
// Make newform 8016.2.a.be in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_2_a_be();".
// 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_8016_2_a_be();".
// 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 := [242, -5687, 292, 6584, 272, -2470, -151, 417, 22, -33, -1, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-7710406, -38238885, 34805559, 74577279, -15128561, -23458607, 2748114, 2378035, -193391, -70944, 2623], [-2478234, 19499297, 11595133, -25269067, -7692731, 9091923, 1417390, -1226719, -65533, 55664, -1379], [34175614, -220910235, 124569801, 195304833, -92700271, -52505737, 21384366, 5477813, -1989097, -195192, 64985], [39160638, -427193567, 176633261, 387400045, -109201699, -110447061, 25094846, 12499993, -2608397, -492056, 101765], [20257270, -163661007, 86939145, 142378425, -87108655, -43356601, 25357890, 5468837, -2909401, -242652, 116141], [-65756350, 928868747, -618583841, -850380841, 481600775, 263695489, -124595270, -32908293, 13457169, 1421984, -522481], [-13787246, 1176264339, -743357145, -1213924881, 514603631, 379578137, -132016230, -46952653, 14695769, 2015712, -596041], [47745918, -908813755, 385046857, 789881825, -278511527, -223268481, 71466478, 25611773, -7769641, -1032256, 303385]];
    Rf_basisdens := [1, 1, 1, 10323192, 1876944, 10323192, 10323192, 5161596, 20646384, 20646384, 10323192];
    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_8016_a();"
function MakeCharacter_8016_a()
    N := 8016;
    order := 1;
    char_gens := [3007, 2005, 5345, 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_8016_a_Hecke(Kf)
    return MakeCharacter_8016_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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 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], [1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, -1, -1, 1, 0], [1, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0], [2, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 1, 1, 1, -1, 1], [1, 0, -2, 0, 0, 0, -1, 0, 0, 1, -1], [2, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0], [-1, 1, -2, 0, -1, 0, -1, 0, 1, 0, 0], [1, 0, 0, 0, -1, 2, -2, 0, 1, 0, 0], [4, 0, 1, 1, 1, -1, 0, -1, 0, -1, 1], [0, 0, 0, -1, 1, 0, 1, 0, 0, 0, -1], [0, 0, 0, 0, 0, 2, 2, 1, 0, 0, 1], [1, 0, 0, 1, 0, -1, -1, -1, -1, -1, 0], [2, 0, 1, 0, 1, -2, 0, -1, -2, -1, -1], [2, 0, 0, -1, 1, 0, 1, -1, 1, 0, 0], [1, 0, 0, 2, 1, 1, 0, 2, 1, -1, 1], [-1, 0, -1, 1, 0, -2, 0, 1, -1, 0, 0], [1, 2, 1, -1, -1, 2, 3, 1, 2, -1, 0], [1, -2, -2, -1, 0, 0, -3, 0, 0, 0, -1], [3, 1, 0, 0, 0, 0, 0, -1, 1, -1, 1], [-1, 2, -1, 0, -2, 3, 0, 1, 1, 0, 0], [-1, -2, 1, -1, 0, -2, 0, -1, 1, -1, -1], [1, 0, -1, 0, 0, -2, 0, 0, 1, 2, 0], [2, 1, 0, -2, -2, 2, -1, 1, 2, 