
// Make newform 8034.2.a.bb in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8034_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_8034_2_a_bb();".
// 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_8034_2_a_bb();".
// 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 := [-2048, -7296, 10816, 42456, -3280, -39390, 678, 14851, -847, -2601, 269, 207, -29, -6, 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], [118321024199296, -1166460932605168, 769318123450260, 243900222298282, -351416224121502, 467406199485803, 21533806125636, -175798632679800, 8353374678764, 21896289725388, -1278700159484, -1083388884247, 48838930211, 17211124339], [823088103896320, -1493240738430736, -4474034344329852, 3191603596475186, 4166130863706398, -1866871026854413, -1384162777559138, 480284475172790, 203902612897310, -60950374089966, -13680291423778, 3738713297463, 341475795883, -88374380939], [3910063812049920, 2930798928626144, -10523459905050216, -3273111940991512, 9639491681032898, 1434806545518450, -4294335680262885, 21324319463753, 861495599563087, -82526311620203, -75016193619169, 11004973389543, 2304692170158, -406907271083], [-1049016746216704, -2681914257270768, 5009595167008196, 3738614769502962, -5028642484757934, -1727833814125777, 2001792019065248, 263847579785132, -364296872907848, -852019922224, 29836393456848, -2323893522087, -883549115785, 113401884803], [-3240111054661120, 3859293469715616, 27368166174349864, -6353469675911824, -27089707111959846, 4328705466068094, 10222081364164867, -1550184318259343, -1765360355157129, 283131019369069, 138185242548231, -23999796454509, -3946026327910, 736531555801], [-804113198053376, -796238968943584, 24319897537607016, 2303549139588352, -28653365696975870, 149793013139294, 11978958293225447, -971532240127523, -2216963035494389, 294059953736009, 182412406997691, -30269645318433, -5401357076630, 1012895827421], [821129125068864, -339551833683136, -5548224775659980, -672606967463486, 5633078350696404, 260702734356941, -2274108863937751, 80934994695967, 415409487608205, -39248816007673, -33836085473159, 4568699998010, 993100465781, -161946406606], [2100534074540032, 525183514160720, -12224135566363796, -3969078366908066, 11755366222327694, 2237814174755737, -4744161357889636, -215439634992760, 882287181466036, -47754006020300, -73385354065404, 8513285647275, 2193728871065, -339042867519], [6787275004993536, 