
// Make newform 7865.2.a.bg in Magma, downloaded from the LMFDB on 13 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_7865_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_7865_2_a_bg();".
// 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_7865_2_a_bg();".
// 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 := [13, -244, -232, 2336, 302, -7174, 1412, 9722, -3493, -6506, 2845, 2346, -1116, -466, 230, 48, -24, -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, -5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-260518903, 235219025, 3512649622, 2631275561, -12524935506, -8175831325, 17930439166, 7246068909, -11998140144, -2979480143, 4216579796, 647633186, -804843398, -75066214, 78986990, 4173952, -3120436, -75937], [377659019, -4078307360, -7751210935, 26423815642, 30445979549, -55308542311, -45564816858, 55439368502, 31917106936, -29932263727, -11759091107, 9200071185, 2354259028, -1612880311, -242415395, 150185219, 10048517, -5759455], [107187709, -746386993, -7094781570, 10940332416, 36265806242, -31222875760, -61089951123, 39925892868, 45349078274, -25634443942, -17265886335, 8939371379, 3528989120, -1725257705, -368550030, 173262749, 15435763, -7059337], [14229697, 117728382, 185921844, -943277341, -1401788939, 2486628376, 2630456861, -3147135123, -2064870155, 2038651868, 824467630, -716854046, -176658605, 138678278, 19321024, -13880438, -844855, 561419], [24431179, -3992261897, -1322223827, 27399577702, 8851017565, -68122356247, -15102892868, 80130619207, 11041485666, -49305330649, -4357686310, 16738576093, 974205071, -3150971202, -114503139, 307847249, 5426375, -12166646], [161709347, 2280430025, 2943069707, -19725753251, -26484620498, 55342389936, 52786374918, -72169995771, -42083613603, 47149268022, 16779389088, -16581246780, -3566083779, 3197760766, 386058112, -318599412, -16713999, 12816433], [390858704, -4463643227, -7983504087, 34274601553, 32401680108, -93352847441, -50682420425, 117817687823, 36652071120, -75768015704, -14107723981, 26483924515, 2999097008, -5095807281, -331303152, 507036320, 14776968, -20365912], [264756856, -4499215503, -9188534811, 39591266345, 40425106977, -113703372580, -64596179697, 147314976039, 46679557207, -95848552352, -17826996064, 33670705571, 3755566723, -6491417984, -411647521, 646260624, 18254432, -25954856], [356988614, -1855558038, -6899949974, 27937591168, 28996592698, -97092950953, -46203889769, 138063083602, 33505963999, -94506800437, -13100260635, 34215540379, 2873527808, -6720052625, -330036289, 676765154, 15304993, -27371792], [-169826443, 7180198339, 12120909981, -54733777647, -51925341919, 143890543284, 78083589478, -175038553467, -54449431941, 109804615406, 20333627227, -37777989206, -4217651167, 7194390442, 457098561, -710862889, -20100839, 28411617], [-1980170, 25218753, 56477733, -216894881, -236507426, 607177264, 365612377, -770926104, -260649685, 495773164, 99041242, -172970564, -20827836, 33198936, 2280684, -3294623, -101010, 131997], [23033125, -352696333, -978447819, 2865332611, 4517686556, -8344659876, -7389443042, 11275734301, 