
// Make newform 8008.2.a.y in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8008_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_8008_2_a_y();".
// 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_8008_2_a_y();".
// 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 := [-128, -1032, 538, 6417, -2804, -7275, 2996, 3378, -1306, -750, 273, 78, -27, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16683138390, 99775185884, -180363225628, 81567084, 239059287029, -11970806879, -120227610741, 3080027798, 27555851639, -1105297503, -2843591948, 202633696, 106614016, -11074521], [264264384876, 297904178158, -437413157891, -1032089287458, 431622615285, 685142582580, -265081043224, -161921785412, 76649830082, 12008354815, -9589273014, 429387489, 415732687, -58724453], [-403244409496, -443394945668, 1859229248461, 342029081890, -1451593421773, -158509912178, 415744840748, 64820773162, -46010891430, -16590727105, 1134603604, 1990802275, 56007919, -84573421], [-148105342534, -467654431587, 1262115684952, 1580032475418, -1505481346651, -1291874241299, 681805991838, 424769570879, -143598407974, -62980268748, 14135363599, 4042783649, -519312690, -86197394], [51792914190, 207142479617, -465460641758, 492483408208, 362130604330, -640067051370, -104822338105, 252001600285, 14040705209, -43636307733, -872022317, 3433412919, 20088978, -100221529], [-12295329000, -32398485750, -319648593677, 562295462232, 721519658405, -618802526998, -471825438700, 259840610114, 127831458804, -53506698243, -15034911330, 5370665871, 624487815, -204529955], [306825120252, 331065135556, -3075140922419, 436856651496, 3310520475965, -832031771460, -1330673265748, 385128276528, 245864340184, -75443571221, -21109853824, 6654223007, 681733073, -217441493], [-9943373676, 10035841931, 434273622092, -522691012619, -721447644136, 630297354346, 415135249248, -282276492982, -105792814085, 58834255396, 12048495611, -5735162063, -491561333, 208926672], [-141640358616, -668143324930, 1348441940209, 267466107302, -1837139952593, 540080988466, 1033264260798, -418235692408, -271168142180, 110106082525, 31978291166, -12295355455, -1340879233, 488425337], [816058514, -245962834998, -211254743701, 1239699101363, 424114073096, -1152974583333, -309731889602, 449176063003, 93824334855, -86063793263, -11978040794, 7980519403, 525449102, -282478821], [105424374118, -180279716217, -1405372442586, 1215628784765, 1851226663240, -1367160452244, -976096110556, 607199002972, 241925316093, -127270246922, -27398869301, 12534518457, 1119246863, -461482718]];
