
// Make newform 4008.2.a.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4008_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_4008_2_a_m();".
// 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_4008_2_a_m();".
// 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 := [12144, -72184, 112850, -14532, -72240, 26911, 16968, -7582, -1879, 901, 99, -49, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1211669325949820, -4373968597429142, 1650853947740980, 1724063123712248, -1037037612348531, -196205928405368, 201417980697138, 5016057310831, -16627326024925, 295838273673, 576861363065, -12640530894, -6207348777], [3605634089719788, -3707736754746094, 598916455112988, -993417294727940, -714691137288825, 1012871433621172, -90263735212158, -193085396549923, 48247617829913, 14015645726951, -4532098501229, -351643713690, 126516435801], [-261781210799648, 719430531278666, -61985566741568, -552823809736903, 129168526434186, 149114772131121, -42939671149010, -18714943276923, 5918945839452, 1102737562371, -363301449706, -24439858116, 8155241826], [-9603799562310, 1829362384230014, -2780474603245950, -967840084997036, 1772258179746447, 236986143576626, -406570644479846, -30214877144177, 43420560752365, 1857640717639, -2192557864275, -42904580362, 42253210649], [-455409403401910, 1829362384230014, -2724748902766000, -967840084997036, 1772258179746447, 236986143576626, -406570644479846, -30214877144177, 43420560752365, 1857640717639, -2192557864275, -42904580362, 42253210649], [-2901404480931816, 10417867358098218, -6707972583182866, -5762733210647085, 4728856566083805, 1217996115089036, -1131795821929059, -121899916781369, 127066606088584, 5771355044103, -6791649579897, -105435570025, 138385219703], [-629228580752360, 6347787481933558, -5320985228785100, -4408101894281592, 3686070397711199, 1109495790415652, -912294162144422, -129559753690019, 104000386651945, 7117312024823, -5533191476745, -148746957474, 111023509273], [6802024064231936, -25352491816897518, 14675109344372636, 16707098673252820, -11231871461142475, -4297098906328266, 2761108370208174, 524566352898419, -309203328646339, -30174775561853, 16238231918487, 655210507370, -323942240303], [-1313465392184157, 7530986189108326, -7596810398586632, -3276596851987855, 5522030332256130, 348943217873482, -1435122721494978, 10485608743617, 173349897433318, -3015289235129, -9810853937919, 101272023955, 208422301596], [-20950363454455596, 