, 3, -3, 3, -1, -1, -1], [3, -1, 1, -1, 0, -1, 2, -1, -3, -1, -1, -2, 0, 0, 0, 1, 0, -2], [3, 1, -1, 1, 0, 1, 0, 1, 1, 1, 0, 0, -3, 0, -2, 1, 1, 0], [-4, 3, 1, 0, 0, -4, -1, 0, -4, -2, 1, 0, -1, 1, 0, 0, -1, 1], [0, 2, -2, 3, -1, 0, -1, -5, 2, 0, 3, 1, -3, -2, 1, 0, -1, 0], [3, -4, 1, -1, 1, 1, 1, 0, 3, -2, -1, -1, 1, 0, 0, -1, -1, -1], [6, 4, 0, 1, 0, 0, 2, -3, -1, 1, 1, 1, -1, -2, 2, 0, 0, 0], [5, 0, -1, 0, -1, 4, 2, 0, 4, 1, -1, 0, 2, -2, 1, -1, 1, 1], [0, -1, 1, 0, -1, -1, -2, 3, -2, -1, 0, -1, 1, 3, -1, 0, -1, 0], [3, 1, -1, 1, -1, 2, 1, 0, 3, 1, 0, 0, 1, -1, 1, 1, 1, 0], [6, -2, -3, -1, 0, -3, 1, 1, 4, 1, -2, 1, 0, 1, -1, 2, 1, 1], [-4, -3, 2, -1, 1, 1, 1, 3, 2, -1, 1, -2, 0, 1, 2, 0, 0, -1], [-2, 5, 2, 0, 4, 0, 0, 0, 2, -1, 0, 2, 0, -1, 3, 0, -3, 0], [5, -3, -1, 2, -2, 7, 2, -2, 4, 1, 0, -1, -2, -3, -1, 1, 2, 0], [6, -1, -2, 3, -2, 0, 2, -4, 0, 0, 0, -2, -2, 1, 0, 2, 1, 1], [-5, -1, -3, 2, -1, 2, -1, 0, -4, 1, 2, 1, -4, 0, -2, 1, 2, 2], [-6, 1, 2, 1, 1, 1, 3, -1, -5, 1, -2, -2, 3, 1, -2, 2, 1, -1], [0, -2, 1, 1, 0, 2, 1, -1, -3, 0, 2, -1, -2, 2, -1, 2, 2, -1], [-1, -1, 0, 1, -2, 1, -1, -2, -1, -1, 0, -2, -2, -1, 0, -1, 0, 1], [1, 1, -2, 1, 0, 1, -1, -3, -1, -2, 0, 0, -3, 0, -1, 1, 0, 1], [0, 0, 4, -3, 1, -3, 1, -1, -4, -4, 1, -1, 2, 0, 2, -2, 0, -2], [4, -1, 0, 0, 1, 2, 1, 2, 0, 0, -1, -1, 3, 1, 1, 0, 0, 1], [3, 2, 4, -2, 4, -1, 1, 2, 3, -3, -2, 0, 2, 0, 2, -1, -1, -2], [-4, -1, 0, 2, -1, 6, -1, -1, 2, 1, 2, -1, -1, -2, -2, 0, 1, 1], [3, 0, -4, 3, -2, 2, 1, -3, -1, 2, 0, 0, -5, 2, -2, 3, 2, 1], [1, 0, 1, 0, 1, 3, 1, 3, -1, -1, 0, -1, 2, -3, 1, -1, -2, -2], [2, -5, -1, 0, 0, 4, -1, -2, 0, 0, 1, -2, 1, 1, -2, 0, 1, 1], [2, -4, 5, -2, 2, 3, 2, 2, 1, -2, -3, -2, 5, 2, -1, -2, 1, -3], [3, 2, 0, 1, 0, 0, 3, -1, 5, -1, -1, 1, -1, -1, 0, -1, 0, 0], [5, 4, 0, 1, -2, -2, -1, -2, 2, -1, 0, 0, 0, -1, -1, -1, -2, -1], [2, -3, -4, 1, -3, 2, -1, -2, 0, 2, 0, 0, -1, 2, -2, 0, 0, 1], [0, -3, 1, 2, -3, 4, 1, -4, -5, 1, 0, -3, 0, 2, -2, -1, 2, 0], [-8, 0, 0, 2, 0, 1, -2, 1, -3, 0, -1, -2, -2, 2, -3, 2, 0, 2], [2, 0, 3, 0, -3, 0, -2, 0, 0, -1, 3, -2, 0, 3, -1, -2, 