    Rf_basisdens := [1, 1, 1, 15572760974, 31145521948, 31145521948, 15572760974, 15572760974, 31145521948, 31145521948, 15572760974, 31145521948, 15572760974, 15572760974];
    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_8008_a();"
function MakeCharacter_8008_a()
    N := 8008;
    order := 1;
    char_gens := [6007, 4005, 3433, 4369, 4929];
    v := [1, 1, 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_8008_a_Hecke(Kf)
    return MakeCharacter_8008_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, 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, 1, 0, 0, 0], [1, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0], [1, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1], [-2, 1, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1], [-2, 2, -1, 1, 0, 0, 1, 0, 1, 1, 0, -1, 0, 0], [-2, 2, -2, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 1, -1, 1, 0, 0, -1, 0, 1, -1], [-1, 1, -1, 1, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0], [-5, 0, 0, -1, 0, 0, -1, 2, 0, -2, -1, 0, -1, 0], [0, -2, 1, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, 0], [-3, 2, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 1], [-3, 3, -1, 1, 1, 0, 1, 0, 0, 0, -1, 1, -1, 2], [3, -5, 2, -2, -1, -2, -2, -1, -3, -2, -1, 0, 2, -1], [1, -2, 1, -1, -2, -2, 0, -2, -2, -1, -2, 0, 0, 0], [-5, 1, -2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0], [-1, 2, 1, 1, 1, 0, -1, 1, 0, 1, 1, 1, 0, 1], [-2, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, 1], [-4, 2, 0, 1, 0, -1, -1, 2, -1, -1, 0, 1, -2, 1], [-2, 1, -2, 1, -1, 1, 1, -1, 1, -1, 0, -1, -1, 2], [-1, 1, 0, -1, -1, 1, 0, 1, 0, -1, 2, 1, 0, 1], [2, -1, 1, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1], [-4, 3, -1, 0, 0, 1, 0, 2, 0, -1, -1, 1, -2, 0], [-2, 0, -1, -1, -2, 0, 0, 1, -1, -1, 1, 0, -1, 1], [0, 0, 1, 1, 2, 0, 1, 0, 1, 1, 0, 1, 0, 0], [1, 2, 1, -1, -1, 0, 0, 1, -1, 0, 1, 0, -1, 0], [-4, 1, 0, 1, 1, 1, 0, 0, 1, 0, -2, 0, -1, -1], [0, -1, 1, 0, 0, 0, 0, 0, 1, 0, 1, -1, 2, -1], [0, 4, -1, 2, 2, 2, 0, 2, 3, 1, 1, 0, 0, 0], [-3, 