943866195323488, -53935123111369352, -19724650983886152, 62275142320830146, 12427081549409178, -28137851717749085, -1212936140282751, 5544743002953591, -307896247879923, -474790047743417, 54849889967511, 14412109193718, -2206595272859], [10136073926334976, 9764388104189088, -83761116115370808, -35303846107753504, 85869291262886602, 18558566709942790, -35077355093355085, -2119780149955167, 6499156388583719, -265156769412035, -536833570831273, 56053557441147, 15946662074178, -2312493819663], [564375014073344, 774956085111656, -4698153606819416, -2867839136841206, 5085371693596568, 1611592294003569, -2204586185386584, -227141704110620, 425935692445960, -10207803685344, -36139437565114, 3542117432817, 1092105641867, -155979666211], [-17591589253353984, -9119618453491232, 105396875559095672, 23513032611045888, -103939479614198202, -9221457804581926, 40700141428094181, -504552281455193, -7281353341457903, 596543424673451, 586439935888449, -75341966167987, -17117209536866, 2734835840775]];
    Rf_basisdens := [1, 1, 165587393151104, 165587393151104, 662349572604416, 165587393151104, 662349572604416, 662349572604416, 82793696575552, 165587393151104, 662349572604416, 662349572604416, 41396848287776, 662349572604416];
    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_8034_a();"
function MakeCharacter_8034_a()
    N := 8034;
    order := 1;
    char_gens := [5357, 1237, 5773];
    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_8034_a_Hecke(Kf)
    return MakeCharacter_8034_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 := [[-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, -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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0], [-1, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0], [0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0], [-1, 1, -1, 0, 0, -1, -1, 1, 0, -1, 0, -1, 1, 0], [-3, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1, 0, 0], [-1, 1, 1, -1, -1, 1, 0, 0, -1, 0, 0, 0, 0, -1], [-2, -1, 1, -1, 0, 1, 0, 0, -1, 1, 0, 1, -1, -1], [-1, 1, -1, 1, -1, 0, -1, 1, 0, 0, 0, 0, 0, -1], [1, -1, 0, -1, 0, 1, 1, -1, -1, 1, 1, 2, -1, 1], [-1, 1, 0, 1, -2, 1, -1, 1, -1, 1, -1, 1, -1, 0], [0, 0, 1, -1, 1, 2, 2, -1, -1, 0, 0, 2, -1, -2], [-2, 0, 0, 1, 2, -2, 0, 0, 2, -3, 0, -1, 3, 1], [-2, 2, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, 1, 0], [-1, 0, 2, -2, 1, 0, 0, -1, 0, -2, 1, -1, 