1, -1], [1, -4, -1, -1, -1, -1, 1, -2, -3, 0, 0, -1, 0, -2, 2, -1, 2, -1], [0, 2, 1, 1, -1, 3, 1, -4, 3, 0, 2, -1, 0, 1, -2, 0, 2, 1], [5, -6, 0, -2, -1, 3, -1, 3, 1, 0, -2, -2, 2, 5, -3, 0, 0, -2], [2, 2, 0, 1, 0, -2, 3, 1, 1, 1, -1, -1, 3, -2, 0, 0, 0, -2], [3, -4, -1, -1, -1, 0, 1, 0, 0, 1, -2, -1, 2, 1, -4, 1, 1, -1], [6, 5, -2, -1, 2, 0, 2, 0, 2, 2, -3, 2, -1, -1, 0, 2, -2, -1], [-9, -7, 2, -2, -1, 3, -1, 1, -8, -2, 0, -1, 1, 1, -2, -2, 2, -1], [6, 1, 0, 2, 3, 1, 3, -4, -1, 2, 0, 1, -1, -3, 2, 2, 0, 1], [2, 4, 1, -1, 1, -1, -3, 0, -1, 0, 2, 3, 2, -1, 2, -2, -2, 3], [4, -4, 3, 0, 2, 4, 7, 3, -1, 2, -3, -1, 6, -2, 1, 0, 1, -2], [3, -8, 5, -2, 1, 0, -1, 3, -1, 0, -1, -2, 3, -1, 0, -1, 1, -1], [6, -3, -3, 0, -2, -2, -3, 1, 5, 0, 0, 1, -4, 1, 1, -2, -1, 0], [6, -1, 4, -2, 0, 0, 1, 4, -3, 0, 0, -1, 1, 4, 1, -2, 2, 1], [3, 2, -1, 3, -2, -1, 2, -6, 3, 2, -2, 0, -3, 0, -3, 5, 0, 0], [6, 0, -3, 0, 0, -1, -2, -3, 4, 1, 4, 4, -7, 1, 2, -1, 0, 5], [0, -4, -1, 4, -3, -1, 0, -5, 0, 1, 1, 0, -1, -5, 2, 2, 0, 3], [-10, 0, 2, -3, 2, -1, 0, 0, 5, -3, 1, -1, 2, -4, 3, -2, -1, -2], [6, 3, -3, 1, 0, -4, -1, -2, -2, 2, 0, 2, 1, -2, 2, -2, 0, 2], [-7, -5, -3, 0, -2, 1, -4, 0, -2, 0, 3, -1, 2, 0, -2, -1, 3, 2], [-2, 2, -1, 2, -2, 0, -1, 0, 2, -1, 1, 0, -1, 2, 1, -2, -2, 1], [-5, 0, 2, 0, 3, 5, 2, 0, -9, 1, -2, 0, 1, -1, -2, 1, 0, -2], [-2, -4, -5, 3, 1, 4, 5, 0, -4, 3, -2, 1, -2, -2, 0, 2, 1, 1], [6, 1, -1, 2, 0, 2, -1, 1, 5, 0, 0, 1, -4, -3, -3, 0, -1, 0], [6, 2, -5, 2, -1, -8, -2, -2, 6, 3, 3, 4, -2, -3, 5, 0, -1, 3], [-8, 1, -1, 1, 2, 1, 1, 3, 8, -2, 0, 1, 0, 0, 2, 0, -2, -1], [10, -2, -5, 2, -2, 4, -2, -5, 11, 0, 3, 0, -6, 2, 0, 0, 0, 3], [1, -3, -3, -2, 1, -2, 0, 4, 8, -1, -2, -1, -2, 2, 0, -1, 0, 0], [11, -4, 3, 1, -2, 4, 2, 1, 1, 2, 0, -1, 4, -1, 1, -1, 2, -3], [2, 7, 4, 0, 0, -1, -2, 3, 4, -4, 1, -1, 5, -4, 6, -4, -4, -1], [2, -5, 0, 0, -3, 2, 3, -1, -6, 1, -1, -1, 4, -4, 0, 0, 2, -1], [-5, 2, 3, 1, 2, -4, -3, -2, 1, -5, 3, 1, -4, -2, 2, -1, -3, 1], [-5, 3, 1, -1, -1, -3, 1, -1, -2, 1, 1, 0, 3, -5, 2, -3, 1, -2], [-3, 5, 5, -3, 3, 4, 3, 4, 2, 1, -3, 2, 4, -5, 