6, -1, 1, -1, 1, 0, 2, 0, -1, 0, 2, -1, 2], [-6, 2, -1, 1, 1, 1, 1, 2, 2, 0, 2, 0, -2, 0], [-3, 2, 0, 0, -1, 2, -1, 1, -1, 0, -1, 1, 0, 0], [-2, 1, -2, 0, -1, 1, 1, -1, 1, 0, -1, -1, 1, 2], [-2, -2, 0, -1, -1, 1, -1, -1, -1, 0, 1, -2, 1, 1], [1, 2, 0, 1, 1, 1, 1, -1, 1, 1, 1, -1, 1, -1], [-1, -1, 1, 0, 1, 1, 2, -2, 1, 3, -1, 0, 3, -1], [1, -1, 1, 0, -2, -2, 0, -1, 0, -2, -2, 0, 0, -1], [1, -3, 2, -2, -2, -2, -1, 0, -3, -2, 1, 1, -2, 0], [0, 0, 0, 1, 1, -1, -1, 1, 1, 0, 0, 0, 1, -1], [-2, 1, 0, 0, 1, -1, 0, 0, 0, -1, -1, 2, 0, 2], [0, -4, 1, 0, 1, -2, -1, -2, -2, 0, -5, 1, 2, -1], [-2, -1, 1, 0, 0, -1, 0, -1, 1, 0, 1, 1, 1, 1], [1, -1, 2, -1, 0, -1, -1, 1, -1, 0, 0, 1, -3, -1], [-1, 0, 0, -1, 1, -2, -1, 0, -4, -1, 0, 1, 0, -1], [6, -4, 1, -2, -1, 1, 1, -1, -1, 2, 0, -1, 1, -3], [-5, 3, -3, 0, -1, 1, 2, 1, 2, -3, 0, 1, -3, 2], [-7, -2, 0, -1, 0, -1, 0, 0, -1, -1, 1, 1, -2, 1], [-2, 0, -1, 0, 0, 1, 1, -1, 1, 0, -1, 0, -1, 2], [-1, 5, -1, 2, 0, 1, 2, -1, 2, 0, 2, 0, -4, 3], [4, -2, 1, -2, -1, 1, -1, -2, -2, -1, 1, -2, 2, -2], [8, -4, 2, -1, -4, 1, 0, -1, -1, 1, 5, -3, -1, 0], [-1, 3, -3, 0, -1, 1, 2, 2, 1, -2, -1, 2, -2, 3], [1, -3, 1, 2, -1, -1, 0, 1, 0, 2, 1, 0, 1, -1], [-2, 0, -2, 2, 3, 1, 2, 1, 3, 1, 0, 1, 0, 1], [-5, -6, 0, -1, 2, -1, 0, -1, 0, -2, -1, 0, 1, 0], [1, 2, 1, 2, 3, 1, 1, 0, 3, 2, 3, 1, 0, 2], [-6, -3, 0, -1, 0, -1, -3, 0, 0, -1, 1, 1, -1, 0], [-1, 6, -4, 1, 1, 3, 2, 1, 2, -1, -2, -1, 0, 1], [-5, 5, 0, 0, 2, 0, 1, 3, 4, 1, 3, 1, -2, 0], [-9, -6, 1, -2, -2, -1, -3, 1, -4, -3, -2, -2, 3, -2], [3, 5, -1, 3, 1, 1, 3, -1, 4, 1, -1, 0, -1, 2], [-4, 1, 1, 0, 1, -1, 1, -1, 1, 3, 0, 0, 1, 0], [-5, 1, -2, -1, 2, 3, 0, 2, 3, 1, -3, 0, 1, -2], [-1, 0, 2, 0, 0, -3, 0, -1, -1, -1, 1, 1, -1, -2], [2, 0, 3, -1, 0, 1, -2, -2, 0, 1, 5, 0, 0, -1], [-2, -4, 1, -1, 0, 0, -1, 1, -2, 2, 2, 1, 0, 0], [0, -6, 2, -1, -1, -2, -1, -1, -3, 1, -1, -1, -1, 0], [1, -3, 1, -1, 0, -2, -2, -1, -2, 0, -3, 0, 4, 0], [-5, -6, 2, -1, 0, 0, -2, 0, -1, 0, 0, -2, 4, -2], [-3, 1, -1, -2, -1, 1, -1, 2, 1, -2, 3, -1, -1, 1], [-8, -5, 0, -1, 0, -2, -1, 0, -2, -1, -1, 3, -1, 0], [2, -4, 1, -2, -1, -2, 0, -2, -1, 1, -4, -3, 0, -3], [3, 1, 0, -2, 0, -2, 3, 0, -1, 1, 0, 0, 0, 1], [-1, -1, 4, -2, 0, 1, -2, 1, -1, 1, -1, 1, 1, -4], [2, -5, -1, 1, -2, -1, 0, -1, -1, 0, -1, 0, 2, 3], [-8, 10, -4, 4, 4, 3, 2, 