0, 0], [-3, 1, 0, 1, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1], [-3, 1, -2, 1, 2, -1, 0, -1, 1, -2, 0, -1, 1, 0], [3, -1, 0, 0, 1, 0, 0, -1, 0, 1, 2, 0, -2, 0], [-2, 1, -2, 2, -2, -2, -2, 1, 0, 1, -1, 0, 1, 1], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 1, -1, -1, 0, 0, 2, 0, 1, -1, 1, 1], [-1, 0, 0, -2, 2, 2, 0, 0, -1, -3, 0, -1, 0, -2], [0, -1, 0, 0, 1, -1, -2, 1, 1, 1, 0, 0, 0, 3], [0, 2, -1, 1, 1, -2, -2, 0, 1, -1, 1, -3, 2, 1], [-5, 0, 2, 0, -1, -1, -2, 1, -1, 0, 0, -1, 0, 1], [1, -1, -1, 0, 1, 1, 1, -2, 0, 0, 1, 2, -1, 1], [0, 0, -1, 2, 1, 0, -1, 0, -1, -1, -2, 0, 1, -2], [-2, -2, 0, -1, -2, 0, 0, 1, -2, 2, 0, 2, -1, 1], [-8, 0, 1, -3, -1, 0, -2, 1, -1, 2, -1, -2, 0, -1], [-1, 2, 1, 0, 2, 1, 0, -3, 1, -3, 0, -1, 1, 0], [0, -1, 1, 1, 2, 1, 1, 0, 0, 0, 0, 1, 0, 1], [-6, 1, 1, 0, -1, 1, 2, -1, 0, 2, 2, 0, -2, 0], [0, 0, 0, 0, 3, 0, 2, 0, 4, 0, 1, 0, 2, 1], [-5, 1, 0, 1, 2, -1, -1, 2, 0, 1, -1, 1, 0, 0], [-1, -1, 2, -3, 2, -1, 1, 1, 1, -1, 0, -2, 2, 0], [-2, -1, -2, 1, 0, 2, 2, -1, 0, -1, -1, 0, 1, -2], [-6, 1, 1, 3, -2, -1, -2, 2, 0, 1, -1, 0, 2, 2], [-3, -1, -1, 0, 2, -2, -1, 0, 2, -1, 0, -2, 1, 1], [0, -1, -1, 2, -2, -1, 0, 2, -2, 2, -3, 2, -1, 1], [-4, 1, 0, -2, -1, -1, 0, -1, -1, 1, -1, -1, -1, -2], [3, -3, 0, 0, 1, 1, 2, -2, 1, 1, 1, 1, 1, 0], [0, -1, 2, -1, 1, 0, 2, 1, 0, 0, 1, 1, 1, 0], [-1, -1, -1, 1, 2, 0, 1, 0, 0, 0, -2, 1, 0, 0], [0, -1, 2, 0, 1, 0, 3, -2, 0, -1, 1, -1, 1, 1], [-9, 1, 0, 1, -4, 0, 0, 1, -2, 1, -1, 0, -2, 0], [-7, -1, -1, 4, -2, -3, -1, 2, -1, -1, -3, 0, 3, 1], [0, 1, 1, 0, 2, -1, 1, 0, 2, 0, 2, -1, 0, 2], [3, 1, -3, 2, -1, 2, 2, 0, 0, 4, 0, 5, -3, 0], [-5, -3, 1, -1, 1, -1, 1, 1, -1, 0, 1, -2, 2, 0], [0, -3, 0, -1, 1, 1, 1, -1, -1, 1, -2, 2, 0, -1], [-7, 3, -3, 1, -1, -1, 0, 0, 1, 1, -1, 0, 2, 2], [1, 0, -2, 2, 1, -2, 1, 1, 3, 0, -1, -1, 0, 1], [-7, 0, 1, 0, -2, 0, -2, 3, 0, 2, -1, 0, 0, 0], [1, 2, 2, -5, 0, 2, 0, 2, -2, 2, 1, -1, 0, 0], [-4, 1, -2, 1, -2, -2, 1, 1, 0, -2, 1, -1, 2, 0], [9, -3, -1, 0, 2, 1, 2, 0, 0, 1, 0, 2, -4, 0], [0, -1, 0, 0, 1, 0, 5, 0, 2, -1, 1, 1, -1, -3], [4, 0, -1, -1, 0, 2, 1, 0, -1, 1, -1, 1, -3, -3], [0, 1, 0, -1, -2, 2, 0, 0, -1, -1, 3, 1, -2, 0], [-1, -2, -1, -1, 2, 0, -1, -2, 1, -2, 1, -1, 3, 1], [8, -2, 0, -1, 1, 0, 3, 0, 1, -2, 1, 1, 0, 0], [2, -2, -2, 2, -2, 0, -1, 2, -1, 4, -1, 0, -1, 1], [-5, -3, -1, 4, 2, -2, 0, -2, 3, -5, 0, 0, 3, 2], [-3, -2, -1, 1, -2, -1, -4, 2, -2, 0, -2, -3, 