5432960486, -7595611422, -2093401731, 2739996136, 441208765, -539472495, -48089038, 54651963, 2113411, -2227866], [-1715961, 30227098, 55842057, -251867259, -256757975, 710439052, 420797500, -917662457, -309860583, 598122797, 120241699, -210623720, -25661912, 40682766, 2839626, -4055172, -126695, 162984], [-143663255, 8877280464, 10351203962, -75135244357, -54359012341, 207665229294, 93550538438, -264276419849, -70072978271, 170843640884, 27505997081, -59860223820, -5943124870, 11520830516, 666445102, -1145005672, -30127742, 45899239]];
    Rf_basisdens := [1, 1, 1, 1, 79208321, 79208321, 79208321, 4659313, 79208321, 79208321, 79208321, 79208321, 79208321, 79208321, 315571, 4659313, 315571, 79208321];
    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_7865_a();"
function MakeCharacter_7865_a()
    N := 7865;
    order := 1;
    char_gens := [3147, 3511, 1211];
    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_7865_a_Hecke( : Kf := Kf)
    return MakeCharacter_7865_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]];
    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, 0, 0, 0, 1, 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, 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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 1, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0], [-2, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 1, -1, -1, 1, 0, 0, 0, 0], [-3, -1, 0, 0, -1, -1, 0, 1, 1, -1, -2, 1, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 2, 0, 1, 0, 0, -1, 2], [1, 0, 1, 1, 1, 1, -1, 0, 1, 0, 1, 0, 1, -1, 0, 0, 1, -1], [-1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1], [-1, 1, -1, 1, -1, 0, 0, 0, -1, 1, -2, 0, -1, 1, 0, -1, -2, 1], [-2, 1, 0, 1, -1, -1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0], [0, 1, 0, 0, -1, 1, -1, -1, -2, -1, 0, 0, 1, -2, 0, 0, 1, 0], [2, 0, -1, 1, 0, 2, -1, -1, -2, 0, 0, -1, -1, 0, -1, 0, 0, 0], [-3, 0, -1, 0, 1, -1, 1, 1, 1, 0, 0, 1, 1, 0, -1, 1, 0, -1], [-1, 1, 1, -1, -2, -1, -1, 0, 0, -2, -2, 0, -2, 0, -1, 0, 0, 0], [2, 1, 1, 0, 1, 0, 0, 0, 0, 0, 2, -1, 0, 1, 0, -1, -2, 1], [0, 2, 2, -1, 0, 0, 0, -2, -1, 1, 0, -2, 0, -1, 0, -1, 0, 0], [-3, 0, 1, 0, 1, 1, -1, 0, 0, -1, 2, -1, 0, 0, -1, 1, 3, -1], [-1, -1, 0, 0, -1, 0, -2, 0, 1, 0, -2, 1, -1, 1, 0, 1, 0, -1], [2, 3, 0, 0, -1, 0, -1, -2, -2, -1, -2, 0, -2, 0, 0, 0, -1, 1], [-2, 1, 1, 0, -2, -2, 1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0], [-3, -1, 1, -2, 1, -1, 1, -2, 1, 1, 1, 0, -1, 0, 1, -1, 2, 1], [0, -3, 0, 0, 0, 0, 1, 0, 1, -1, -1, 2, 0, 0, -2, 1, 0, 1], [-2, -2, 0, -1, -1, -1, 3, 0, 1, -1, 0, 0, 0, 1, 1, 0, 0, 2], [1, 0, 0, 0, -2, 1, 1, 0, -1, 2, 0, -2, 1, -1, 3, -1, 0, -1], [0, 1, 1, 1, -5, -1, 1, 3, -3, -1, -3, 0, 0, -1, 1, 0, -2, 0], [1, 3, -3, 0, 0, 1, 2, 0, -4, 2, 1, -3, -1, -1, 1, -3, -4, -1], [-6, 1, 0, 0, 2, -1, 0, 0, 1, 1, 2, 0, 0, 1, -2, 0, 0, -2], [0, 1, 1, -1, 0, 1, 0, -1, 1, 1, 1, -1, 1, -1, -2, 1, 2, 0], [-5, 0, -1, 1, -2, 1, -1, -1, -1, 0, 0, -1, 1, -1, 0, 1, 1, -1], [-3, 0, -1, 2, 1, 1, -3, 0, 2, -2, -2, 1, 2, -1, -1, 3, 3, -2], [-2, -2, 1, -3, 0, -2, -1, -1, 1, 1, 0, 2, 0, 