0, 4, 2, -4, 3, -1, 3], [1, -5, -1, -1, -2, 0, -1, 1, -2, -2, 0, 2, 1, -2], [-4, -7, 4, 1, 0, -2, -1, 2, 0, -4, 0, 0, -3, -1], [-8, 5, -3, 2, 0, 2, 2, 1, 2, 3, 1, 0, 1, 0], [5, 1, -1, 2, 3, 2, 0, 0, 1, 1, 3, 3, 2, 2], [0, -2, 3, -1, 1, -3, -1, 0, -5, -2, -2, 1, 0, -2], [-10, -3, 1, 1, 1, -2, -1, 4, 1, -1, 2, 3, -3, 2], [9, -4, 0, -3, -2, -1, 1, -2, 0, 0, 5, -3, 0, -1], [-8, 6, -2, 4, 0, 3, 3, 2, 5, 2, 6, 0, -3, 2], [5, -1, 2, -1, -1, 0, -1, -2, -1, 0, -2, -3, 4, -2], [-5, -1, 2, -1, -1, -1, -1, -1, 1, -1, -4, -2, 0, -4], [-8, 2, -2, 0, 1, 3, 1, -1, 0, -1, -1, 1, 1, -1], [-8, -6, 3, 0, -1, -3, 1, -2, -3, -2, 0, -1, -1, -2], [-6, -1, -4, 1, -1, 2, 2, 1, 2, 1, 1, -2, -3, 3], [-10, -2, 2, -3, 0, 0, -2, 3, 0, -4, -1, -2, -1, -1], [-1, 3, 0, 0, 4, 1, 0, 2, 0, 0, -3, -1, 1, 0], [-8, 8, -2, 1, 2, 1, 3, 0, 4, -2, 1, 2, -1, 1], [-4, 9, -6, 2, 4, 3, 5, 0, 5, 2, -2, 0, -1, 0], [5, -6, 0, -2, -2, -2, -4, -1, -3, 0, -1, 0, 1, 1], [-12, 7, -1, 1, 3, 3, 3, 3, 2, 1, 0, 3, -4, 2], [-4, -1, 1, -1, 3, 1, 1, 1, 2, -2, -1, 1, 1, -2], [7, 0, -1, 4, -3, -1, 2, -1, 0, 1, -1, -2, 0, -1], [-1, -1, 1, 1, 0, 2, 2, 1, 1, 2, 8, 0, -2, -2], [2, -4, 5, 1, -1, -2, -2, -1, -3, 0, 2, 0, 1, -3], [-17, 7, -3, 0, 1, 1, 2, 0, 3, 0, -1, 0, -2, 3], [5, -8, 3, -1, -1, 0, 1, -3, 0, -1, 4, 0, -1, 0], [5, -1, 2, -2, -2, -1, -1, -3, -2, -2, -3, -4, 0, -3], [4, -7, 4, 0, 3, 2, 0, 1, 0, 3, 1, -1, 2, -3], [-14, 8, -3, 4, 4, 2, 1, 7, 5, 0, -1, 3, -4, 3], [2, 4, -2, 3, 2, -2, -2, 4, 1, -2, -2, 5, -1, 2], [0, 0, -1, -2, -4, 2, -2, 0, 1, -1, 5, -4, -2, 0], [-9, 4, -2, 1, -1, -2, 2, 1, 0, -1, -2, 0, -3, 0], [1, -2, -2, 0, 2, 3, -1, 1, 3, 3, 2, 1, 2, 2], [2, -2, 0, -1, 1, 2, 0, -3, 1, 2, -2, -3, 3, -6], [-1, -3, 3, -1, -3, -1, 1, -5, -6, 0, 1, -2, 2, -2], [-3, -3, 0, 0, 0, -3, 3, -2, 3, 0, 2, 3, -2, 4], [-15, 4, -3, 2, 3, 2, 1, 1, 4, -2, -6, 1, -1, 0], [-2, -3, 0, 0, 0, -2, 1, -1, 2, -2, 2, -1, -1, 3], [-2, 4, -4, 0, 2, 2, 2, 2, 0, -2, -3, 0, 0, 2], [11, 0, 1, -2, 0, 0, 1, 0, 4, 2, 0, -3, -1, -2], [-12, 9, -1, -1, 4, 4, 2, 4, 5, 1, 0, 2, -2, 1], [2, -4, 1, -1, -4, -4, 1, -6, -6, -3, -10, 1, 2, 0], [-14, 4, -1, 4, 4, 2, -1, 1, 5, 3, 3, 1, -3, 3], [5, 0, 1, 3, -3, -1, -1, 2, -1, -2, -3, 2, -3, 2], [1, 2, -5, 1, -3, 1, 0, -3, 3, 2, 3, 1, 0, 1], [-12, 0, -1, 1, 3, -1, 1, -1, 4, 2, -1, 3, -3, 4], [7, 4, -2, 2, 3, 2, 3, 2, 1, 