1, -1], [-5, -2, 1, -2, -3, 4, 2, -1, -2, 3, -2, 1, -5, -3], [1, 0, -2, 1, 2, -3, -1, -1, 2, -4, 0, -3, 1, 1], [7, -6, 0, 1, 0, 3, 4, -3, 0, 2, 0, 4, -5, 1], [1, -2, 2, -1, 1, -2, -1, 1, -1, 1, 0, 0, 2, 1], [-3, -3, 0, 2, 4, -4, 1, 2, 3, -2, 0, -1, 5, 2], [5, -4, -1, 0, 3, -1, 4, -1, 2, -1, 0, 1, 0, 2], [-2, -1, 2, -6, -2, 1, 0, 1, 0, -2, 1, -5, -1, -3], [-8, 0, 1, -1, 3, 0, 0, -1, -1, -1, 0, 0, 0, -2], [6, -3, 2, -8, 3, 3, 1, 0, -1, -1, 3, -1, -2, -2], [-2, -3, 2, 1, 0, -2, 1, 2, 2, -3, 0, -1, 3, 3], [1, 0, -2, 4, -1, -2, -1, 1, 3, 0, 2, -1, 3, 2], [-2, 0, -4, -1, -2, 1, 2, -3, 0, 0, 0, 3, -4, -2], [2, -3, 3, 0, -3, 2, -1, 0, -4, 0, -2, 1, -2, -3], [0, 4, 0, -1, -2, 4, 2, 1, 0, -4, 2, -2, 3, -3], [-1, 0, -2, 3, -2, 1, 2, 0, 0, 4, -4, 0, 0, 0], [-4, 1, 3, -2, 3, 2, -2, -2, 0, -7, 0, -2, -1, -1], [5, -2, 1, 0, 1, 0, 2, -3, 3, 2, 1, -2, -1, 0], [-3, -5, 0, 0, 4, -3, -3, -3, 1, -5, -2, 1, 0, 1], [-3, -6, 2, -2, 3, -2, 0, -1, 2, -1, 0, 2, 2, 4], [2, 3, -3, 2, -2, -1, 0, 0, -1, 2, 0, 1, 0, 2], [-7, -1, 3, -5, 3, 6, 4, -1, -4, -1, 0, 4, -1, -4], [-5, 6, 0, 1, -5, 5, -2, 1, -3, 1, -2, 0, -3, -2], [7, 1, -2, -2, 3, 0, 3, 0, 1, -1, 1, -1, 0, 0], [-4, 0, -3, 3, 1, -6, -2, 2, 3, 0, -1, -3, 5, 3], [-2, -7, 1, -1, 1, -3, -3, 2, 0, 0, 1, -1, 2, 3], [6, -4, -3, 1, 2, -3, 0, 0, 2, 1, -1, 1, 0, 2], [0, 0, 2, -1, 0, -4, -4, 1, 2, -2, 4, -3, 1, 5], [3, 5, -2, 7, -3, 0, 0, 0, -1, 1, 0, 3, -1, 2], [-2, 2, 0, 3, 0, 2, -1, 3, 1, -4, 0, -1, 1, 1], [0, 3, 0, 3, -2, 2, -5, 3, -2, 1, -2, 0, 0, 0], [-8, 1, -2, 2, 1, -1, -2, -4, 1, -2, 0, 0, 0, 1], [-7, 6, 1, 0, 4, 0, 3, -3, 2, -3, 2, -1, 2, -1], [-3, 4, 3, 0, 0, 2, -6, 2, -1, 1, 1, 0, 2, 2], [-7, 1, 1, 2, 0, 2, 2, 1, 1, 2, -1, 2, 4, 2], [2, 4, -1, 2, -1, -1, -5, 0, 1, -8, 1, -3, 6, 1], [3, -2, -1, 3, 1, -1, 0, -3, 1, -5, -1, 3, -1, 2], [-3, 2, -3, 1, 3, 0, 4, -2, -2, 0, -2, 5, -5, -4], [3, 2, -2, -3, 2, 1, 4, -2, 2, 0, 1, 0, 1, -1], [2, -3, 2, 1, 1, 1, 0, -1, 0, 1, 2, 2, 0, 2], [1, -1, 5, -7, 5, -2, -2, -1, 3, -4, 7, -8, 3, -1], [7, 0, -1, -2, -2, 4, 1, 0, -1, 5, 1, 4, -6, 0], [0, -1, 0, -4, -2, -3, -1, 1, 1, -2, -1, -3, 3, -4], [2, -3, 2, -3, -2, 2, 5, -1, -3, 5, -2, 4, -4, -4], [-5, 2, 0, 4, -3, -2, -2, 2, -1, 0, -1, 0, -3, -1], [6, -1, -7, 4, 0, -2, -2, 1, 4, 0, 0, 2, 2, 3], [3, 0, 0, 2, 1, -6, -4, 1, 1, -4, -2, 1, -1, 4], [0, 6, 2, -1, -3, 1, -4, 3, -6, 0, 0, -1, -1, 0], [7, -5, 1, -4, -1, 0, 4, -1, -1, 6, 1, 2, -5, 0], [-10, -1, -2, 6, -6, 