1, -1, 0, -2], [-5, -2, -2, 1, -2, -4, 1, 0, -1, 2, 3, 1, -4, 2, 3, -1, 0, 2], [-8, 0, 0, 5, -1, 1, 1, -5, 3, 2, 1, -1, 0, -4, 0, 0, 0, 2], [-8, 3, -1, 2, -1, -1, -4, -8, -11, 2, 3, 3, -1, 2, 0, 0, 0, 4], [0, -1, -2, 3, -5, 3, 2, 1, 0, 0, 0, -4, -6, -1, -1, 0, 3, -1], [-1, -1, 2, -4, 1, 1, -3, 5, 0, 0, 0, 3, -3, 3, 0, -2, 0, 1], [-5, -2, 0, 0, 1, 1, 3, -1, -2, 0, -4, -4, 0, -1, -3, 3, 2, -2], [2, -8, -2, 4, 0, 4, 1, -7, -6, 3, 1, 2, -4, 0, -1, 4, 1, 2], [6, 0, 3, 0, 2, 2, 1, 0, 12, -1, 1, 2, -1, -4, 1, -4, -2, -1], [4, -1, -3, 0, -1, 0, -2, 4, -3, 3, -1, 2, 1, 0, 1, 0, 2, 1], [10, -7, -6, 4, -3, 4, 1, -5, 1, 5, -1, -3, 0, 0, -6, 6, 4, 1], [2, -2, -4, -1, -1, 1, 1, -3, 14, 4, -1, 1, 0, -4, 0, 0, 2, 0], [8, -3, -3, 3, -3, 4, 2, -9, -5, 0, -1, -2, -3, 1, -6, 4, 0, -2], [3, 1, 0, 0, 2, 3, 4, 0, 2, 7, -2, 1, 5, 1, 3, -1, 1, 1], [2, 3, -1, 4, 1, -3, -3, 2, 1, -2, -3, 1, -3, 1, 2, 3, -4, 2], [-1, 7, 0, -1, 2, -3, 0, -2, 3, 1, 0, 0, 0, 1, -2, 3, -2, 1], [-7, 0, -1, 1, 1, 0, 5, 1, -8, 0, 0, -3, -1, 0, -1, 5, 5, -1], [4, 4, -1, 1, 1, -2, 4, -1, 8, 0, -1, 3, 0, -1, 1, 2, -2, -1], [-10, 5, 5, 2, -1, -4, -4, -7, 0, -3, 4, -1, 0, 1, 2, -2, -4, 0], [5, -2, 3, -4, -1, 1, -1, 7, -2, -1, -1, -2, 3, 4, -2, -1, 2, -3], [-7, -7, 6, -2, 0, 3, 3, 4, -1, -1, -5, -3, 0, 0, -1, 3, 2, -3], [-1, 4, -4, -5, 0, -4, -4, 1, 1, -3, 1, 1, -3, 2, 0, -3, -2, 0], [-18, -2, -1, 0, -1, -6, -6, 4, -7, 3, 0, 0, 1, 1, -1, -2, 0, 1], [7, -8, -4, 2, -2, 2, 3, -6, 4, 4, 0, 2, -2, -5, 0, 1, 2, 2], [-8, 3, 3, -3, 1, 1, 0, -3, -13, 0, 1, 3, 0, 1, 2, -2, -2, 1], [14, 3, -4, -3, -1, -4, -7, -3, 4, -2, 2, 4, -1, -6, 0, -2, -2, 3], [7, 4, -2, 1, 0, 1, 1, 3, 6, 0, -1, -1, -7, 6, -2, 3, 1, 0], [-9, 11, 0, 2, 0, -1, 0, -2, -3, -1, 2, 3, -1, -5, 5, 0, -1, 3], [-11, -1, 2, -2, -1, -2, -4, 1, -4, -2, 3, -5, 1, 6, -1, -1, 2, -1], [6, -1, -2, -1, 2, 7, 4, 4, 7, -2, -3, 1, 4, 1, 2, 0, -2, -4], [10, -5, 2, 0, -2, 1, 2, 0, 5, 3, -3, -1, 2, -3, -1, 0, 2, -1], [-5, 6, -1, 5, 0, 1, 0, -4, 1, 4, 2, 0, -3, -4, -3, 3, 2, 2], [11, -2, 0, -1, 0, 6, -3, 6, 8, 2, -1, 1, 2, 0, 3, -5, 0, 2], [4, 2, 1, 0, 2, 0, 3, -6, -4, 