1, -2, 1, 2, 0], [-2, 2, 1, -2, 0, -3, 1, 2, -1, -1, 0, -1, 0, -1, 0, -2, -1, -1], [3, -1, 0, 0, -1, 2, -1, 0, -1, 1, 0, -2, 1, 0, 1, -1, -1, 0], [-3, 0, 0, -2, 2, 1, 1, -2, 3, 3, 0, 0, 3, -1, 4, -1, 3, 0], [1, 1, 3, 0, 0, 0, 1, 2, 1, 3, 0, -2, 2, -1, 2, -1, -3, -1], [0, 2, 0, -2, 3, 0, -1, -2, -1, 0, 1, -4, -3, 0, 1, -1, 1, 1], [2, 3, 0, 0, 2, 2, -2, -3, -2, 1, 2, -2, 2, -2, 0, 0, 0, -2], [0, -4, 2, -2, 0, 0, -1, 0, -1, 2, -1, -1, -2, 1, 1, 0, -3, 1], [-2, 1, 0, 2, 3, 1, -1, 4, 2, -1, -1, 0, 1, 0, -1, 1, 1, -4], [0, 4, 0, 2, -1, -1, 2, 4, -2, -2, 0, 1, 1, -1, -2, 0, -1, -2], [-4, -2, 0, -1, -1, -1, 1, 3, 1, -4, 3, 2, -2, 0, 0, 1, 1, 0], [-6, 6, 2, 0, -1, 0, 0, 0, -2, -1, -1, 0, 0, -2, 2, -1, 1, 2], [-1, 3, -2, -3, 1, 0, -1, -2, -1, 1, -1, -2, -2, -1, 2, 0, 0, -1], [-7, -4, 0, 0, 0, -3, 1, 0, 4, 0, 0, 2, -1, 4, -2, 2, 1, 0], [-4, -6, 0, -3, 1, -1, 0, 1, 6, -1, 1, 0, 1, 3, 1, 3, 3, -2], [1, -3, 1, 0, 0, 1, -3, -3, 1, -2, -2, 3, -1, 0, -3, 1, 1, 3], [-3, -2, -1, 1, 2, 1, -3, -2, 0, -1, 0, 2, 2, -2, 2, -2, -2, 1], [-4, 7, 1, 2, -2, 0, -3, 0, -4, 0, -2, -2, 1, -3, 0, -1, -1, -3], [1, 3, -3, 1, 2, 3, -3, -2, -1, 2, -2, 1, 4, -1, -4, 2, 2, 1], [4, 0, -1, 1, 0, 1, 1, 0, 0, 3, 1, -4, 1, 0, 0, 1, -1, -1], [0, 0, 0, 1, 1, 3, 0, -1, 0, 1, 1, 0, 3, 2, -3, 2, 1, 0], [-4, 4, 0, 2, 3, 1, -4, -2, 1, -3, -2, 1, 1, -1, -5, 0, 2, 1], [-4, -1, -1, -2, -2, 0, 0, 1, 0, 0, 2, 0, -4, 1, 3, 1, -1, 1], [-4, -3, 4, -6, 3, -3, 0, -2, 6, 1, -1, 1, -1, 4, 1, 1, 3, 3], [0, -5, 1, -1, 2, 3, 0, 0, 2, -1, 4, 1, 2, 1, 1, 2, 3, -1], [-1, 2, 2, -2, -2, -2, -1, -2, -2, -2, 0, 0, -4, 0, -1, 0, -3, 2], [3, -1, 0, 1, 3, 1, 1, 0, 2, 2, 5, -1, 1, 3, 0, 0, 2, -1], [-3, -3, -2, -2, 1, 0, 0, 0, 2, -1, 2, 3, -3, 4, -2, 2, 0, 1], [3, -1, -1, 3, 2, 1, 0, -2, 0, 4, -1, -2, 2, 0, 3, 0, 1, -2], [0, 7, -2, 1, -4, 2, -3, -1, -3, -1, -5, 0, -1, -4, -1, -1, 2, 0], [3, -2, 2, -1, 1, 2, -3, -3, 0, -2, 0, -1, -2, 0, 2, 2, 2, 4], [3, 0, 3, -1, -4, -2, 1, 1, -3, 2, 2, -1, -2, 1, 1, 0, 1, 0], [-3, 4, 2, 0, -2, -2, 2, 2, -2, 0, -3, -3, 1, -1, -1, 0, -3, -2], [2, 3, 5, -3, 0, -2, -1, 1, -5, 0, 1, -2, -2, 0, -4, -1, -3, 1], [-9, 0, -2, 0, 1, -3, 2, 2, 4, 0, 2, 3, 0, 3, -1, 1, -1, -1], [-8, 0, 1, 0, 3, -1, 2, 2, 5, -3, 1, 4, 0, 3, 1, 2, 0, -1], [-6, 4, -1, -1, -1, 1, -2, 0, -3, 3, 4, -2, -1, -2, 3, -2, 0, 0], [-4, 4, -1, 0, 0, 1, -1, -3, -3, 2, -2, -3, -1, -2, -4, -1, 2, 1], [0, -5, -1, -2, 5, 1, 0, 0, 1, 1, 1, -1, 1, -1, 5, -3, 0, 1], [-2, 2, -3, 2, 0, 2, -4, -2, -2, 2, -4, -2, 1, -2, -4, 1, 3, -4], [-10, 1, 3, -3, -1, -3, 2, 1, 2, 0, 1, -1, 3, -3, -2, -1, 3, -3], [7, 6, 1, 2, 0, 1, 3, 0, -2, 0, 0, -3, 1, -3, 2, -3, -4, 3], [5, -2, -4, 3, -1, 1, 2, 2, -1, -1, 2, -1, 1, 2, 0, 2, 2, -4], [-3, -1, 2, 0, -1, -1, 0, 1, 2, 2, -1, 0, -2, -2, 2, 1, 2, -1], [3, -2, 2, 2, -3, 1, -2, 2, -2, 0, 2, -3, 2, -3, 3, -1, -1, -3], [3, 7, 3, -1, 2, 0, 5, 0, -4, 3, 2, -3, -2, -1, 7, -3, -6, 2], [5, -2, 0, 0, -1, -2, -1, -1, -1, -1, -2, 0, 1, 0, -4, 0, -1, -1], [1, -1, 3, 0, 1, 2, 2, -2, -1, 2, 4, -2, 0, 3, 1, -3, -1, 2], [2, -2, 0, 4, 0, 3, 0, -1, 3, 0, -1, 2, 1, 2, 1, 2, -1, 2], [-5, 0, 1, 3, 0, 2, -5, -2, -1, -4, -7, 5, 