1, -3, 0, 1, 3], [-15, 3, -5, 5, -1, 1, 7, 1, 2, 1, -1, 2, -3, 4], [-4, 7, -3, 2, 0, 1, 2, -3, 4, 2, 3, -1, 3, 3], [0, -4, 4, -4, 0, -1, -4, 4, -2, -4, 2, 1, -1, -1], [-8, 8, 1, 0, 0, 2, 2, 0, 3, 4, 1, -2, 1, 0], [7, -7, 7, -2, 1, -2, -4, 1, -3, 2, -5, 3, 1, -1], [-5, -5, 1, 0, -1, -2, -6, 1, 0, 1, 4, 0, 2, -1], [5, -3, -1, 1, -3, 0, 4, -3, 1, 3, 4, -3, 1, 1], [-5, 2, -4, 0, -1, -2, 3, -4, -1, 2, -1, -1, 1, 0], [-1, 3, -1, 1, 0, 1, 0, 2, -3, -2, 2, 4, -1, 0], [3, -2, 3, 3, -2, -2, 2, -2, 0, 1, -5, 2, -2, 2], [1, 5, 2, -1, 0, 3, 0, -2, 1, 3, 1, 0, 1, -4], [-2, -5, 2, -1, 1, -4, -5, 0, -3, -3, -8, 0, 2, 0], [3, 1, -4, 1, -4, 0, 1, -4, -2, -2, 1, -2, 3, 0], [1, -5, -2, -2, -1, -1, 1, 1, -4, 1, 4, 1, -3, 0], [6, 2, 2, 1, 3, 2, 0, -4, 1, 4, -1, -2, 5, -5], [7, -2, -1, -1, -1, 1, -1, 1, -2, 2, -4, 1, 5, 3], [-13, 4, -2, 4, 3, -2, 1, 1, 2, 1, -9, 1, 1, 0], [3, 5, -6, 0, -2, 0, 3, 1, 4, -2, 3, -1, -1, 6], [-12, 7, -2, 2, 2, 2, -2, 5, 2, 0, 3, 3, -1, 4], [-1, 8, -2, 3, 0, 1, -1, 0, 0, 2, 1, -1, 4, 1], [-20, 9, -6, 4, 1, 0, 0, 1, 5, -3, -3, -1, 0, 1], [13, -2, 2, -4, -2, 3, -1, -5, 0, -1, -3, -2, 5, -1], [13, 0, -1, -1, 1, 4, 0, -1, -1, -2, -1, 0, 1, 2], [10, -9, 6, -1, -1, -6, -1, -4, -6, -2, -5, 1, -3, -2], [-22, 2, -6, 1, 2, 1, 0, 4, 3, -2, -8, 2, 0, 5], [-11, 6, -3, 5, 2, 3, 2, -3, 3, 5, 0, 0, 1, 0], [7, 2, -1, 1, 1, -1, 1, -6, -1, 4, -3, 2, 5, -2], [13, 0, 3, -1, -4, 1, 0, -4, -2, -1, 2, -2, -2, 2], [0, -7, 1, -2, 0, -5, 0, -4, -3, -2, -9, 1, 4, -1], [10, -10, 2, -3, 0, -1, -2, -4, 0, -3, -1, -1, 1, -5], [7, -7, 5, -1, -1, 1, -3, -1, 0, 3, 5, -2, 1, -4], [9, 2, 3, -1, 0, -1, 3, -3, 1, 2, 3, -4, -1, -2], [5, -1, -1, 2, -2, 0, 1, 1, -1, -1, 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[-5, 8, -6, 2, -1, 4, 3, 3, 0, 2, 11, 1, -2, 2], [26, 4, 8, 0, 0, -3, 1, 0, -3, 4, -3, 1, 0, -3], [-13, 3, -6, 1, -3, 3, 5, -2, 2, -2, -6, 0, -2, 2], [15, -9, 0, -4, -3, -4, -1, -5, 0, 2, 8, -4, 1, -1], [7, 2, -2, 0, -7, -4, -4, 1, -3, -4, -6, 3, -1, 4], [-4, -13, 8, 1, 2, -4, -5, 0, -2, -5, -2, 4, 0, 3], [-15, 13, -5, 5, 3, 4, 2, 8, 4, -3, 1, 10, -5, 8], [9, -5, 3, 4, 4, 3, -1, 3, 3, 0, 10, 2, 1, -1], [15, -5, 1, 0, 6, 1, -3, 0, 2, 1, 7, 3, 2, 2], [-10, -10, 3, -3, -3, -2, -3, 2, -6, -8, 2, 1, -1, 1], [3, -4, 2, -5, -1, -5, -5, -2, -4, -1, -8, -2, 3, -8], [6, 8, -3, 4, 6, 5, 1, 9, 5, 4, 