3, -1, 0, 5, -2, -3, 2, 0, -1], [9, -5, 0, -2, 1, 8, 4, -4, 2, 2, -3, 1, 1, -6, 1, -1, 4, -3], [-1, -9, -7, -1, -1, 6, 2, -5, -3, 6, -3, -2, -3, 2, -4, 5, 6, 2], [9, 5, -2, 3, 0, 2, 5, 1, 6, -2, 0, -2, 0, -2, 4, 1, -1, -1], [10, 2, 0, -5, 1, -7, 0, 2, 1, -1, -3, 3, 3, 3, -1, 2, -2, -4], [9, -4, -3, -3, -1, -1, -1, 0, 11, -1, 3, 1, -8, 1, -1, 1, 3, 3], [25, 9, -9, 8, -2, -2, 0, -5, 13, 6, 2, 5, -8, -4, -1, 3, -1, 4], [8, -8, -1, -3, -2, 1, 3, 1, -3, 3, -1, 0, -1, 6, -2, 2, 1, 0], [-1, 6, -3, 1, -1, -1, 1, -1, 2, -1, 2, -1, -5, -3, 1, 1, 2, -1], [-18, 3, -3, 3, -5, 1, -2, 1, -6, -1, 3, -4, -3, -2, 0, -2, 3, 2], [2, 6, -2, -1, 2, -7, -4, 1, 2, -2, 5, 5, 3, -5, 6, -4, -5, 0], [-2, -4, 7, -4, 3, 5, 2, 3, 0, 1, -5, -2, 7, 3, -2, 2, 2, -1], [6, -6, 4, 1, 0, 2, 1, 0, -8, 0, 0, -3, 3, 2, -1, 1, -1, 0], [-7, -1, -1, -6, 1, -7, 1, 8, -5, -2, -4, -1, 0, 5, -4, 1, 3, -2], [-4, -5, 4, -2, -1, 7, 1, 0, -9, -1, 3, -7, 5, 4, -1, 0, 3, -1], [-7, 8, -2, 1, 1, -8, 2, -8, -2, 0, 3, 3, -5, -3, 3, 1, -3, -2], [-4, 1, 2, 0, 0, 2, 0, 4, 4, -2, 1, -3, 2, 4, -1, -2, 1, -1], [3, 7, 0, -3, 1, -6, 0, 3, 3, 3, 0, 4, 2, -2, -1, -1, 0, 3], [1, 0, 4, 3, -1, 3, -2, -2, 11, -1, 3, 3, 3, -6, 5, -5, -2, 0], [9, -2, -7, 0, -4, 1, -2, -3, 13, 3, 1, -2, 0, -3, -4, 1, 1, 3], [-6, -4, 0, 1, -4, -2, -4, 2, -2, 0, -1, -1, -6, 9, -3, 2, 2, 6], [10, -3, -1, -2, -1, -5, -4, 3, 15, -5, 0, 1, -1, -1, 1, -2, -3, 0], [-5, -4, 4, 2, -2, 4, -1, 3, -10, -3, 3, -5, -4, 1, 0, -1, 2, -1], [8, -6, 7, -1, -3, 6, 0, -2, -2, -5, -2, -11, 2, 4, -6, -2, 3, -3], [0, -10, 2, 2, -3, 9, 1, -6, -12, 2, 0, -4, 0, 1, 1, -2, 2, 2], [-1, 10, 2, -4, 6, 0, -1, 2, 2, -2, 2, 4, 4, -3, 4, -1, -4, 0], [-3, -2, 8, -5, 1, -6, 0, 6, 8, -2, -3, -1, 7, -1, 3, -3, -3, -2], [5, -6, 2, -1, 2, -1, 1, -4, 4, 5, -2, 5, -1, 3, -1, 3, 0, 0], [-14, 6, 6, 0, 2, 1, 2, 7, -15, -2, -3, -2, 4, 2, -1, -2, -2, -2], [0, -1, -1, -2, 4, 6, 7, -1, 7, 0, -2, -1, 4, -9, 7, -2, -1, -4], [-18, 3, 4, 2, -2, -4, 0, 0, -13, -2, -2, -3, -1, 0, -3, 0, -2, -3], [-18, -8, 3, 1, -1, 6, 1, -2, -14, 1, -2, -5, 2, -1, -4, 2, 5, 1], [-2, -1, 4, -2, 6, -4, 3, 0, -14, 