3, -2, -1, 1, -3, 5], [-2, -4, 0, -1, 2, -1, 3, 0, 6, -3, -2, 5, -1, 2, 3, 0, 2, 1], [5, -3, -3, 0, 0, -2, -1, 3, 1, -1, -6, 5, 1, 3, 0, 1, -5, 2], [7, 9, -4, 5, 3, 4, -4, -4, -3, 0, 4, -3, 1, -1, -2, -1, -2, 1], [2, -5, 3, 4, -1, 0, 2, 0, 1, 0, 0, 1, 2, 3, 0, 3, -6, 3], [3, -5, -6, 0, 0, 2, -1, 2, -2, 0, -2, 2, 1, 0, 3, -1, -2, -3], [2, 4, -4, 4, -2, 1, 2, -1, -4, 3, -2, -3, -2, -2, -4, -2, -2, -1], [-5, 1, 0, 4, 5, 0, -1, -1, 1, -2, 0, 0, 2, -1, -4, 0, 4, 0], [-9, -6, -1, -1, -1, -1, 3, 2, 2, 2, 2, -1, 2, 3, 2, 0, 2, 0], [3, 2, 5, 1, 0, 4, -3, -3, 1, 1, 5, -2, 4, -6, 3, -3, 4, -3], [6, 7, -7, 5, 1, 4, 2, -1, -6, 1, 0, -5, 3, -4, 1, -2, 1, -1], [5, 0, 4, -1, 0, 3, 0, -1, -3, 0, 7, -4, 2, -2, -3, 0, -3, -2], [-3, 3, 1, 0, 0, -4, 0, 1, 4, 2, -2, -2, 1, 0, 0, 0, -3, -1], [-1, -4, -2, 0, 4, -4, 1, 2, 3, -2, 5, 5, -4, 3, 1, 2, 1, -3], [-4, -9, -4, -3, -4, -2, 1, 0, 1, -2, -1, 1, -5, 5, -6, 1, 3, 1], [-10, -4, -3, 0, 0, 0, -1, -1, 2, 0, 0, 4, 0, 4, -5, 3, 3, 0], [-6, 3, 0, 1, 1, -2, -2, 1, 3, 1, -1, 4, -2, -1, 1, -1, 1, -1], [-4, 3, 4, -3, -5, -3, 0, 0, -2, -3, -7, 1, -3, -1, 1, -3, -1, 2], [0, 3, 0, -3, 0, -4, 3, 5, 0, 4, 5, -4, 1, 1, 2, -1, -4, -5], [0, 1, 1, 2, -3, 4, -7, 1, -4, -3, -3, 1, -2, -2, 1, 0, -4, -3], [-14, -10, -2, 3, -1, 1, 2, 2, 2, 1, -1, 3, -1, 6, 3, 2, -6, 2], [10, 4, 1, 0, -3, 3, 2, -1, -2, 1, 3, -5, -1, -5, 2, -2, 0, 2], [0, -5, 0, -1, 4, 1, 5, 3, 2, 2, 3, 1, 0, 0, 8, -3, -2, 4], [3, 7, 0, -2, -6, -1, -1, -2, -5, -2, -5, 0, -3, -3, -3, 0, 3, 2], [-7, -8, 1, 0, 2, -3, 3, 2, 6, -3, -3, 5, 0, 3, -2, 5, 1, 4], [-7, 1, 2, -3, 3, -3, 4, 4, 6, 1, 7, -3, 1, 3, -3, 1, 1, -4], [-12, -3, -1, 0, 3, -1, 0, 1, 4, -1, 3, 0, 2, -1, -1, 1, 5, -3], [2, 2, 2, 2, -4, 0, 2, 4, -2, 4, 1, -7, -3, 2, 3, 1, -6, -1], [3, -8, 2, -1, -2, -1, -2, -5, 2, 0, -1, 5, 1, 0, 3, 2, 4, 7], [1, 1, 2, 3, 0, 3, 1, 5, -1, -2, 7, -4, 0, -2, -1, 1, -1, -3], [5, -1, -1, -2, -3, -3, -1, 1, -4, 3, -3, 3, -2, -1, 2, -2, -7, 3], [3, 2, 1, 3, -2, 2, -7, -4, -4, 0, 4, -1, 2, -3, -1, -3, -4, 1], [6, 2, 4, 1, 0, 2, 2, 5, -3, -2, 5, -3, 2, -3, 0, 1, -1, -4], [0, -11, -3, 1, 1, 0, 4, 5, 3, 1, 2, 3, 3, 2, 1, 3, 1, -6], [0, -9, 2, -3, 0, 1, 2, 0, 4, 3, 5, 2, 2, 4, 6, -1, 3, 1], [0, -5, 1, -2, 2, 0, 7, 7, 1, 6, 2, -4, 0, 0, 6, 1, -1, -7], [1, -8, 3, 1, 2, 0, -2, 1, 3, -1, 6, 2, 3, -2, -1, 2, 1, -2], [1, -5, 3, 2, 5, -1, 3, 3, 4, -1, 7, -4, 1, 3, -2, 3, 2, -3], [-1, -8, 0, 4, 0, -2, 1, 5, 1, -6, -2, 9, 4, 0, 1, 2, 2, 0], [-7, 0, 2, -3, -5, -6, -3, 1, -2, -2, -2, 3, -4, 4, -6, 2, 0, 0], [3, -9, 0, 3, 0, 0, -3, 0, 4, -4, -4, 3, 6, -2, -5, 0, 3, -3], [-4, 4, -1, -5, 0, -1, 1, -3, -1, 2, 5, -9, -6, 2, 5, -4, -1, -1], [7, -4, 3, -2, 4, 3, 4, -1, 2, 4, 3, -3, -2, 1, 4, -2, -2, 2], [-2, 7, 2, -2, 0, -1, 3, 0, 0, 1, 2, -3, 1, -3, 2, 1, 4, 2], [-4, -4, 3, 1, -7, -5, -1, -4, 0, -3, -12, 2, -1, -2, -1, -2, 1, 4], [-6, -8, 2, 3, -1, 0, -6, -4, -3, -4, -3, 8, -1, 0, -4, -1, -3, 4], [-2, 9, 1, 1, -5, 1, 0, 1, -9, 3, 0, -2, 0, -1, -4, -4, -10, -1], [-2, -3, 0, 1, 4, 1, 1, 2, 5, -1, -1, -1, 3, -2, 6, -1, 4, -1], [-1, -11, -1, -3, 7, 5, 5, -4, 7, 2, 4, 0, 5, 2, 2, 1, 8, 3], [2, -7, 6, -7, 1, -2, 1, 2, 0, 