67210663867287138, -31856351450553896, -44471075457654780, 26745397139049795, 11473935236033836, -6930554296860074, -1397243777337779, 807810238699029, 79678278406623, -43690158347557, -1710596413190, 889692540773], [-2219186123392628, 10312125439960266, -10980877487070868, -4406202874244528, 7475418034130291, 558179494985656, -1812887391077730, -13254732421323, 205661421513157, -1496522255749, -11028950583521, 63390661414, 224267911141]];
    Rf_basisdens := [1, 1, 111451400959900, 557257004799500, 27862850239975, 55725700479950, 55725700479950, 139314251199875, 111451400959900, 278628502399750, 139314251199875, 557257004799500, 111451400959900];
    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_4008_a();"
function MakeCharacter_4008_a()
    N := 4008;
    order := 1;
    char_gens := [3007, 2005, 1337, 673];
    v := [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_4008_a_Hecke(Kf)
    return MakeCharacter_4008_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], [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, -1, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0], [2, 0, 1, 0, 1, 1, -1, -1, -1, 1, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, -1, 0], [0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1], [0, 0, 0, -1, -1, -1, 0, 1, 1, -1, 0, -1, -1], [-1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, -1, -1], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [2, -1, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0], [3, 0, 1, 0, 0, 0, 0, -1, -1, 1, 0, 1, 1], [2, -2, 0, 0, 1, 1, 1, -2, -1, 0, 1, 1, 0], [1, -1, 0, 0, 0, -1, 0, 0, 1, -1, -1, -1, 0], [4, -1, 1, 0, 1, 0, 0, -1, 0, 1, 0, 0, 1], [1, -1, 0, 0, 1, -1, 0, 1, 0, 0, -1, -1, 0], [4, -2, 0, 0, 1, -1, 0, -1, 0, 1, 0, 1, 0], [3, 0, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0], [3, -1, -1, 0, -1, 1, 0, -1, 0, -2, 2, -1, 0], [0, 0, -1, 0, -1, -1, 0, 1, 0, 0, -1, -1, 0], [1, 0, 1, 0, 0, 1, -1, -1, 0, 1, 0, 1, 1], [1, -1, 0, 0, 0, 1, 0, -1, -1, 1, 1, 0, 1], [4, 0, 1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0], [2, 1, 0, 2, -1, 1, -1, 0, -2, 1, 1, 1, 0], [-1, 0, -1, 0, -1, -2, 1, 2, 1, 0, 1, 0, -2], [3, -2, 1, 1, 1, -1, 0, 0, -1, 1, 0, -1, -1], [0, 1, -1, 0, -1, -1, 1, 2, 2, -1, 0, -1, -2], [-1, 0, -2, 0, 1, 0, 0, 3, 1, 2, 0, 0, -1], [1, 0, -1, 0, 0, 1, 0, -2, 0, -1, 0, 1, 2], [3, 0, 1, 0, 1, -2, -1, 1, 0, 0, -1, 0, 0], [2, 0, 0, -2, 0, -1, 0, 0, 