1, -1], [0, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1], [-2, 0, -1, -1, -1, 0, 0, -2, 2, 2, 1], [2, 2, -1, -2, 0, 2, 1, 1, 0, 0, -2], [0, -3, 0, 0, 2, -2, -1, 0, -2, -2, -1], [5, 4, 0, 0, 1, 1, 2, -1, -3, 2, -1], [2, -1, 1, -1, 2, -2, 0, -1, 0, -1, -1], [6, 0, 2, 1, 1, -2, -1, -2, 0, 1, -1], [2, -3, 2, 2, 2, -4, -1, -1, -2, 2, -1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, -2, 0, -1, 1, -2, -1, 0, 2, -1, -1], [-2, 0, 1, -1, 2, 2, 1, 0, 1, -1, -1], [3, 2, 0, 3, -1, 2, 0, 2, -2, 1, 1], [0, 2, 2, 0, 1, 0, 1, 0, 0, 1, 0], [-2, 2, -1, -3, -2, 4, 3, 1, 3, -1, 1], [9, -1, 0, 3, -4, 2, -4, 1, -1, 3, 1], [2, -2, 2, 1, 3, -2, 1, 1, -3, 2, 0], [-3, 2, 0, 1, -3, 4, 2, 2, 0, 3, 2], [-1, 0, -3, 2, -2, 0, -4, 1, 3, 1, 0], [-5, 0, 1, 1, 2, 0, 0, 2, -3, 0, 2], [2, 0, 2, 1, -1, 2, 1, -1, 4, -3, 0], [3, 2, -1, 1, -1, -2, -1, 0, -1, 0, 1], [-1, 2, -3, 2, -3, 4, -3, 0, 2, 1, 1], [0, -2, 3, 0, -1, 0, 2, 0, 4, -1, 1], [-1, -4, 0, 1, -1, -2, -4, 1, -1, 0, -1], [10, 3, 1, 0, 3, -2, -1, -3, -2, 0, -2], [-1, 4, -2, 0, -3, 4, -4, 1, -1, 2, -1], [6, -2, -3, 0, 0, -3, -1, 2, -1, 1, -1], [3, 0, 1, 2, 0, -1, -2, -2, -2, -3, 1], [0, 4, -3, 1, -4, 4, -1, 1, 2, 1, 0], [0, 2, -2, 0, 1, -2, 3, 0, 2, 1, 1], [2, 0, 0, 3, -1, 0, -3, -2, -2, 3, -1], [5, 6, -4, 1, 1, -2, 0, 2, -1, 0, -1], [-3, 1, -2, -1, -3, 2, -1, -1, 1, 2, -2], [3, -2, 1, 1, 3, -6, -1, -2, -5, 0, -1], [2, -2, -1, 1, -2, -2, -5, -2, 0, 0, 2], [9, -2, -2, 2, -2, 0, -1, 2, 2, 0, -1], [7, 0, 5, 0, 7, -3, 5, -4, -4, -1, -2], [-3, 4, -1, -1, -1, 2, 3, 0, -2, -1, -2], [-10, 0, -4, -4, -6, 6, 0, 3, 5, -2, -1], [-2, 2, 2, -1, 1, -2, 1, -2, -1, -1, 0], [4, 2, -1, 2, 0, 2, -1, 1, 0, 2, 4], [-2, 4, 1, 2, 3, 0, 6, 1, 2, 1, 1], [-3, 0, -2, -2, -2, 0, -1, 0, 3, 1, 0], [-6, 4, 0, -2, 2, -4, 6, 1, 3, 0, 1], [4, -5, 2, 4, 6, -6, -1, -1, -3, -1, 0], [1, 0, 4, -1, 5, -2, 4, 1, -4, -5, 1], [8, -3, 1, -4, 0, 0, 0, -1, 1, -1, 2], [-1, 0, 3, -2, 3, -2, 9, 2, -1, -4, -2], [3, 1, -2, 1, -3, 2, -5, -1, -3, 1, 2], [-3, 0, -2, 3, 3, 0, -6, -1, -4, 1, -1], [-5, 0, -2, 0, -6, 6, -3, 4, 4, 0, 4], [-5, 4, 2, -1, -1, 0, 4, -1, -2, 1, -2], [3, 2, 2, 4, -1, 0, 0, 0, 2, -1, 0], [1, 0, 0, 0, -1, 0, 0, 3, -3, 2, 0], [6, 5, 1, -3, 2, 2, 2, -1, 1, -2, 2], [3, -2, 1, 6, -1, -2, -5, 1, 1, 1, 3], [-1, 2, 4, 0, -1, 0, 8, 3, 4, -1, 4], [-4, 0, 5, 4, -1, 4, 2, 1, 1, 0, 4], [3, 0, -3, -1, 0, -2, 0, -4, -1, 3, -2], [4, 2, 0, -1, 2, 5, 0, 0, -2, 1, 2], [0, 0, 1, 5, 6, -4, -5, -3, -2, 0, 0], [-3, 2, 3, 0, 5, -4, 5, -1, 1, -4, 3], [2, 2, -2, 3, -2, 3, -4, 2, -3, 2, 1], [3, -2, -2, -1, 5, -6, -2, -5, 1, -2, -2], [-1, 2, 1, -2, -3, 3, 1, 6, -1, 0, 1], [-1, 2, -1, -3, -6, 6, 0, 2, 7, -1, 1], [4, 0, 1, 1, 6, -2, 1, 0, 0, -2, 2], [1, 2, 0, -1, 4, -1, -1, -6, -3, 0, 0], [-4, 6, -7, 1, -4, 0, -1, 1, 1, 3, -5], [-7, -4, 0, -2, -2, -2, -5, -3, -2, 1, -1], [7, -4, 