5, 10, -3, -5, 1, 9, -1, 0, -4], [0, 0, 0, 4, 2, 3, -5, 0, 1, -2, -8, 5, 5, -5, -2, 5, 7, -6], [-1, -5, -2, -6, -2, -6, 5, 8, 10, 0, 4, 5, -2, 9, -1, 4, -1, -2], [-3, 2, 1, 0, 5, 5, -5, -2, 0, 0, -1, 3, -1, -2, 1, -1, 5, -3], [-5, 1, -6, 5, 2, 0, 1, 0, 5, -7, -2, 5, 5, 1, -9, 2, 3, 2], [12, -11, 0, 1, 0, 2, -5, -5, 2, -2, 3, 5, 0, 2, -6, 3, -1, 5], [-1, -3, 3, -8, 7, -3, -3, 1, 3, 2, 3, 1, -4, 6, -5, -1, -1, -1], [-7, -9, 1, 3, -2, 1, 3, 5, 0, -3, 0, 4, 1, 1, 1, 0, 4, 0], [2, 3, 2, -5, -3, -4, 3, 2, -4, 3, -4, -5, -4, 2, -3, -3, -3, -2], [-12, -10, -2, 3, 6, 2, -5, 3, 6, 0, -7, 4, 3, 2, 1, 4, 6, -6], [2, -1, 2, -2, -5, -7, 5, 6, 3, -2, 2, 5, -2, 5, -4, -1, -3, 1], [-2, -5, 3, 0, -1, 2, -3, -2, 3, -6, 8, 2, 0, 0, -1, 3, 0, -1], [4, 5, 0, -2, -2, -2, 0, -5, -4, 4, -6, -4, -4, -1, -2, -4, -3, 4], [3, 5, -4, 1, 1, 5, 0, -2, -9, 1, -3, 0, 1, -5, 6, -5, 0, 1], [18, -2, -5, 3, 2, 4, -5, -6, 0, 1, -1, 3, 4, -4, 3, 2, 3, -2], [-5, -4, -2, 3, -1, 1, -4, -7, -4, 1, -9, 3, -1, -5, 3, -1, 1, 2], [6, 2, 9, -4, 4, 3, -6, -7, -3, 2, 3, 1, -1, -1, -2, 0, -1, 1], [-9, 6, 0, 4, -5, 3, -1, -4, -5, 2, -9, 0, 2, -4, 0, -5, -2, 4], [0, -6, 3, -8, 3, 0, 1, 0, 5, 5, 2, -4, -6, 4, 3, -3, 2, 0], [-10, 8, 0, 1, 2, 0, 0, 4, -2, -3, -3, 2, -4, -5, 1, -3, 3, 1], [-2, -2, -3, -3, 3, 3, -1, 1, 3, 3, 6, -5, -1, 2, -1, 2, 4, 0], [-8, 4, -1, 3, 4, 2, 1, -3, 0, 4, -3, -1, 3, 0, -4, 1, 3, 0], [-1, 7, -6, 1, -2, 4, 1, -4, -6, 2, -3, -3, 3, -3, -3, -3, -5, 5], [2, -6, -5, -3, 4, 7, -7, -8, 0, 1, -1, 1, 1, -2, 4, 0, 6, -1], [5, 0, 2, -1, 5, -1, 6, 1, 2, 5, 7, -2, -1, 4, -9, 0, -2, -3], [5, -7, -2, 0, 7, 4, 0, 0, 11, 6, -2, -2, 5, 2, 9, 5, 9, -3], [-6, -7, -3, -2, -2, 1, -7, -8, 3, 2, -2, 5, 1, 2, 2, 3, 3, 6], [5, -4, -1, -2, 2, 1, 1, 5, -3, 1, 8, 1, -3, 0, 6, -1, -5, -8], [7, -5, 0, 4, 5, 10, -6, -7, 2, 0, 2, -1, 5, -2, 1, 3, 10, 1], [-5, -3, -6, 0, -1, -5, 3, 4, 2, -6, 2, 4, 2, 3, -7, 1, -2, 4], [4, -12, 0, 4, 9, 6, -3, -3, 7, -4, 5, 6, 9, -2, 1, 7, 11, -2], [-9, -4, -9, 7, 1, 4, -4, 2, -4, -3, -4, 6, 0, 3, -1, 2, 0, -1], [-4, 2, 0, -2, 1, -5, -3, -1, -5, -4, -5, 1, -4, 0, -2, -2, 2, 0], [-2, -3, 3, 3, 0, -3, 2, -2, 4, -1, -4, 2, -5, 2, -1, 0, -2, 3], [-8, -7, -1, -5, 7, 0, 5, -2, 9, 1, 4, 2, -2, 9, -3, 3, -3, 1], [-10, -9, -4, -1, -5, -5, 2, 6, 5, 3, 2, 2, -1, 11, -5, 0, -4, -4], [4, 9, -3, 2, 3, 7, 1, -1, -5, 2, 0, -4, 6, -4, 6, -3, 1, 3], [-11, 2, 1, -4, 1, -5, 10, 4, 0, 6, 4, 1, -7, 7, -7, -2, -4, 5], [8, 1, -7, 0, 5, -2, 0, 0, 3, 0, 4, -3, 1, -1, -6, 2, 4, -4], [-9, 3, -2, 1, 0, -4, 4, 2, 3, 5, -1, -1, 2, 2, -8, 2, -3, 2], [10, -1, 0, 3, -6, 5, -11, -4, -6, -2, -6, -4, 0, -5, 1, 1, 2, -2], [-15, -4, -4, 0, -2, -3, 0, 7, -3, -2, -1, 0, 5, -4, 1, -1, -3, -9], [-10, 5, 1, 6, -5, 0, -3, 5, -1, -4, -8, -4, -4, -2, 1, 1, -2, -4], [-21, 1, -1, -6, -4, -5, 8, 1, -3, 4, 2, -2, -4, 6, 0, -5, -6, 5], [-10, -5, -10, 0, -2, -6, 4, 3, -1, 4, -2, 2, -1, 5, -2, 1, -5, 1], [-2, 4, 0, -1, 2, 5, -6, -1, -1, 2, 0, -11, 1, -3, 6, 0, 6, -6], [3, 9, -3, 4, 1, 4, 5, 0, -5, -2, 4, 1, -2, 1, 8, -1, -5, 5], [-3, -11, 4, -7, -8, -6, -1, 0, 2, -4, -9, 2, -4, 3, 4, 2, -1, 4], [12, -4, 2, 0, 6, 2, 4, -1, 1, 