2, -1, -1, -1, 1], [3, -2, 1, 0, 1, 1, 0, 0, 0, 1, -2, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -1, 0, 2, 0, -1, -1, -1, 1], [2, -4, 2, 2, 0, 2, -1, -3, -2, 0, -1, 0, 0], [0, -1, 0, -2, 1, -2, 0, 1, 3, 1, -1, -1, 0], [-3, 0, -2, 0, -2, 0, 0, 0, 0, 0, 1, 1, 1], [3, 0, 0, 0, -1, 0, 0, 2, 2, -1, 1, -2, 0], [-3, 0, 0, 1, 0, 4, -1, 0, -1, 1, 1, 2, 0], [-1, 1, 0, 0, 0, 3, 1, -1, -1, 1, 0, 1, 0], [3, -3, -1, 2, 1, 0, 0, -1, -2, 0, 1, 0, 1], [-1, 1, 1, -2, 1, 1, 1, -3, 0, 2, 1, 1, 0], [0, -2, 1, 0, -1, 0, -2, 0, -1, -2, 0, 0, -1], [5, 0, -1, 0, 0, -1, 0, 3, 1, 0, -1, 0, 0], [3, -1, 2, 0, 1, -1, 1, -4, -1, 0, 2, 1, 2], [-4, 0, 1, 0, 0, 1, 1, -2, 1, -1, -1, -2, 0], [4, -1, 0, 0, 1, 1, -1, 1, -1, 2, -3, 1, 0], [-1, 0, -1, 0, -2, -2, -2, 2, 1, -1, -2, -2, 0], [0, 0, 0, -2, 2, 2, 0, -3, 1, -1, 0, 0, 2], [0, 0, 0, 0, -1, -1, -1, 0, -2, 0, -2, 1, 1], [-1, 0, 1, -2, -1, -2, -2, 1, 0, -2, -3, 0, 1], [-1, 3, 0, -2, -1, 0, 1, 0, 0, 1, -1, 0, 3], [3, 0, 0, -2, -3, 3, 0, 0, 0, 0, -1, -1, 0], [-1, 0, -1, 2, 1, 2, -1, 0, 1, 2, -1, 2, 0], [0, 0, 1, 0, -2, 0, -2, 0, 1, -2, -1, -2, 0], [-1, -2, -1, 2, 2, 0, 2, -1, 1, -1, 2, 0, -2], [2, 0, 1, 0, 1, -2, 2, -1, 0, -5, 0, 0, -1], [-7, 1, -1, -2, 1, -4, 1, 2, 2, -1, -2, -2, 1], [6, -1, 0, 0, 1, 2, -1, 1, 1, 1, 2, 0, -3], [-6, -3, -1, 0, 4, -1, -1, 1, 0, 0, 2, 1, -2], [-2, 0, -1, -1, 1, 1, 0, -3, 0, -1, 3, 3, -1], [3, 0, 0, 0, -2, -1, 1, 0, 0, 1, -1, -1, -1], [-5, 2, -2, 0, -1, -1, -1, -1, 0, -2, 2, 1, 0], [6, 2, 0, -2, 2, 0, 0, 0, 2, 0, 1, 0, 1], [3, -1, 2, 0, 4, 2, 0, -4, -3, 3, 2, 2, 1], [0, -1, 0, 0, 3, 1, 1, -2, -3, -2, 1, 1, 2], [-5, 2, 0, 0, 2, -2, 0, 1, 2, 1, -1, 0, 0], [4, 2, -1, 0, 0, 0, 2, 1, 1, 2, 0, 0, -1], [-3, 1, -1, 4, -2, -1, -2, 4, 1, 0, -1, -1, -3], [-6, 3, 1, 0, -1, 1, -2, 0, -2, 2, -1, 0, 1], [-5, -2, -1, 3, -3, 3, 1, 2, 0, -2, 1, -1, -3], [5, -1, -1, -2, 0, 1, 0, -2, 0, -1, 0, 0, 1], [1, 3, 0, 0, -1, -2, -1, 1, 0, -3, 1, 0, -1], [4, 1, 1, 0, -1, -3, 0, 0, 2, 4, -3, -2, -1], [5, -1, 1, 2, 0, 2, -1, -3, -4, 0, -2, 3, 2], [5, -1, -1, 0, 3, -4, 2, 0, 2, 0, 3, -2, -1], [-14, 3, -2, 2, -2, -1, 3, 1, 1, -1, 5, -1, -2], [3, 0, 3, 0, -3, -1, -1, -2, 1, -3, 0, 0, 2], [-3, 1, -1, 2, 0, 0, -1, 3, 1, -1, 0, -2, -1], [5, 0, 1, -2, 2, -1, -1, -3, -1, 2, 1, 3, 2], [0, 1, 0, 0, 2, 1, -1, 0, 1, 1, -1, 1, -2], [10, 2, -1, 0, 0, 0, 1, 2, -1, 2, -2, 0, 2], [-3, 1, -2, -2, -3, -2, 1, 2, 3, -4, -1, 0, -1], [0, -1, -2, 0, 3, -4, 2, 2, 0, 6, -3, 0, 1], [0, -1, 