1, 2, 3, 2], [-4, -3, 3, 0, 1, 1, -5, -2, -2, -5, -6, 1, 1, 3], [16, 1, 3, -2, -4, -2, -7, 4, -4, 1, 7, -1, 2, 1], [-14, 13, -2, 5, 2, 3, 6, -3, 4, 4, 4, 0, -9, 5], [8, 5, -5, -2, 1, 3, -2, 7, 8, 0, 2, 0, -5, 4], [-10, 21, -10, 4, 5, 9, 8, 7, 10, 8, 0, 3, -4, 2], [-22, -1, 6, -7, 3, 2, 0, -3, 2, 0, -3, -2, 2, -7], [20, -12, -2, -1, -6, -3, -3, -1, -9, -5, 2, 1, -2, 2], [9, 2, 4, 2, 1, 2, 1, 5, 4, -1, 8, -1, -7, -2], [-9, -11, -1, 0, 2, -3, -1, 1, -5, -3, -6, 0, 4, 1], [-14, 9, -17, -1, 1, 5, 7, -1, 6, 0, -3, -1, -3, 2], [-7, -1, 1, -7, -3, 4, -2, 0, -4, 3, 9, 0, -1, -2], [-2, 15, -5, 6, 5, 7, 2, 4, 10, 7, 5, -3, 2, 3], [1, -5, 0, 2, 4, -2, 1, 4, 0, -4, -6, 0, -3, 3], [-3, -3, -11, 1, -1, -1, -2, 1, -3, -4, -5, 3, 0, 5], [3, -10, -2, -9, -6, -1, -6, 3, -8, -1, -3, -6, 0, -6], [-11, 7, -5, 3, 3, 7, 1, 4, 8, 1, 4, -1, -3, 3], [-20, 6, -7, 5, -3, 0, 3, -3, 6, 1, 2, -3, 4, 0], [6, -4, 9, -5, 3, 1, -9, -2, -7, 4, -2, 1, 4, -6], [0, 11, -10, 1, 0, 0, 7, 0, 6, 4, 10, -4, -1, 3], [3, 5, 0, 0, 2, 2, 0, -3, 5, 1, -11, -4, 6, 1], [-9, 10, -3, 1, -3, 1, 7, 1, 5, 2, 15, -2, -7, 3], [-8, 11, 2, -4, 6, 0, 1, 0, 5, 0, 8, 1, -2, -1], [-11, 16, -10, 9, 4, 6, 1, 2, 11, 4, 2, -1, -3, 10], [-4, -17, 5, -6, -1, -1, -6, 5, 0, -2, -4, 2, 3, -4], [4, 1, 6, -2, 3, -1, -5, -6, -1, 7, -3, -3, 6, -3], [-7, -6, -4, 0, -2, -4, -2, 7, -5, -5, -3, 6, -2, 9], [-6, -4, 11, -5, 2, -1, 1, 0, -1, -3, 4, 0, 1, -3], [7, 6, -4, 1, -5, 1, 1, -1, 3, 0, 0, -1, -2, 7], [-4, 6, -14, 6, 5, 2, 5, -3, 10, 1, 3, -4, -2, 2], [1, -1, 2, -1, -2, -2, 3, -3, 3, 1, 5, 2, -3, -2], [5, -9, 0, 2, 1, 3, -2, -2, 4, 6, 10, 0, 1, 3], [10, -5, -1, -2, 0, 3, -2, 3, 4, -6, -3, -3, -3, -3], [13, 8, -1, 1, 7, 3, -1, 4, 3, 7, 7, 0, -4, 2], [7, -4, 2, 6, 6, -3, 1, -10, -1, 8, -9, 2, 9, -5], [-20, 5, -5, 0, -2, -4, -4, 0, -3, -10, -7, 3, -2, 7], [19, 5, 0, 1, -2, 0, 0, -5, -6, 3, -2, 4, -2, 2], [-5, -20, 0, -4, 0, 1, 2, -5, 1, -1, -9, -3, 3, 0], [9, -3, 2, 6, 4, -1, 1, 3, 2, 4, 3, 8, 0, 5], [-25, -7, -5, -2, -3, -1, 6, -3, 2, -4, -8, 3, 4, 4], [-15, 4, -7, 3, 9, 5, 4, -1, 10, 3, 1, 1, -3, -1], [-9, -10, 0, -5, -8, -6, 0, -1, -10, -5, 5, 3, -1, 3], [2, 1, 0, 2, -2, -5, -1, 2, -1, 4, -6, -2, 5, -6], [24, -13, 5, 0, -4, -2, -2, -3, -6, 2, 3, 0, 2, -4], [8, 3, -4, 2, -1, -2, -1, -4, -1, 0, -11, -6, 2, -2], [-22, 0, -6, -3, -1, 2, 3, 0, 2, 