5, 10, -8, 3, 0, -3, 0, 3, 2], [8, 2, 2, 5, -2, 3, 0, 4, -4, 7, -2, -3, 3, 0, 7, -2, -8, -2], [-1, 2, -2, -1, 2, 0, -2, 0, -5, 2, 3, 2, 3, -4, 4, -1, -5, 0], [-1, -4, 1, -4, 8, 3, -3, -3, -1, 1, 12, -1, -1, -2, 0, 1, 1, 0], [-16, 12, 0, 0, -2, -6, 7, 2, -1, 3, -2, -1, 1, -3, 3, -4, -2, -6], [14, -3, -8, 2, 3, 4, 2, 1, 0, 4, 5, -2, 6, -3, 4, 1, 1, -3], [2, -5, 3, 2, -2, -3, -2, 4, -1, -5, -5, 9, -6, 6, -6, 2, -8, 5], [-4, -1, 5, -2, -2, 5, -1, 1, 5, 0, 2, -5, 1, 0, 7, 0, 8, -5], [0, 4, -4, 4, 0, -5, 3, -3, 1, 2, -4, 7, 1, 5, -1, 0, 2, 3], [-6, -7, -3, -2, 2, -4, 6, 3, 5, -2, 7, -1, 3, -1, 5, -1, 2, 0], [2, 11, -6, 9, -7, 6, -8, 0, -6, -2, -10, -2, 7, -5, -7, 1, -1, -8], [-9, -7, -3, -2, 8, 2, -5, -9, 7, -2, -4, 7, 3, 1, 7, -2, 4, 8], [-6, -4, -5, -1, 0, -2, 1, 1, 2, -2, -4, 3, -5, 0, 11, 0, 0, 0], [15, 8, 5, 2, 0, 4, -3, 3, -4, 4, 2, -7, 0, -7, 5, 0, 1, -6], [-12, -5, -1, -7, 0, -4, 4, 1, 4, -2, -10, 4, -4, 2, -3, 0, 2, 6], [-4, 12, -4, 3, 0, 4, -2, -3, -7, 3, -10, -6, 2, -5, 4, -5, -4, 3], [6, 10, 1, 0, -2, 1, -2, -3, -9, 4, 7, -3, 1, -6, -2, -3, 2, -4], [3, 8, -4, 7, 3, 6, -2, -8, -4, -4, -9, 3, 5, -6, -5, -5, 0, 3], [2, -1, -2, -3, -1, -2, -4, -9, -3, -3, -3, 4, -8, 3, -7, 3, -1, 5], [-2, 3, 1, -7, 7, -5, -2, 4, 3, 5, 4, -2, -4, 3, -3, 5, 3, -10], [-22, -5, 7, -3, -6, -6, 7, 2, 0, -2, 5, 0, -1, -3, 5, 0, 0, -1], [4, -9, 9, -2, 6, -4, 3, 7, 10, -4, 3, 7, 5, 4, -8, 6, 8, -5], [8, 15, -7, 5, -1, 7, 1, -3, -3, 5, -1, -3, 7, -6, 5, -3, 7, 1], [0, 5, 5, -4, 0, 0, -1, -5, -5, 3, 6, -3, 1, -2, 0, -4, 4, 3], [2, -4, 6, -2, 1, -1, -3, -2, 4, -2, 7, 3, -1, 3, -3, 6, -1, 1], [17, 8, 7, 1, -2, 1, -9, -2, -4, 0, -5, -6, -2, -7, -9, -1, 1, -4], [9, 2, -8, 1, -2, -1, 2, -2, -2, 4, -5, 3, -1, 2, 5, 0, 6, 2], [7, 7, -3, -1, 1, 4, -3, -5, 0, -1, -4, 2, -3, -4, 5, -1, 2, 4], [3, 4, 3, -5, -6, 0, -4, -6, -5, 5, -2, -6, -7, -4, 16, -5, -6, 5], [14, -4, -5, 1, 5, 2, 1, -2, 5, 4, 8, -2, 10, -3, -2, 2, 12, -6], [8, 1, -3, 9, -1, 1, 1, 3, -7, 4, -4, 2, 9, -4, 0, -3, -4, -2], [2, 9, 2, -3, 9, -2, -2, -7, 9, 9, 4, -2, 3, 2, -2, 2, 8, -2], [7, 0, 0, -9, 10, -4, -1, -4, 0, 1, 8, -4, -1, 0, -7, -4, -1, 2], [1, 0, -1, 2, -2, -4, 4, -3, -6, 2, -4, -3, -3, -2, 0, -6, -7, 6], [-13, 4, 0, 7, 0, 1, 4, 6, 3, -5, 3, -1, 1, -2, -1, 5, 4, -1], [-11, 5, 4, 6, -3, -3, 4, 6, 0, -3, 2, 4, -1, 1, 0, 1, -8, -1], [-2, 11, -2, 6, 2, 1, 2, 1, 2, 2, 2, -2, 3, -3, -2, 2, 1, -2], [2, -6, 10, 2, -3, -3, -5, 2, 2, -6, -5, 6, -1, 4, -9, 4, -4, -1], [5, -3, 2, -4, -6, 1, 0, -3, 5, 1, -8, 4, -6, 0, 3, 1, 0, 7], [-5, -16, 5, 1, 4, -1, -5, 3, 12, -6, 4, 8, -1, 5, -4, 11, 7, -1], [-9, -5, 3, -9, 10, -3, 4, -5, 8, 3, 6, 0, 0, 5, -3, -5, -1, 5], [4, 18, 6, -3, -2, -2, 3, 1, -4, -4, 10, -3, 2, -4, -3, -3, 1, -4], [-13, 0, 6, 1, -5, -6, 5, 5, 1, 4, -1, 4, 1, 2, 2, 2, -7, -1], [4, 4, -1, 2, 4, 6, 2, 1, -2, 2, 2, -1, -3, 1, -4, 2, 2, -2], [12, -7, -2, -2, 0, -3, -6, -4, 2, -5, 4, 1, -6, 1, -9, 0, 0, -2], [-10, 0, 3, -3, -1, 1, 8, 3, -3, -1, 1, -5, 1, 1, 4, -3, -7, 0], [-6, 11, 7, -3, -1, 0, -2, 4, -1, -4, -2, -1, 2, -5, -4, 0, 0, -4], [-3, 7, -4, -2, -9, -5, -1, -6, -6, -1, -5, -1, -5, 2, 0, -1, -8, 7], [-13, 