0, -4, 1, 7, -3, 7, 1, -2, -1], [-2, 2, -1, 0, -2, 1, 5, -4, -8, 2, -2, -3, 2, 0, -2, 4, 5, 0], [-21, -6, 1, -2, -3, 0, -4, 4, -11, -3, -2, -4, 5, -1, -1, 1, 0, -1], [6, -2, -1, 1, -1, -6, -3, -9, -1, 0, -1, -1, -4, 4, -5, 5, 4, 3], [-10, -5, 6, 1, 1, 4, -1, 5, -3, 1, -3, -4, 7, 5, -1, 0, -1, -3], [11, 4, 2, -4, 1, -8, 2, -3, 1, -1, 2, 2, 6, -9, 5, -1, 1, 0], [12, 5, -2, -2, 2, -7, -1, 6, 10, 1, 0, 3, -1, -3, 3, 0, -1, -3], [-2, -6, -5, -1, -4, 0, -5, -2, -10, 3, 4, 1, 5, -5, 2, -2, 2, 3], [-5, -4, 1, -2, 1, 4, 1, -3, 7, 2, 1, 0, 5, -1, 2, -1, -1, -1], [0, -3, 0, -1, -3, 3, -7, -1, 4, 0, 2, -4, -4, 4, -3, -4, 1, 3], [-5, -1, 0, -2, -2, 1, -6, -1, 0, -2, 1, 1, -1, 2, -4, -7, -2, -1], [2, 3, -4, 2, 2, -2, -2, -1, -9, 4, 0, 4, -3, 0, -2, 2, -3, 0], [-4, 6, 9, -7, 4, -5, 2, 8, 7, -6, -3, 1, 4, -5, 6, -5, 0, -3], [4, -4, -3, -3, -3, 4, 4, 4, 0, 1, 0, -5, 0, 2, -4, -6, 5, -1], [-3, -2, 1, -1, 1, -4, 1, 5, -4, -2, -3, -1, 0, 8, 1, 3, 0, -1], [-8, -6, 11, -7, 2, -2, -5, 7, 5, -6, 4, -1, 10, 2, 5, -8, 0, -1], [10, -2, 4, -3, 2, -8, -2, -6, -2, -4, -1, 3, 2, 1, 3, 2, -4, 0], [-10, -6, -3, -2, 2, 11, 1, 7, 3, 1, 1, -4, 2, 2, -2, 2, 3, -1], [7, 0, 1, 0, 1, 3, -1, -1, 4, 1, -1, -2, 1, 8, -2, -1, 0, -1], [13, 0, 2, 2, -4, -2, 3, -6, -4, 4, -2, 0, -4, -1, -4, 1, 2, 2], [-7, 4, -4, 5, 0, 0, -1, -1, -5, 7, -1, 5, -5, 5, 0, 1, 2, 6], [-5, -6, 3, -2, 0, 3, 0, 1, -14, -2, 0, -1, 2, 0, 0, 1, 1, -2], [1, 0, -3, 1, -5, 6, 2, 0, 2, 3, 4, -3, -4, 5, -4, 1, 5, 3], [-10, 9, 8, -2, 4, 0, -4, -3, -3, -5, 3, -1, 3, -5, 2, 0, -3, -1], [0, -2, 3, -1, 2, 6, 3, -2, -11, 3, -4, 3, 9, -7, 4, -2, -2, -3], [6, -5, 1, 1, -3, 4, 2, -2, 3, 1, -2, 0, -1, -6, -1, 2, 3, 2], [8, -5, 2, -3, 1, 1, 3, -2, 6, -2, -2, -2, 0, 0, 3, 0, 3, -1], [5, -3, 2, -2, 1, 6, 2, -2, 16, 0, 3, -1, 1, 0, -1, 1, 4, -1], [0, -3, 9, -3, -1, -1, 6, -6, -4, 0, -2, -4, 1, -3, -3, -2, 4, 0], [-15, -1, 2, 0, -2, 2, -5, -7, 2, -3, 2, 0, 0, 1, -3, -3, -1, 4], [-9, 14, -1, -1, 3, -9, -1, -1, 15, -2, 2, 5, 2, -6, 8, -3, -4, 1], [5, 11, 3, 2, 7, -2, 1, 3, -8, -4, -1, 5, 2, -1, 7, -1, -5, 0], [4, -12, -1, -3, -4, -2, -1, -6, 3, 