3, -2, -1, 3, -1, -4, 0, -2, -2, 3, 2], [1, -5, 1, 4, 1, 2, -1, -3, -4, 4, -2, 2, -1], [-9, 4, 3, -2, 1, -3, -1, 2, 2, 1, -4, -1, 1], [-2, 2, 1, 2, -2, -3, 0, 2, -1, 1, 0, 0, -2], [-4, -1, -3, 3, -2, 1, 2, 1, 1, -2, 4, -1, -2], [4, -3, 0, -2, 5, 1, -1, -2, -1, 2, -3, 2, 1], [1, -1, -1, 0, 0, -1, 0, 1, 0, 3, -4, 1, 2], [-4, 1, -1, -2, 0, -3, 1, 5, 3, -1, 0, -1, -2], [7, 0, 1, 2, -1, 3, -2, 2, -1, 2, -3, -1, 1], [-3, 0, -1, 0, -2, 0, -2, 2, -3, -3, -1, 0, -1], [2, 0, 0, -2, 1, -2, 1, 2, 1, -1, -4, 0, -1], [-5, -1, -1, 0, 1, 5, -3, 0, -2, 3, -1, 1, 2], [-3, -3, -2, 0, 2, -3, 4, 2, 1, 3, -1, -1, 0], [-4, -1, -1, -2, -1, 1, -1, -1, 1, -3, 2, 1, 0], [3, 0, 2, -2, 4, -4, 2, 0, 1, 2, 2, 0, 1], [-3, 1, -2, 0, -3, -5, 3, 5, 1, 0, -2, -4, -3], [4, -1, 0, 0, 0, -2, 2, 1, 1, -1, 2, 0, 0], [0, 2, -2, -2, -4, -3, 2, 5, 3, 0, -3, -3, -3], [-8, -2, 0, 2, 1, -2, -3, -2, -2, -3, 1, 2, 2], [2, 1, 1, -2, 1, -3, 0, 6, 2, 1, -1, -1, 0], [-7, 1, -2, 2, -3, 3, -3, 1, 0, -2, 0, 1, -2], [0, 2, -1, -2, -1, 1, -4, 1, 1, -1, -4, 0, 0], [-8, 1, -2, 0, -6, 1, -1, 4, 3, 0, -1, -1, -2], [6, 0, 1, 1, -1, 2, -2, -1, -2, 2, -4, 4, 2], [-15, -3, 0, 2, -1, -1, 1, 2, -1, -4, 0, -1, -2], [4, -3, 1, 4, 1, 5, -2, -6, -5, 1, 1, 1, 4], [3, -1, -1, 1, 2, 1, 2, 0, 1, 1, 1, 1, 0], [-2, -5, -1, -2, 1, 5, -1, -2, -2, -4, 0, 1, 2], [-2, -1, 2, -2, 3, -2, 2, 0, 3, 5, -1, -2, 1], [-9, 1, 2, -2, -1, -5, 1, 4, 3, -1, 1, -3, -1], [4, -1, 2, -3, 3, -3, 0, -1, -3, -1, -1, -1, 2], [3, 2, -4, -4, 0, -1, 1, 0, 2, 0, 1, 1, -1], [3, -5, 0, -2, 4, 2, -1, -2, -1, 0, -3, 2, 3], [8, -3, 1, -1, 3, -5, 1, -3, -2, 1, -1, -1, 4], [-2, 3, 1, -1, -2, -4, 0, 1, 1, -3, 3, 0, -1], [-9, -4, 4, 0, -1, 4, -1, -1, 0, 1, -2, -2, -2], [6, 2, -1, 2, -5, 4, -3, -3, 1, 0, 1, 2, 3], [-5, 0, 0, 2, 1, -1, 1, -3, 0, -4, 3, 1, 1], [-7, -5, -2, 2, -1, -4, -2, 2, 2, -1, 0, -2, -3], [5, 0, 0, 2, 0, 3, -4, 4, -2, 4, -4, 2, 0], [-7, 2, -5, -2, -2, -6, 2, 8, 4, -4, -1, -4, -3], [-6, 3, 0, 0, 2, -1, 1, 0, 1, 5, 5, -1, -2], [1, -6, 0, -4, 0, 0, 2, -4, -1, -4, -3, 0, 4], [-1, 3, -1, 2, -4, 2, -3, 4, 2, 0, -4, 2, -1], [-11, -1, 0, 2, 2, 1, 0, -2, -3, -2, -1, 3, 2], [6, 3, -2, 0, -5, -3, 3, 3, 3, -4, 1, -5, -2], [-4, -3, -2, 0, -5, -2, -2, 5, 2, 0, -4, 0, -2], [7, -5, 2, 5, 0, 2, -2, -4, -4, 0, -3, 0, -1], [-4, 2, 0, 1, -6, 1, 0, -1, -3, 3, 2, 3, -1], [-3, 3, -3, -4, 1, -6, 3, 5, 6, -1, 0, -2, -1], [-6, 6, 0, -2, -1, 1, -4, 0, -1, 0, 0, 3, 4], [-8, -4, 1, 1, -3, -3, -2, 1, 2, -1, -5, 