3, -4, 0, 3, 2, -3, 1, 5, -6, 0, 8, -2, -3, 8, 2, 4, 2], [-6, -8, -3, 1, 2, 3, -4, -2, 1, 0, -10, 5, 8, 0, 1, 3, 6, 3], [-14, -3, 6, -7, -1, -3, 2, 3, 3, 2, 4, -5, -5, 0, 0, -2, 4, 2], [-3, -2, 1, -5, 1, 6, -8, -2, -4, 5, -2, -6, -3, -1, 2, -2, 1, 0], [10, 2, -9, 3, 0, 2, 1, -1, -5, 1, -2, -10, 5, -7, -1, -2, 4, -5], [0, -10, 0, 9, 2, 3, -5, 1, 5, 1, -4, 6, -1, 7, -6, 3, -2, -2], [1, 15, -4, 1, 4, 2, 1, 0, -5, -5, 17, -4, 0, -9, 0, -3, 3, -3], [2, 9, 0, 0, -4, -4, -2, -7, 0, -2, -8, 0, -2, -3, -2, 2, 3, 4], [-3, 7, 2, -5, -11, -7, 4, -2, -6, 0, -6, -3, -8, 4, 5, -5, -10, 8], [-6, -7, -8, 2, 1, -5, 7, -4, 0, 3, 6, 2, -2, 8, -10, 2, -8, 2], [-11, 4, 3, -1, -4, -2, -8, -4, -3, -3, 2, 0, -5, 0, 12, 3, 6, 3], [-5, -4, -5, -3, 1, -1, 7, 0, 5, 4, -9, 2, 5, 3, 0, 2, 5, 5], [-5, -10, 3, -1, -1, -4, 1, -3, -6, 4, 4, 1, 0, -2, -2, -4, -5, 3], [2, -6, 2, -1, 3, 3, -1, -6, 6, 3, 1, 3, -5, 2, 6, -1, 2, 10], [8, 2, 6, -6, 8, 1, 1, 1, 0, 7, 9, -10, -6, 3, 16, 0, 2, -3], [13, -2, -6, 3, -1, -3, 3, -4, -2, 3, -8, 1, 1, -1, 1, -1, 0, 3], [-7, -2, 5, -1, -11, -9, 3, 2, 0, -2, -3, 1, -2, -3, 9, -2, -3, 1], [-12, -10, -5, 0, 0, 0, 3, 1, 3, -3, -11, 8, 3, 1, 7, -4, -5, 6], [9, 6, -1, 3, -2, -1, 2, 2, 4, -3, -1, 3, 9, -2, -15, 5, 11, -2], [-1, -1, 8, -3, 1, -8, 3, 1, -2, 1, 2, -6, -10, 6, 1, -2, -3, 2], [6, -3, -6, 3, 6, -4, 3, -1, 4, 12, 5, -2, 5, 3, -6, 1, -2, -3], [20, 0, -6, -1, -5, 3, 7, 0, -4, 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5, -8, 1, -8, 0, 4, 3, -11], [-14, 2, -4, -10, -3, -7, 2, 4, 16, -6, 1, 2, 0, 1, 3, 5, 13, -1], [-5, 8, -2, 2, -10, -3, -6, 4, 1, -1, -18, 0, -8, 4, 4, -1, -9, 0], [-24, 1, -2, 1, 7, 0, 0, -2, 6, 0, 5, 8, 9, 9, -4, 4, 6, 2], [16, 22, -3, -7, 8, 1, -4, -6, -9, 0, 2, -7, -4, 0, -19, -5, -4, -1], [10, 1, 2, 1, 4, 0, 7, 2, -1, 8, 4, 5, 1, 9, 9, 0, -8, 1], [-10, 6, -8, -3, -2, 11, -4, 0, -12, -2, 10, -7, -9, 0, -1, -1, -3, 2], [-10, -12, -2, -3, -3, -1, 0, 0, 5, -5, -2, 3, 1, 2, 0, 3, -1, 10], [-11, -17, 12, -5, -3, 1, 2, 0, 10, -9, 3, 5, -1, 3, 0, 11, 9, 6], [7, 19, -11, 9, -4, 7, 5, -2, -19, -3, -4, -4, -8, -10, 0, -7, -2, 9], [-5, 11, -1, -2, -4, 4, 11, -3, -10, 7, 0, -7, 0, -7, 12, -8, -9, 7], [11, -4, -2, 1, -10, -1, -2, -4, -8, -6, -4, 9, -5, -2, -8, -6, -7, 9], [-17, 7, -2, 7, -4, -1, -4, 4, -4, -3, -3, 1, 1, -3, -5, -1, -13, -4], [8, 0, 4, 2, -4, 4, -2, -4, 1, 5, 5, -5, 1, -9, 5, -6, 4, -11], [-35, -25, 0, -13, -9, -13, 11, 2, 17, -3, -2, 5, 2, 11, 8, 5, 10, 4], [-9, 4, 0, -2, 0, 1, -2, -5, 3, 8, 5, -2, 3, -1, -6, 3, 2, -9], [9, 5, -1, 17, -9, 3, 2, 2, -7, -1, -5, 6, 9, -3, -1, -4, -10, 0], [20, 11, -4, 5, -1, -5, -1, 9, -2, 0, -3, -2, 5, -3, -5, 1, -2, -9], [4, 13, -4, -3, -7, 1, 10, -4, -20, 11, -5, -7, -3, -3, 11, -13, -13, 2], [14, -14, 3, -2, 0, -4, 5, 3, 16, 5, 5, 3, -3, 9, 1, -2, 0, 5], [-3, -4, -1, 9, 4, 11, -6, -10, -1, -1, 3, 5, 2, -2, -3, 2, 15, 9], [-25, -1, -1, -4, -7, -6, 4, 8, 9, -4, -3, 7, -11, 7, -11, 9, 4, -1], [5, -9, 0, 8, -1, 4, 2, 0, 5, 4, -8, 3, 10, -2, 10, 2, 14, 5], [-6, 7, 10, 4, 1, 1, -4, 5, 0, 5, 8, -8, 3, -6, 8, 4, -1, -5], [16, -1, -4, 1, 7, 3, 3, -2, -2, 2, 21, 3, 0, 7, -3, 7, 