0, -2, -3, -2, 0, -3, -2, 2, 1], [-18, -4, 6, -4, 1, 5, -2, 1, -4, -2, 0, 0, 12, -5, 3, -2, -2, -4], [-11, 1, -1, -1, -1, 2, 3, 0, 5, 1, 2, 2, 3, 1, -1, 1, 1, 2], [5, 8, 1, -2, 0, -10, -4, 0, -14, -1, -3, 0, 1, 1, -1, 3, -4, -1], [14, -4, 2, 3, -2, 10, 3, 0, 0, 2, 0, -3, 5, 0, -3, -1, 5, -4], [18, 9, -1, 6, -2, -2, -3, -1, 23, 4, 0, 1, -2, -7, 3, 2, -3, 6], [-8, -8, -1, 4, -5, -5, -4, -1, -6, -1, 3, -8, -3, 5, -4, 2, 4, 3], [10, -6, 9, 2, 3, -3, -2, -7, 9, -1, 0, 1, -4, 4, 3, 0, -4, 2], [-6, 3, -4, -3, 4, -1, -4, 4, 3, -1, 0, 2, 4, 0, 2, -4, 0, 3], [-1, -6, 1, 1, 0, 4, 2, 6, 0, -5, 2, -7, -1, 4, -2, -1, -2, -3], [12, 7, -10, 3, -5, -8, 0, -3, 3, 5, 3, 2, -5, -8, -1, 3, 1, 3], [4, 6, -9, 2, -4, -9, -3, -6, 3, 4, 1, 6, -5, -3, 1, -2, 1, 5], [-1, 2, 9, 1, -1, 0, 1, 3, 1, -4, 0, -5, -1, -2, -3, -3, 1, -3], [5, 9, -14, 3, -5, -1, 0, -9, -10, 2, 4, 4, -12, 1, -3, 5, 1, 1], [-3, 4, -2, -3, -4, -12, -7, -2, 4, -2, 3, -1, -6, 0, -3, 3, 2, 4], [-16, -6, 11, 0, -3, 3, 1, -1, 0, -3, 3, -8, 1, 3, -4, -2, 4, -3], [-1, -6, 9, -7, 2, 7, 8, 1, -6, 1, -4, -5, 6, -2, -5, 1, 7, -7], [7, 8, -3, 5, 0, -8, 2, -9, -15, 2, -4, 3, 0, -3, -3, 3, -4, -3], [-7, 16, 0, 5, 0, -9, -1, -8, 3, -3, 1, -1, 0, -9, 3, 3, -5, 0], [3, -4, 3, -2, 1, 7, 1, -1, -2, 3, 3, 4, 1, -4, 2, -5, -2, -3], [0, 7, -1, 1, -1, 4, -4, -2, 14, 3, 3, 2, -4, -2, -5, 2, 0, 2], [6, -2, 1, 2, -1, -4, 0, -12, 7, -3, 2, 0, -3, 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-2, 0, -2], [15, -4, 5, -3, -5, 6, -4, 7, -6, -6, -1, -3, 2, 6, -9, -1, 0, -5], [-3, -5, 9, -4, 2, 17, 2, 3, -6, -1, -4, 1, 4, -9, -3, -3, -2, -6], [24, 18, -13, 6, 2, -10, 5, -11, 4, 6, 6, 12, -9, -11, 4, 0, -5, 1], [-8, -4, 0, 5, 2, -8, 4, -9, -9, -1, -3, -1, 3, 4, -2, 8, 0, 0], [-15, -6, 6, 2, -4, -1, 1, 8, -9, -4, 3, -8, 0, 7, -1, -7, 6, -6], [0, -6, -1, -2, -3, 3, -4, -8, -10, 5, -3, 4, 1, 3, -2, 2, 4, 5], [-16, -5, 11, -1, -5, 1, -4, 3, -24, -13, 5, -8, 3, 2, -8, -2, -1, -10], [8, -4, 1, 2, -2, -3, -7, -9, 11, -6, 7, -4, -11, 11, -5, 0, 1, 3], [-18, 5, 7, -3, 6, -12, -1, 6, -22, -1, -3, 8, 9, 5, 3, 0, -9, -4], [-7, -37, 6, 3, -4, 