0, -8, -6, -4, -1], [-4, -5, -4, 1, -1, 4, -1, 5, -3, -2, -2, -6, -3, -3], [-19, 10, -2, -5, -2, 3, 4, 2, 5, -1, -2, -3, -6, 1], [-4, 7, -3, 3, 4, 1, -1, 2, 7, -3, -1, -3, -4, -2], [17, -9, 11, 1, -4, -5, 1, -10, -4, 4, 0, -5, 0, -2], [29, -14, 8, -1, -1, -2, -5, 3, -3, -1, 8, -1, -5, -3], [14, -14, 4, -6, -10, -1, -4, -2, -3, -3, 13, -2, -2, -3], [16, -13, 9, -3, -2, -3, -4, -3, -7, -1, 1, 2, 1, -5], [24, -12, 9, -4, 3, -4, 1, -5, -9, 4, -2, 2, 2, 0], [-10, -7, -2, 2, 3, 1, -2, 6, 6, 2, -7, 4, 1, 2], [-14, 3, -1, 1, 0, 1, -6, 2, -3, -3, -3, 2, 7, 0], [9, -25, 3, -4, -5, -4, -4, -4, -7, -2, -1, -1, -3, 6], [18, 0, 2, -2, -2, -4, -7, 1, -2, -5, 10, 5, -2, 5], [31, 2, 0, -4, -4, -2, -5, 4, -3, -7, -3, 1, -3, 0], [25, -7, 1, -4, 3, 5, -4, -2, 1, 3, 7, -8, 7, -7], [16, -8, 4, -2, -4, 2, -5, -4, -1, 5, -6, -6, 8, -6], [-14, -1, -1, -4, -1, 1, -5, 6, -6, -7, -2, -3, -5, -3], [-5, -7, 2, 0, 6, -2, -4, 4, 2, -3, -3, -2, 3, -1], [6, 11, -1, 6, 2, 4, 2, 5, 4, -6, 7, -2, -4, 2], [10, -2, 5, 6, -4, -5, 3, -2, -3, 6, 6, -5, -1, 3], [-18, 1, 0, 0, 1, -1, 1, -2, -3, -2, 1, 3, -3, 3], [12, 6, -5, 0, -4, 4, 0, -1, 6, -3, -1, 1, -4, -1], [6, 1, 1, -2, -4, -6, 1, -3, -11, 2, 12, 4, -3, 4], [14, -16, 10, 3, 4, -10, -1, -7, -4, 1, 4, 2, 5, -1], [-30, -2, -2, 2, 3, -6, -2, 4, -3, -3, -4, 6, -5, 7], [-21, 20, -12, 4, 0, 5, 3, 2, 7, 2, -2, 0, -1, 9], [-17, 0, -8, 1, -3, 5, 0, 11, -2, -9, -2, 1, -4, 1], [26, -12, -3, -1, -2, -2, 0, 0, 2, -1, 5, -1, 3, 0], [-14, -25, 9, -4, -2, -1, -5, -5, -1, -5, -5, -5, 3, -1], [-19, 3, 0, 4, -1, -3, 1, 4, 1, -3, -3, 6, -3, 11], [26, -12, 8, -2, -2, 0, -5, -6, -9, -5, -3, 0, 4, -4], [-4, 3, 4, -1, 2, 1, -1, 0, 10, 0, 0, -3, -2, 2], [1, -17, 8, -5, 3, -10, -8, -3, -10, -1, -17, 1, 4, -7], [-6, -6, 6, -1, -2, 5, -4, -6, -3, 8, 0, -2, 8, -9], [1, 8, -4, -1, -11, -2, 0, 3, -2, -8, -3, 4, -10, 6], [-19, -8, -1, -3, -1, 0, 1, -2, -1, -1, -1, -1, -7, 1], [-30, 1, 10, -4, -1, -4, -5, 1, -6, -2, -10, 5, 0, -3], [-25, 8, -11, 2, -2, -3, 1, -1, -2, 2, -1, 1, 2, 6], [-25, -3, -2, -3, -1, 3, 3, 0, 4, -3, -6, -3, 4, -5], [-19, -1, -14, 1, 3, 3, 3, 6, -1, -6, 1, 3, -5, 9], [-3, -4, 2, 2, 1, -1, 0, 5, 5, -2, -4, 4, -2, -1], [18, -5, 4, -4, -2, 1, -7, 5, -5, 0, 8, -2, 7, -5], [36, -9, 3, -2, 0, 5, -3, -6, -2, -1, -1, -6, 7, -6], [-3, -16, 1, -6, -2, -5, -4, -1, -6, -3, -4, -5, 