1, -1], [0, -1, 3, 0, 1, 1, 2, -1, 1, 5, -1, 1, -2], [3, 1, -2, 1, 6, -2, 1, 2, -1, 1, 0, 0, -1], [5, -5, 1, 0, 1, 1, 2, -4, 0, -3, 3, 3, -1], [-13, -1, 2, -2, -4, 2, -1, 2, 3, 1, -7, -3, 0], [15, 0, -1, 2, -1, 6, 1, -1, -1, 1, 3, 3, -1], [-2, -3, 3, 2, -1, 5, 0, -7, -4, -1, 3, 1, 2], [6, -4, 2, -2, 3, -3, 1, 1, -2, 1, -2, 1, -3], [-3, 0, 2, 2, 1, 0, 0, 3, -2, 3, 1, -4, 0], [3, 3, 1, 0, 4, 0, -1, -4, -2, 4, -2, 4, 3], [-5, 2, -3, -2, -5, 1, 1, 3, 0, -1, -2, -3, -1], [1, -4, -3, 0, 0, 0, 3, -2, -3, -7, 3, 0, 2], [1, 0, 2, 0, 4, 0, 0, -3, -4, 3, -1, 2, 4], [4, -1, -1, 4, -1, 2, -1, -3, -2, 1, 5, 4, 3], [-12, 1, 0, 4, 2, 3, -1, -1, -1, 1, 1, 1, 2], [-12, 3, -4, -4, -4, 0, 3, 6, 7, -2, 4, -4, -7], [-3, 7, 1, 0, -4, 2, 1, 2, 2, -5, 6, 0, -3], [-5, -6, 2, 0, -4, 1, 1, -2, 0, -6, 3, 1, -3], [-9, 4, 2, 5, 3, -4, 2, 2, -3, 1, 4, 0, -2], [-6, -4, -1, -4, -1, 2, -1, 2, 5, -6, 2, -5, -1], [-5, 0, 0, 0, 2, 2, -4, 2, 1, 6, 1, 0, 0], [-5, -2, 1, 2, -3, 1, 0, -3, -1, -2, 3, -1, 1], [-9, 3, -1, 2, -3, 0, 2, 2, 0, 6, -4, -2, -4], [-13, 7, 2, -4, 1, -6, 1, 2, 2, 1, 1, 0, -3], [-2, -7, -2, -2, -2, -1, -1, 4, 5, -6, -1, -5, -2], [-9, -6, 5, 0, 1, 2, -3, -2, 0, 0, -5, -2, 0], [5, 1, 3, 0, 4, -6, 4, -2, 2, 6, -1, -1, 0], [0, -4, -1, 0, 2, -5, 4, -2, -1, -4, 6, -2, 0], [-16, 2, -4, -2, -3, -2, 3, 4, 6, 2, 3, -4, -6], [7, 5, 2, 2, 2, 3, 0, -2, 1, 1, 6, 1, -3], [-4, 0, 2, 2, 1, 4, -1, -4, -2, -4, 3, 2, 0], [-12, 2, 0, -4, -4, -1, 0, 6, 4, 4, -1, -3, -3], [1, 6, -1, 4, 1, -2, -1, 2, -1, -2, 1, 1, -1], [-12, -3, 1, 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2, 3, 0, 2, 7, 0, 8, 1, 0, -5], [-16, -4, 0, 6, 1, -1, 4, -4, -3, 2, 5, -1, 3], [7, -2, 3, 2, 4, -3, 3, -3, -6, -7, 8, 1, -3], [-16, -1, 0, 4, -1, -4, -1, 4, 1, 6, -2, 2, -3], [-10, 10, 1, 0, -5, 7, -1, -2, 1, 3, 1, 1, 3], [6, 11, 0, 2, 1, 10, -3, -7, -1, -1, 6, 2, 5], [0, -7, -3, 0, -2, -9, 6, 6, 4, -9, 5, -2, -5], [25, 2, -1, 0, -2, -1, -5, -4, -5, -2, -2, 7, 5], [10, -1, 4, 4, 4, 5, -5, -7, -3, 6, -5, 2, 5], [6, 2, -1, 4, 3, -1, -2, -1, -3, 9, -6, 1, 5], [-13, 7, -1, 2, -7, -2, 1, 6, 1, -9, 0, -2, 1], [10, -2, -4, 4, 1, -1, 5, 5, 4, 1, 8, -2, -4], [18, 0, -5, 1, -4, 3, 6, 3, 4, 3, 1, -3, -3], [10, -1, 5, 0, 6, -2, -3, 1, 2, 5, -1, 4, 1], [8, -2, 5, -2, -3, -9, 0, 6, 3, 3, -8, -3, -3], [-6, 3, 2, -8, 0, -4, -5, 6, 6, 4, -4, 0, -1], [1, -5, 0, -2, -1, 5, 1, 4, -1, 5, -6, 1, -2], [9, -3, -2, 4, 6, -2, 4, 0, -7, 5, 2, 0, -1], [6, 6, 5, -6, 1, 2, 3, -3, 