19, 3, 7, -6, 0, -4, -12, 9, 2, -3, -1, 12, -1], [11, -5, 1, 6, -3, 11, 3, -15, 5, 3, 1, 1, 2, -6, 3, -11, -2, 0], [-14, -2, 6, -3, 5, 1, 5, 5, -5, -6, -2, -3, -7, 16, -6, 6, 1, -2], [34, 3, -2, 0, 2, -6, 4, -15, -3, 3, -1, 3, 5, -3, 4, 0, -7, -1], [-14, -4, 2, -3, 2, 2, -5, 13, -5, 3, -3, 5, 3, -1, 2, -2, -6, 2], [13, -6, -2, -9, 1, -13, -1, -7, 1, 8, 2, 9, 5, -7, 8, -7, 1, 4], [9, -1, -6, 1, 0, -8, -6, 10, -8, 0, -4, 4, -8, 8, -4, 7, 1, 7], [-6, 10, -3, -2, 0, 3, 7, 3, -3, 1, 4, 2, 3, -8, 6, -4, 2, -5], [-21, 4, 6, -6, 4, -9, 3, 1, -12, -9, 1, -3, 1, -8, 7, -3, -3, -5], [6, 13, -6, 1, 2, 10, -4, 6, 3, 5, 2, 6, -3, 2, -7, 2, 5, 1], [-10, 4, 0, -1, -4, 10, 6, 2, -3, 1, -3, 3, 5, -6, 0, -2, 0, -4], [-18, 7, 14, 3, 2, -7, -6, -6, 3, -4, 5, -1, 6, 5, 10, -4, -10, 0], [10, -3, 1, 2, -2, 4, -1, 0, -3, 9, -6, 3, 9, 0, -2, 2, 1, 2], [-8, -6, 0, -1, -2, -11, -7, 6, -13, -1, 1, -5, -8, 15, 1, 0, 1, 10], [15, 12, 8, -9, -3, -9, -4, -3, -3, -10, 0, -5, 3, -3, -4, -7, -3, -10], [0, -5, 12, 2, -1, 12, 3, 6, 12, -4, 5, -5, 3, 1, 1, -2, 0, -1], [1, -31, 6, 4, -1, 11, 6, 2, -13, 5, -3, -7, -1, -1, -3, 9, 7, -1], [-3, -6, 9, 4, -3, -1, -1, -7, -17, -2, 1, -10, -2, 9, -7, 1, 8, 5], [20, 10, 2, 7, 1, -1, -1, -6, 24, 7, 1, 1, 1, -1, 7, 4, -4, 4], [-8, 11, 5, -2, 10, -6, 1, 7, -9, -2, 0, 5, 6, 1, 5, 2, -7, 0], [16, -1, -8, 4, -6, 6, -2, -6, 9, -2, 4, 1, -3, -14, 3, -6, -6, -1], [-6, 0, -7, -6, 4, -14, -3, 5, -8, 4, -8, 12, 7, 9, 2, 4, -5, 3], [-9, -17, 10, -4, -5, -8, -4, 7, -4, -4, 1, -3, -5, 12, 1, -1, 0, 5], [-2, 13, 2, -2, 2, -15, 0, -7, -12, -4, 5, 3, -3, -8, 4, 2, -4, -1], [22, 9, -10, 5, -5, -5, -1, -9, 10, -4, 8, 0, 0, -2, -1, -4, -3, -1], [5, -3, -10, 2, -1, 2, 4, 3, -11, 7, -3, 1, -6, 3, -8, 5, 8, 1], [22, -3, -1, 2, 1, 12, 6, 0, 2, 2, -1, -2, 2, -6, 0, 0, 0, 3], [26, 9, -7, 7, -4, 3, -3, -7, 15, 3, 7, 4, -12, 5, -2, 4, 0, 12], [-32, 0, 6, 4, -7, 3, -3, -11, 0, -2, 10, -6, 0, -7, -1, 2, 4, 4], [-19, 10, -13, -1, -2, 2, -6, 4, 10, 4, 5, 10, -7, 3, -2, -5, -4, 8], [-16, 5, 2, 5, 0, 3, -10, -9, -3, 4, 7, 6, -6, -5, -3, 0, 5, 11], [11, 4, 5, 5, -3, -5, 3, -7, 11, -3, 