2, -4], [-18, -2, 7, -9, 1, -8, 7, 5, 6, 2, 6, 3, -4, 3, 4, -2, -1, -2], [-20, 22, 2, 3, -5, -4, 7, 10, -2, -9, 4, 5, 1, -3, -13, 6, -6, -6], [3, 18, 1, 3, -4, -4, -4, 7, 1, -2, -5, 2, 0, 2, 0, -5, -3, -7], [-3, -2, 8, 0, 5, 0, -7, 0, -3, -5, 12, 0, -2, 0, 3, -1, -8, -2], [-4, 13, 3, 0, -2, -4, 3, 14, 0, -4, 6, 2, -3, -1, -10, -1, -3, -9], [10, -11, 8, -3, 9, -4, 2, 3, 10, 9, 4, -3, 5, 2, -4, 5, 1, -6], [-6, -2, -15, -4, 5, -10, 3, 2, -4, 6, 4, 3, -2, 8, -8, -2, -7, -4], [-7, 7, -2, 0, -8, -4, -2, 4, -8, -16, -2, 6, -9, -7, -9, -1, -7, 0], [-5, 2, -5, 6, -3, -2, -4, -5, 0, 5, -8, -4, 7, -2, -11, 3, -2, -2], [6, -5, 10, -8, 9, 0, 6, 5, -1, -6, 12, -7, 1, -3, -11, 5, 4, -4], [-13, 7, -6, 13, 4, 3, 0, 2, 9, -4, 6, 4, 7, 1, 5, 4, 8, 0], [27, 16, 5, 6, 9, 12, 2, -2, -8, 7, 13, -10, -5, -2, 12, -3, -3, -4], [-11, -17, -5, -10, 1, -5, 7, -6, 8, -10, -2, 9, -9, 8, -3, -2, 4, 14], [6, 6, 6, -11, 4, 1, 7, -5, 2, 1, 6, -7, -7, -5, 3, -13, 4, 10], [-10, 4, 4, 0, -2, -8, -3, 6, 12, -3, 4, 7, -7, 5, 0, 2, 0, -6], [-1, 6, -1, -5, 9, 2, -1, -4, -5, 2, 11, -7, -5, -8, -5, -6, 7, 0], [3, 5, 6, -1, 2, 6, -13, -8, -1, -2, -10, 6, 3, -6, -1, 1, 6, 7], [24, 1, -4, 9, -4, 5, -4, -5, -16, 4, 10, -10, -3, -3, -4, -14, -14, 3], [3, 11, 6, -1, -1, 2, 10, 4, 4, 4, 10, -16, 2, 0, 7, -3, 2, -12], [-15, 8, -7, 1, -7, 0, 11, 14, -7, 4, 3, -7, -1, 0, 9, -8, 1, -4], [15, 0, -5, 1, -2, 0, 7, 5, -6, -4, 4, -5, -12, 13, -1, -3, -12, 4], [5, 0, 4, 1, 11, -1, 5, 4, 7, 5, -2, 1, 9, 2, -8, -2, -5, -2], [-12, 18, -7, -2, 2, -1, 3, 9, -3, 2, 5, -5, -1, 2, 6, 1, 6, -11], [8, -3, -7, 12, 8, 5, 4, 2, 6, -1, 10, 8, 9, 0, 12, -2, -7, 4], [-13, 11, -2, -11, 5, -4, 1, -5, 1, -11, 1, -2, -12, 7, -8, -3, 0, 3], [-16, 16, 6, 9, -6, -5, -2, 4, -8, 1, -11, -4, -6, -8, -9, -8, -17, -1], [14, 3, 6, 9, -2, 8, -1, -4, -11, 6, -2, -8, 5, -4, -6, -3, -14, 1], [12, -20, 5, -4, 3, -6, 9, -3, 8, -2, -5, -2, 5, 5, 5, 4, -2, 7], [-24, 1, 7, -9, -4, -3, 1, 1, 3, -2, -12, 3, -5, 2, 9, -2, -10, 5], [11, -17, 1, -1, -4, 11, -9, -5, 7, 0, 0, -1, 1, -4, 0, 4, 16, 3], [-22, -9, 1, -4, 5, -5, 10, 3, -4, 8, 18, -9, -2, 8, -1, 0, 0, -4], [12, 11, -10, 1, -1, -2, -3, -11, 0, 0, -7, 1, 2, 8, -1, 4, 3, 4], [0, 21, 0, 11, 1, 5, -12, -7, -21, -5, -2, 0, 3, -11, 4, -7, -11, 5], [4, 0, -13, 0, -3, -2, 7, -5, 1, 3, -8, 10, -4, 3, 4, -4, -3, 16], [-6, 11, 3, 0, 6, 9, -5, -5, 5, 3, 5, -5, 11, -7, 3, 3, 20, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7865_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7865_2_a_bg();".
// 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_7865_2_a_bg(:prec:=18)
    chi := MakeCharacter_7865_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_7865_2_a_bg();".
// 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_7865_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7865_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![13, 244, -232, -2336, 302, 7174, 1412, -9722, -3493, 6506, 2845, -2346, -1116, 466, 230, -48, -24, 2, 1]>,<3,R![1, 20, -966, -8136, 5584, 26432, -6456, -32010, 1425, 19164, 1347, -6260, -898, 1132, 218, -106, -24, 4, 1]>],Snew);
    return Vf;
end function;