2, -1, -1, -3, 5, -6, 4, -2, 0], [2, -5, 1, -4, 2, 4, 2, -4, 0, 1, 0, 4, -4, -2], [7, 4, -1, 1, 0, 2, -1, 0, -1, -1, 2, -1, -1, -2], [6, -2, 0, -2, -3, 0, 1, 3, -2, 6, 1, 3, 1, -1], [-4, -3, 4, -4, 5, -2, -3, -2, -2, -4, 0, 0, 5, 2], [4, 8, -3, 6, -4, -1, -3, -1, 0, 2, -4, 2, 1, 0], [-3, -3, 4, -3, -5, 5, -1, 4, -6, 7, 0, 3, -4, -1], [-5, -3, -1, 0, 6, -6, -2, -3, 6, -2, 3, -4, 2, 6], [2, 1, -1, 2, -1, -2, 1, -5, 0, 0, -3, 0, 0, 0], [2, 1, 3, -6, 2, 0, 0, -3, 1, -4, 6, -8, 2, -1], [-4, -3, 1, -1, 0, 4, -1, 0, -4, 7, -4, 6, -2, 0], [-5, 1, 2, 2, -3, -1, 2, -2, 2, 3, 2, 3, 0, 1], [-6, -1, -2, -6, 2, 0, 0, 1, 1, 0, 4, -5, 1, -2], [-1, 5, -3, 7, 1, -6, -1, 0, 6, -4, -1, -5, 4, 4], [-1, 0, 1, -3, 4, 5, 4, 0, -2, -1, -4, 5, 0, -7], [-9, 7, -2, 4, 3, -3, -2, 1, 6, -6, 2, -5, 6, 1], [-6, 8, -4, 1, -1, -1, -3, 1, 0, -1, 2, -6, 0, -4], [5, -3, 0, -2, -3, 2, -3, 4, -4, 3, -3, 2, -4, -5], [-7, 5, -2, -1, -6, 2, -5, 2, -4, 1, -1, 1, -3, -2], [-1, 6, -3, 2, -1, 5, 7, -5, 1, -1, 2, 3, -2, -4], [-8, 1, 0, 5, -4, 1, 0, 3, -3, 3, 1, 0, -2, 2], [3, -7, 4, -7, 3, 3, -1, 0, -1, 0, -1, 0, -4, -3], [3, -3, 1, 4, 0, 1, 0, -1, 0, -2, -1, 1, -1, 3], [-9, 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-1, -3], [-8, -12, 9, -6, -5, 6, -5, 2, -11, 5, 0, 6, -5, 3], [-7, 6, 0, 5, -5, 2, 3, 1, 1, 3, 5, 2, -1, 0], [13, -6, 5, -9, 7, -3, 6, -2, -1, 3, 6, 4, -1, 3], [6, 2, 1, -5, -2, 1, -1, 3, -2, -3, 0, -3, 4, -7], [8, 4, -3, 0, -2, -6, -4, 4, 3, 1, 7, -6, 3, 2], [17, -1, -5, 4, 7, 0, -3, -2, 4, -10, -4, 1, 3, -1], [8, -1, 0, -3, 4, -4, 1, 3, 5, -9, -3, -4, -1, 3], [-5, -4, -1, 3, 5, -7, -3, 4, 5, -3, -8, -1, 5, -1], [1, 1, -3, 5, -10, -3, -3, 1, -1, 10, -4, -2, 0, 4], [8, -3, 7, -1, 8, 2, 2, -6, 2, -6, 0, 3, -1, -3], [18, -5, -1, -3, 0, 2, -2, -8, -5, -2, -2, 3, -1, -3], [18, -16, 6, -6, 1, 0, 5, -3, -3, 6, 1, 1, -3, -3], [-1, 0, -2, -1, -5, -10, -9, 2, 0, 4, -6, -5, 2, 3], [7, 5, -1, 0, -4, 0, 5, 4, 0, 5, 3, -4, 4, 2], [9, 5, 1, 1, -4, -4, -5, 8, 0, -6, -2, -10, 10, 3], [6, -9, 3, -4, 5, 4, -2, 1, -2, -4, 1, -2, -4, -2], [22, -4, 4, -9, 4, 8, 7, -3, -6, 0, -1, 5, -2, -12], [-4, 8, 1, -6, -5, 5, -1, 2, -3, -7, -4, -8, 0, -8], [-11, 0, 4, -5, -5, 11, -1, 2, -6, 10, -7, 4, -8, -9], [-4, 3, 1, -8, 1, 1, 1, -2, 1, 0, 3, 1, -2, 0], [24, 0, 12, -12, 6, 3, -6, 0, -3, -2, 8, -3, 3, 1], [14, 3, -2, 2, 6, -6, 0, 1, 3, 0, 2, -9, 7, 2], [3, -5, 1, -4, -3, 11, 12, 2, -2, 8, 6, 3, -6, -6], [7, -3, 0, 0, 2, 3, 4, 