7, -9, -4, -1, -1, 3, 5, -1], [3, 2, -8, 2, -3, 4, 6, 9, 10, 5, 2, 4, 0, -11, 5, 3, 3, 0], [-7, 0, 0, 2, 2, -1, 5, 1, -11, 6, 1, 8, -2, -11, 13, 1, 2, 6], [-2, 5, -7, 8, -6, 4, -5, -5, -11, 8, 4, -1, 0, -3, -1, 2, 9, 6], [1, 0, -12, -1, 0, 6, -3, 7, 17, 2, 0, 8, -1, -4, 2, 1, 0, -1], [3, -6, 0, 7, -3, 8, 6, -7, 16, -1, 5, -3, -6, 2, 8, 3, 1, 4], [-8, 24, -1, 8, 5, 3, 4, 0, 12, -2, 3, 8, -2, -10, 9, 2, -6, 1], [0, 1, 3, 6, 3, 0, -2, 2, 3, 1, 1, 4, -15, 8, -3, 6, -2, 1], [-15, 1, 2, 8, -4, -3, -5, 0, 8, 3, 10, 2, -3, 1, 12, -9, -4, 10], [-8, -2, -5, -4, 0, -19, 0, 3, -4, -7, 4, -2, -7, 3, -2, 3, 6, 1], [-17, -8, 8, 4, -5, -4, 4, -8, -8, -2, -2, -10, -5, 8, -12, 5, 5, 0], [-19, -19, 13, -10, 0, 18, 10, 7, 1, -6, -6, -13, 8, 0, -7, 1, 7, -12], [10, 2, -8, -9, -1, -6, -3, 3, 10, -4, 7, 5, -1, 4, 5, -8, -1, 2], [-26, 5, 6, -5, 3, -11, 5, 4, -15, -6, -1, -1, 11, 2, 8, -8, 1, -2], [-12, 3, 2, -7, -1, -8, -9, 1, -3, -1, 4, 8, -8, -1, 1, 0, 0, 7], [14, -10, -4, -5, -1, -7, -3, 4, -5, 3, 6, 3, 2, -7, 5, -6, 3, 8], [-5, 12, 8, 0, 7, -13, 4, -7, -16, -2, 2, 6, 0, 2, 7, -1, -6, -2], [24, -4, -5, 6, -4, 0, -5, -10, 16, 1, 3, 2, -13, -8, -3, 2, -4, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4235_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4235_2_a_bp();".
// 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_4235_2_a_bp(:prec:=18)
    chi := MakeCharacter_4235_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_4235_2_a_bp();".
// 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_4235_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4235_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-176, -16, 3048, 3146, -9817, -10744, 14618, 14224, -12411, -9228, 6290, 3158, -1874, -580, 317, 54, -28, -2, 1]>,<3,R![4400, -16240, -31284, 115244, 62219, -224597, -57876, 191761, 21138, -85889, -132, 21207, -1687, -2832, 384, 190, -33, -5, 1]>,<13,R![156496, -2810576, -7005504, 36595538, 67479449, -63191848, -82015293, 55611044, 13201425, -11951605, -323926, 1076987, -56740, -46981, 4291, 986, -111, -8, 1]>],Snew);
    return Vf;
end function;