6, -10], [-20, 10, -1, -3, 6, 5, -3, 8, 1, 2, -5, 2, 0, -3], [15, -11, 5, -6, -4, -1, -4, 4, -5, 0, 2, -3, 2, -6], [22, 0, 4, -4, -3, 0, 4, -9, -2, 0, -4, -2, 8, 0], [0, 23, -2, 4, 1, 2, 1, -2, 5, 4, 1, -2, -2, -2], [4, 14, 3, 9, 11, 7, 7, -10, 11, 14, -4, -2, 5, -1], [-11, 15, -5, -1, -3, 2, 7, 1, 4, 7, -9, -1, -2, 3], [-2, 4, -6, 0, 8, 3, 0, 6, 6, 0, -10, 5, -3, -1], [-24, 7, 1, 3, -4, 0, 1, -2, 1, -1, 7, -4, -5, 1], [6, 12, 3, 3, 5, 3, 1, 0, 3, 8, 2, -3, 0, 4], [5, -13, 8, -9, 3, -1, -10, 6, -1, -3, 2, -1, -1, -9], [-18, 10, -7, 7, 3, 0, 6, 4, 4, 2, 6, 0, -10, 1], [2, 1, -2, 4, 11, 3, 1, 4, 2, 3, 0, 0, 4, -1], [-25, 17, -3, 8, 4, 7, 5, 1, 9, 9, 12, 2, -4, 3], [-27, 11, -6, 7, -5, 1, 7, -1, 11, 10, 12, -2, -3, 4], [-18, 3, 3, -6, -1, -3, 2, 6, -3, -2, 4, 7, -5, 7], [21, -7, -4, -1, -2, -3, -3, -13, -3, 4, -3, -1, 9, -3], [-25, 17, -4, -1, 5, -1, -3, 2, 0, 2, -7, -1, 3, -1], [10, 3, 0, 5, -5, -5, 0, -3, 2, -3, -5, -1, 0, 4], [3, 15, -6, 3, -2, 8, 1, 7, 1, 4, 2, 0, 0, 0], [-24, 3, 2, 1, 2, -4, 2, 2, 5, -1, -6, -4, 3, -7], [0, 15, -11, 0, 4, 1, 3, 2, 10, 1, -1, 4, -1, 4], [6, -13, 11, -4, 4, 4, -7, 7, -4, -6, 7, 3, -1, -2], [-16, 15, -6, 0, -5, 5, 1, 6, 3, -8, -3, 1, -5, 11], [20, 5, -7, -4, 9, 7, 4, 0, 6, -2, -12, -2, 7, -1], [-14, 15, -6, 1, 4, -3, 3, 1, 3, -3, -7, 2, -3, 1], [21, -7, 5, -3, -2, 1, 3, -8, -3, 3, 12, -7, 6, -7], [-28, 8, -8, 1, -1, 3, 7, -7, 3, -1, -10, 0, 6, 5], [-26, -10, 0, -1, -3, -6, 5, -5, -1, 4, 11, 5, 0, 0], [-30, -8, -5, -2, 1, -1, -4, -1, -2, -5, -4, -5, 1, -1], [0, 4, -13, -4, -5, 1, -2, 8, -5, -4, 0, 0, -2, 0], [6, -9, -1, 7, 3, 3, 7, 0, 5, 5, 2, 0, 2, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8008_2_a_y();".
// 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_8008_2_a_y(:prec:=14)
    chi := MakeCharacter_8008_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_8008_2_a_y();".
// 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_8008_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-128, 1032, 538, -6417, -2804, 7275, 2996, -3378, -1306, 750, 273, -78, -27, 3, 1]>,<5,R![360, 2934, 2635, -16489, -2561, 21765, 500, -10984, -472, 2321, 229, -201, -28, 6, 1]>,<17,R![-1063168, 5080832, -4912892, -4563644, 3832443, 2366025, -699242, -532476, -8372, 34742, 3506, -811, -113, 6, 1]>],Snew);
    return Vf;
end function;