1, 3, -7, -1, 0, 3, -3], [19, -3, 2, -4, 9, 0, 7, -5, -1, 1, 0, 3, -2, 2], [8, -7, -6, 7, 6, -11, -7, 0, -2, -7, -3, -1, 5, 1], [-1, -14, -3, 5, -7, -1, -4, -2, -5, 3, -5, 6, 0, 0], [-16, 6, 3, 6, -5, -3, -9, 4, 0, 1, -2, -1, -2, 5], [-17, -2, -1, 1, -2, 3, 3, 3, 1, -3, -1, -5, -2, -8], [3, -12, 3, 4, -4, -2, -7, 3, 0, 9, -4, 7, -10, 5], [-10, 1, 1, -1, -2, 5, 2, 1, -7, -2, -1, 0, -6, -4], [25, -7, -9, 4, 3, -1, 2, 0, 6, -2, -4, 1, 0, -4], [22, -1, 0, -8, 4, 4, 8, 2, -3, -1, 3, -3, 2, -4], [-6, 5, -4, 1, -3, 0, 4, -5, -4, -5, -1, 2, -6, -3], [16, 5, -1, 0, 7, -4, -7, -4, 9, -7, 7, -5, 4, 10], [10, 3, 2, 3, -1, 0, 6, -2, 2, 5, 4, 6, -4, 5], [27, -4, 5, -3, -1, 0, -1, -2, -6, 4, -4, 8, -7, 2], [2, -6, 2, -6, -1, 13, 3, -5, -3, -5, -1, 6, -3, -2], [10, -10, 7, -9, 10, 4, 6, 1, -5, -4, 1, 0, 7, 2], [1, -7, -1, 0, 1, -3, -7, 3, -3, -4, 0, 0, -3, 4], [-11, -11, -1, 1, 0, -5, -3, -1, -8, 6, -7, 7, -3, 4], [-14, 0, 2, -2, -3, 6, 3, 0, -7, -3, -6, -1, 4, -14], [13, -6, -7, 8, -7, -7, -1, 2, 2, 9, -2, 4, -5, 2], [-15, 9, -4, 2, -5, -5, -10, 4, -4, -3, 2, -5, 6, 0], [2, -8, 1, -1, -3, 2, -1, -5, -1, 11, -2, 7, -16, -4], [14, -9, 8, -7, -3, -1, 2, -1, 0, -1, 6, -4, -2, -2], [12, -6, 11, -6, -1, 9, 4, -1, -9, -3, 5, 7, -2, 3], [27, -2, 5, -6, 4, 12, 8, -9, 3, 0, 0, 4, -7, -1], [4, 0, -2, 7, 2, -2, -5, 1, 1, -4, -6, 3, 1, 5], [14, -5, 1, 2, 1, -2, 0, 0, 1, 2, 4, 7, 0, 8], [-7, -3, 0, -2, 3, -2, 10, -3, 6, -5, 3, -6, -1, -7], [1, -6, 5, -8, -9, 6, -1, -4, -6, 1, -1, -2, -5, -7], [-14, -1, 4, -8, 3, 4, 10, -5, 2, 5, 10, -4, -7, -4], [9, -7, 1, -10, -1, 0, 10, -5, -2, 5, 1, 1, -4, -6], [7, -2, 1, 1, -7, -4, 2, -3, -1, 4, 1, 4, -3, 5], [-22, 9, 2, 1, -4, -2, -8, 8, -2, -3, 0, -7, 1, -5], [-16, 3, 2, -6, 4, -1, 2, 0, -3, 0, 3, -2, -4, -3], [7, -7, -2, 3, -3, 1, 2, 5, -7, -1, -8, 6, 1, -6], [23, -2, 0, -8, 3, 3, 4, 2, -1, 6, 0, 4, -5, -3], [-14, 7, -2, -6, -5, -4, -3, 6, -1, -1, 1, -10, 9, -1], [1, 2, 0, -6, 4, 4, 4, 0, 1, -9, 3, 1, 1, -2], [18, -4, 2, -8, 3, 4, 4, 5, 0, 2, -3, 7, -1, -6], [-14, 6, -2, 2, -5, -2, 0, 3, 4, 3, 2, -4, -1, -4], [5, -2, 3, 4, 3, -2, -2, 4, 6, -5, -6, -4, 2, 1], [29, 3, 1, 10, 1, 6, 10, -4, 6, -3, 5, 4, -3, 2], [-7, -1, 7, 0, -6, 3, -7, 1, -2, 6, 2, -2, 0, 0], [7, 1, -6, 11, -5, 0, 1, 2, 2, 5, -7, -1, 0, -2], [-1, -1, -5, 4, 4, -5, -2, 6, 2, -6, 1, -3, 8, 1], [-2, 2, 8, -3, 1, 1, 7, -8, 3, -7, 6, -8, -1, 4], [-7, -6, 0, 7, -11, -1, -4, 7, 2, 6, -9, -1, -1, 2], [-3, 