1, -4, -4, -6, 2], [-6, 2, 1, 4, 0, -2, 3, 8, 3, 0, 2, -2, -2], [29, -6, 7, 4, 5, 5, -9, -8, -10, 3, -7, 7, 6], [-1, 1, 1, 2, 2, -1, 0, 4, 1, 10, -4, 1, 2], [1, 11, -1, -2, 2, -8, -1, 6, 2, 4, -2, 0, 1], [10, -6, 1, -6, 0, -1, 3, -12, -3, 3, 7, 1, 1], [7, 3, 1, 0, 1, -7, 1, 1, -4, 5, -5, 1, 2], [13, -1, 0, 4, -2, 3, 0, -9, -3, -7, 3, 3, -2], [-40, 2, -1, 0, -2, 5, -4, -1, -1, 7, 0, -1, -1], [9, 0, 4, 4, 5, -1, -6, 3, 0, 8, -4, 5, 1], [-4, 0, 1, 8, 4, 1, 0, 1, -5, 11, -2, -1, -1], [1, -5, -2, 2, 6, 0, 3, -1, -5, 4, -5, 0, -1], [-10, 7, -5, 2, -4, -11, 1, 8, 3, -1, -3, -1, -5], [18, -4, 1, 4, 3, 5, -2, -10, -6, 0, 4, 5, 7], [10, 0, -9, 6, -4, -1, 2, 11, 3, -1, -1, -5, -6], [18, -2, -2, 2, 4, 3, -4, 2, 1, 0, 1, 3, 5], [-14, 1, 4, 6, -5, -6, -4, 3, -5, 2, -5, -4, -1], [-11, 8, 3, 0, -5, 7, -4, 5, 1, 4, -2, 1, 0], [-9, -3, 0, -4, -3, 5, 7, -8, -4, -6, 4, -3, -2], [3, 12, -2, 0, -2, -5, -4, 16, 8, 5, -4, -4, -6], [22, 1, 1, -4, 9, 1, 2, -5, 0, -6, 5, 7, 4], [6, -1, 3, 2, -1, 10, -6, -6, -5, -4, 2, 4, 1], [16, 7, 0, 4, 4, 5, -1, -4, -7, 8, 1, 1, 2], [-5, 6, 6, -2, 0, 7, -3, -3, -8, 2, 1, -2, 6], [-1, 8, -2, 0, 1, 1, 4, 0, -5, 15, 7, -1, 0], [-9, -10, 0, 2, -3, 0, -1, -7, 0, -7, 6, -4, -2], [1, 12, 1, -2, -4, 0, -1, -7, -2, -6, 2, 4, 1], [-2, 5, 0, -6, 1, -3, 3, -4, 0, 5, 1, 1, 2], [-34, 6, 1, -2, -2, 3, -2, 3, -1, 10, 0, 3, -1], [-6, -10, -7, 2, -9, 11, 1, -5, 1, -11, 8, -3, -2], [6, -1, 5, 3, 3, 6, 1, -9, -1, 11, -2, 6, 5], [20, -1, 8, 4, -3, -2, 3, -2, -3, 3, -2, 2, -3], [8, 7, -5, -8, 0, 1, 4, 2, 4, -2, -3, 1, 4], [-6, 1, 1, -4, -2, 9, 2, -2, -5, -5, -3, 1, 0], [-2, -6, -6, 0, -8, 7, -3, 3, -2, -5, 1, -1, 0], [20, -11, 6, -1, 0, 4, -5, -5, -11, 0, -7, 4, 7], [0, -2, -1, 6, -2, -1, 0, 3, 3, -1, 0, -5, 3], [-3, -1, -2, 6, -2, -1, 2, 0, -3, -14, 7, 4, -5], [-4, 2, -2, -8, 4, -4, -1, 8, 5, -3, 1, 0, 2], [12, 4, -3, -2, -3, -2, 3, 4, -1, -2, 0, 0, 0], [8, -8, 3, -2, 2, 8, -2, -2, -3, -2, -1, -2, -2], [11, -4, -1, 0, 3, -2, -1, -4, -11, 0, -1, 2, 6], [-4, 1, 0, -4, -10, -7, -2, 6, 8, -11, -6, -1, -2], [12, -3, -4, 2, -7, 7, -1, 9, 3, 1, 1, -3, -4], [-5, 5, 2, 2, 3, -3, 1, -6, -5, 1, 4, -1, 4], [-3, -6, 5, -6, -1, -4, -1, -4, -1, -5, -3, 4, 4], [-18, 6, -5, 0, -2, -2, 1, 0, 4, -1, 4, -2, -6], [14, 0, 7, -4, -4, 4, -7, -2, -7, 4, -3, -4, 3], [7, -3, 1, 4, 8, 7, 2, -7, -8, 7, 10, 3, 2], [17, -5, 0, 0, 6, 4, 6, -10, -1, 4, 2, -2, 3], [14, 10, -2, -4, -7, 1, 0, -5, 0, -7, 1, 1, -1], [-5, 11, 