2, 4, -5, -4, 4, 0, 3, -7, 12, 0, 3, -4, -3], [-17, -4, 5, -2, -7, 4, 2, -5, -2, 9, 1, 1, -8, 2], [11, 0, 1, -7, 2, 1, 8, -4, 3, 0, 3, -1, -6, -4], [3, 14, -5, 9, -6, 0, -3, -5, 2, -3, 1, 5, -3, 2], [4, 3, -5, -2, -5, 0, 3, 0, -3, 10, 2, 9, 0, 5], [-1, 5, -1, 12, -1, -3, -1, 0, 3, 6, -6, 1, -4, 4], [-20, 9, -5, 9, -3, -6, -8, 7, -1, -2, -5, 2, 6, 1], [11, -2, 5, 1, 4, -2, 3, -5, 4, -10, -3, -6, 6, -1], [5, 0, -1, -5, -1, -2, 8, 0, -5, -3, 3, -2, -1, -5], [11, -1, 4, -4, -10, 19, 2, 3, -11, 8, -3, 16, -12, -10], [-7, -8, -6, 3, 1, -8, -6, 6, -2, 2, -10, 4, 6, 5], [-11, 7, 1, -4, 2, 0, -1, -1, -3, 4, -4, -4, -1, -4], [8, -9, 3, -4, 3, 7, 12, -6, -5, 4, 1, 9, -10, 1], [3, -4, 1, 6, 0, 2, 7, 0, 5, -6, 0, 8, 0, 5], [10, 7, -2, -2, 2, 6, 2, -1, -4, -1, -9, 2, -3, -6], [10, -5, -4, -3, -1, 0, 3, 2, -3, 15, -2, 4, -2, 0], [-7, -3, 1, 5, 2, -8, -8, 7, -2, -3, -1, 3, 5, 9], [12, 3, 5, -8, 3, 13, 15, -4, -2, -3, 8, 2, -4, -2], [-2, -9, -1, 0, 7, -2, 1, -4, 4, -2, 6, -4, 0, -4], [5, -5, -2, -1, -4, -1, 7, 1, 0, 9, 2, 10, -8, 6], [-3, -5, -3, 6, -4, 10, 8, 4, -3, 2, -3, 7, -9, -8], [-22, 6, -3, -1, -5, -4, 0, 6, -3, 12, -3, -3, 1, -1], [-10, -3, 0, 3, -2, 0, 5, 6, -1, 1, -1, 5, -2, 6], [3, -1, -2, 4, 10, 0, 4, -1, 3, 5, -1, 0, 0, 2], [10, 2, 1, -1, 3, 0, -1, 7, 0, -7, 0, -9, 14, 0], [-1, 5, 4, -1, 8, -3, 9, 0, 8, -1, 5, -7, 8, 7], [2, 3, -8, 9, 0, -7, 0, 1, 8, 1, 0, -4, 9, 2], [12, 0, -3, 6, -4, -1, 0, 0, 4, 5, 1, -6, 0, 1], [-1, 4, -5, -4, -2, -7, -8, 1, 3, 5, 2, -5, 1, -4], [-13, -10, 3, -1, -5, -7, -2, 1, -1, 1, -7, 1, 1, 2], [-4, 5, -4, 0, 1, 4, 0, 3, -5, -1, -7, 4, 0, -13], [24, 3, 0, -2, 3, -3, 4, -1, -5, -11, 3, -1, 7, -2], [5, 1, -1, 1, 1, 8, -1, -4, -5, 2, 2, 9, -7, 1], [11, 1, -3, 1, 8, 2, 2, -6, 2, -16, 4, -2, 2, -2], [-11, 2, -3, -2, 10, -5, -5, -2, 4, -8, 4, -9, 7, -4], [-7, -14, -3, 1, 6, -8, 10, -4, 6, 6, 0, 4, 0, 8], [4, -4, 4, -6, -4, 2, -9, 2, -1, 11, -3, 5, 4, -1], [-20, 5, -4, 11, -6, -11, -5, 3, 3, 3, -1, 2, -2, 13], [-9, 1, 1, 0, -3, 12, 12, -7, 0, -1, 0, 5, -3, -1], [3, -4, 5, -4, -1, 0, -1, -8, 1, -3, 7, -2, -1, 7], [-30, 6, 2, 1, -7, -5, -9, 4, 1, -3, -2, -5, 5, 0], [-4, 5, 1, 2, -3, -1, 2, -2, -1, -2, 4, -5, 11, 6], [-14, -2, 5, 4, -3, -3, -2, 6, 0, -8, 6, -7, 5, 6], [7, 5, -1, -3, 7, 3, 6, -2, 1, -8, 5, 1, -1, -7], [9, -3, -1, 6, 1, -8, -5, -2, 6, -4, 2, 3, 3, 1], [-7, 13, 1, 6, -5, -1, 0, 4, -6, 6, -3, 7, -3, 3], [-5, -1, 3, -11, 2, 1, 1, -2, -4, 10, 6, 