3, -2, -2, 7, -1, -5, 2, -5, 7, 5, 0], [-4, 4, -5, 2, -3, 3, -1, 1, 1, -1, -3, 3, 1], [5, -4, -6, -2, -1, 4, 0, 4, -3, -2, -4, 0, -1], [-31, -7, -1, -2, 3, -6, 5, -1, 4, -8, 4, -2, -1], [-12, 0, 6, 8, 5, 3, -4, 3, -2, 11, -7, -3, -5], [-17, -1, -2, -2, 7, -8, 4, 3, 1, 5, 2, 0, -7], [21, 4, -1, 0, -4, -2, 3, 1, 3, -10, 1, 0, -4], [-15, -6, 2, 0, 1, -3, 4, -4, 3, 1, 7, -3, 0], [3, 4, 2, -2, -4, 10, 4, -5, -1, 2, 8, 4, -7], [-9, 3, -6, 2, -7, 3, -2, 7, -3, -11, 5, -1, -4], [2, -5, 4, 4, -4, -5, -3, 8, 3, -2, -3, -3, -2], [18, 9, 7, -4, 7, -7, 3, -5, -3, 9, -4, 5, 4], [-12, -2, -2, -8, 4, -1, 2, -3, 5, 6, -3, 3, -1], [14, 3, 2, 2, 4, 0, -1, -7, 2, 2, 3, 2, 0], [-3, 3, -6, 2, -2, 8, 3, 2, 4, 5, 7, -2, -1], [16, 5, 4, -2, 6, -1, 1, -2, -1, -4, 4, 7, 1], [13, 7, 0, -4, -6, -8, 3, 1, 0, -5, 3, -8, -5], [-20, -2, 0, -6, -5, -3, -4, 9, 7, -4, -6, -7, -6], [-20, -1, 3, 4, -1, 5, 0, 3, -4, -2, 1, -1, 0], [8, -9, 3, 2, 3, -6, -3, -2, -8, -5, -9, 0, 5], [-14, -3, -1, 1, 3, -7, 5, -5, -1, -6, 5, -3, 6], [-10, 2, 0, 4, 5, -2, 9, 3, 0, 6, 3, -3, -1], [3, 10, 0, -6, -5, 4, -2, 2, 0, -11, 1, -2, 6], [26, -7, 7, -2, 8, -5, -2, -2, -2, 2, -11, -3, 8], [3, 2, 5, 6, -1, -6, -3, 5, -1, 1, -9, -2, -2], [1, -8, -3, 2, -4, 3, 3, 1, -3, 2, -6, 3, 1], [24, -4, -3, 0, 2, 5, 0, -1, 1, 6, -4, -1, 5], [12, -1, 2, 6, 1, 5, 3, -1, -3, 3, 7, 5, 2], [4, -3, 0, 2, 3, 2, 0, -10, -3, -6, 5, 3, 6], [4, -7, 5, -1, -3, 7, 1, -9, 1, 6, -5, 1, 4], [-7, -12, 1, 0, -2, -1, -3, -3, 5, 2, -1, -5, 2], [2, -1, 1, 4, 8, 1, 2, 0, 2, 7, 3, 2, -3], [-3, 8, -1, 0, 1, -4, 3, -2, 3, -11, 3, -2, -4], [-2, 8, 1, 2, 5, -1, -1, 5, -5, 10, -7, 4, -2], [-6, 0, -4, 1, 2, 1, 6, 1, 1, 5, 8, 5, -5], [15, -2, 1, 4, 5, -2, 3, -5, -7, -3, -5, 3, 3], [-1, 9, 1, 0, -2, 1, 4, -3, 0, -8, 6, 1, 2], [-18, 2, -7, 4, -8, 0, -2, 4, 3, -2, -1, 4, -4], [25, -9, -1, 0, 7, -1, 0, -2, -4, 7, -2, 3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4008_2_a_m();".
// 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_4008_2_a_m(:prec:=13)
    chi := MakeCharacter_4008_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_4008_2_a_m();".
// 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_4008_2_a_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![12144, -72184, 112850, -14532, -72240, 26911, 16968, -7582, -1879, 901, 99, -49, -2, 1]>,<7,R![-6016, -2176, 30480, -1584, -44740, 13575, 19342, -8413, -2082, 1147, 81, -58, -1, 1]>],Snew);
    return Vf;
end function;