5, -5, -3], [-16, -1, -5, -3, 0, -9, -7, 8, 10, -4, -1, -15, 9, -1], [6, 2, -2, -1, 8, -1, 2, -12, 4, -4, 8, 4, -5, 2], [-20, -2, 2, -6, -3, 2, -2, 1, -10, 0, -3, -2, -5, -3], [-11, 8, -5, 0, -4, 2, 3, -1, -4, 12, -1, 4, 0, -1], [30, 2, 3, -10, 2, 10, 6, -4, -1, 5, 11, 4, -11, -2], [-46, 6, 0, 3, -1, -6, -7, 4, -3, -1, -3, 3, 1, 6], [-3, 4, -2, 3, 2, -12, -8, 0, 5, 1, 1, -9, 1, 12], [-6, 13, -5, 2, -6, 0, -5, 7, -2, 6, 2, -5, -2, -5], [-3, 5, -9, 10, 6, -9, -7, 9, 5, -2, -4, 0, 8, 0], [-7, 3, 3, 4, 5, -3, 0, 0, 5, 2, 2, -3, 0, 12], [8, 2, -2, -3, 4, -2, -4, 6, 0, -9, -1, -3, 0, 4], [-24, 6, -1, 2, -6, 5, -3, -5, -10, -4, -4, 2, -7, -6], [-11, -2, 7, -7, -2, 4, 8, -6, -9, 7, -3, 6, -10, -5], [-16, -12, 2, -3, -2, -4, -2, 4, -8, 10, -10, 8, -4, 0], [-4, -6, 6, 1, 7, 2, 3, -1, -5, 4, -4, 7, 0, -2], [0, 3, -2, 4, -3, 3, 9, -4, 9, 1, -5, -3, 2, -2], [5, -2, -5, 1, -7, 7, -3, 1, -5, 15, -7, 14, -9, 0], [-24, 14, -1, 2, 3, 1, -12, -8, 2, -13, -5, -2, 3, -6], [4, -4, 0, 6, 2, -3, 8, 3, 4, 8, 5, -1, -3, 6], [11, 2, -1, -9, 0, 2, -4, -4, -8, 3, -2, 0, -9, -8], [-8, 5, -12, 14, -12, -3, -11, 11, 3, 13, -3, 1, 2, 6], [12, -4, 5, -9, 4, 0, 5, 3, 3, 5, -1, 0, 6, -5], [-4, 0, -6, 7, 5, -13, -13, 5, 13, 0, 4, -1, 11, 11], [-10, 2, 1, -8, 6, -1, -5, -1, 1, 5, 7, 1, -5, -4], [-14, 9, 2, 4, -1, -2, -12, 7, -3, -9, -1, -8, 6, -1], [-11, -13, 4, -5, 5, 5, 10, -6, 7, 4, 5, 3, -2, 5], [3, -10, -2, -9, 3, 3, 14, -6, -1, 8, 9, 8, -4, 1], [-15, 1, 10, -4, -10, 11, -6, 4, -11, 6, -10, -2, -6, -15], [8, 1, -4, 13, 6, -2, -1, 5, 4, 0, -7, 5, 2, 2], [12, -4, -8, 8, -12, -2, -6, 0, 2, 16, -4, 2, -4, 3], [15, 2, 0, 4, -4, 5, 6, 2, 1, -5, -5, -1, 6, 0], [-5, -4, 11, -11, 1, 22, 12, -2, -12, 10, -5, 15, -14, -13], [-11, 1, 9, -9, 6, -2, -7, 1, 0, -12, 7, -18, 18, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8034_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8034_2_a_bb();".
// 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_8034_2_a_bb(:prec:=14)
    chi := MakeCharacter_8034_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_8034_2_a_bb();".
// 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_8034_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2048, 7296, 10816, -42456, -3280, 39390, 678, -14851, -847, 2601, 269, -207, -29, 6, 1]>,<7,R![-10496, 47936, 37432, -129678, -49359, 101391, 27271, -30365, -7497, 3920, 963, -214, -53, 4, 1]>],Snew);
    return Vf;
end function;