6, 0, 3, -1, -2, -3, -5, 0, 1], [11, -4, -3, -3, 2, -2, 2, 2, -3, 3, -4], [0, -6, -1, -2, 1, 0, -6, 1, -4, 3, -1], [-4, 1, 2, -5, -1, 0, 0, -2, 4, -4, 2], [-3, 1, -3, -1, 0, -2, 1, 1, -2, 4, -2], [2, 2, 1, -2, 2, -2, 1, -2, 2, 0, 1], [-7, -2, 1, -2, 3, 0, 9, 3, 2, -6, 0], [-6, 1, -3, -5, 0, 6, 6, 3, 4, -3, -1], [1, 6, -6, -1, 0, 0, -1, -3, 1, -1, 0], [-14, 8, -2, -1, 0, 0, 2, 2, 5, -3, -1], [3, -3, -2, 0, -6, 2, -6, 3, 1, 2, -1], [0, -4, 5, -2, -1, -4, 2, 4, 1, -2, 1], [-11, 0, 0, -1, -1, 2, 4, 3, 0, 1, -2], [-1, -2, -1, -3, 0, 3, -2, 0, 4, -2, -1], [13, 0, -7, 3, -3, 0, -7, -2, 1, 2, 1], [0, -3, -3, 8, -1, -4, -5, 2, -1, 1, 1], [-4, 0, -1, 0, 3, -1, 6, 2, 4, -3, 1], [-6, -2, -3, -1, 0, -4, 1, 1, 0, 0, -4], [0, -4, 4, 0, 0, 2, 6, 4, -1, -1, -3], [-5, 4, -1, -1, 2, -6, 2, -4, 0, 1, -1], [-2, 6, 6, 3, 2, 2, 2, -2, -4, 1, -2], [-9, -2, 3, -6, -1, 0, 3, -2, 2, -1, 0], [16, -4, 2, -2, -2, 2, -2, -2, -5, 1, -1], [8, -2, -1, 1, 1, 4, -2, 3, -3, -1, -2], [5, -2, 3, -3, 2, -1, -2, -2, 1, -3, 2], [4, -2, 4, -2, 6, -4, 2, -7, 0, -2, 0], [-15, 4, 4, 0, 8, -6, 9, 2, 0, -5, 1], [4, -6, -4, -7, 1, -4, 3, -1, 6, -1, -3], [5, 5, -3, -2, -3, 4, 2, 1, -2, 4, -1], [-10, 4, 0, -2, -2, 2, 6, 0, 2, 2, 1], [-5, -2, 4, -1, -6, 5, 1, 1, -1, 1, 4], [13, -2, 0, -2, 3, -2, 4, -1, -2, -1, -2], [9, -1, -1, 4, -3, 6, 2, 5, 7, -3, 0], [-4, -4, -2, 5, -1, -2, -5, 2, 1, -4, 0], [-6, 6, 1, 3, 3, 0, 2, 1, 0, -1, -3], [-6, 2, 1, 6, 7, -2, 2, -2, 2, -1, 3], [2, 1, -1, 1, -6, 6, -6, -1, 4, 0, 3], [0, -5, 2, -1, -2, 0, -5, -2, -3, -1, -2], [2, 6, 0, 0, 2, 1, 0, -2, 4, -4, -2], [-5, -4, -3, -3, 3, 0, 3, -1, 7, -6, -3], [3, -1, 2, 9, 5, -6, 3, 0, 1, -4, 2], [7, 0, 1, 0, 7, -7, 3, -2, -4, 3, -2], [4, -2, 8, -7, 4, -3, 6, -3, -2, -1, 0], [4, 0, 0, 0, -5, 2, -1, 3, 2, 2, 1], [1, 0, 4, -1, 3, 0, 0, -3, 1, 1, 3], [19, -4, 2, 0, 3, -4, -6, -7, -8, -1, -3], [-13, 0, -8, 0, -4, 2, -5, 2, -1, 5, -1], [-6, -4, 0, -5, 6, -4, 2, -3, -5, 1, -3], [-1, 4, 4, 3, -5, 2, -4, 2, 3, -4, 7], [10, -2, -3, 7, -6, 2, -5, 2, -1, 1, -3], [-12, -6, 2, 3, -3, -2, -1, -2, 3, -2, 2], [-2, -5, 3, 6, -1, -4, 1, -3, 0, 0, 2], [-7, 2, 1, -2, 1, -4, 3, -2, -2, 1, -8], [9, -8, 0, 0, -2, 2, -3, 0, 1, -3, 2], [-2, 0, 1, -2, -1, -6, 2, 2, 4, -3, -3], [13, -10, -4, 2, -3, 2, -10, 0, 0, 3, 0], [-1, 6, -6, -1, 2, 0, 3, 3, 2, 4, -3], [-6, 4, -4, 2, -1, 4, 7, 6, 5, 0, 0], [-3, -2, -1, -6, -9, 8, 1, 2, 2, 5, -4], [-8, -5, -4, -5, -4, 0, -3, 0, 4, -2, -5], [-14, 0, 4, -2, 6, -4, 8, 0, 0, -6, -1], [-3, 7, -3, 4, -5, 10, -4, -1, 1, 1, 0], [2, 0, -5, 7, 0, -4, 1, 0, -3, 3, -3], [-1, -2, -6, 2, 0, -4, 1, 3, -5, 4, -2], [7, 8, -5, -2, 2, -5, 0, -4, -4, 5, -3], [1, 2, 2, -5, 3, 2, 6, 5, 2, -5, 2], [4, 2, 2, 2, -1, 1, -9, -2, -7, 7, -4], [-21, 6, -3, -5, -6, 8, 4, 10, 7, 1, -1], [3, -8, 2, 0, 7, -8, 4, -7, -3, 0, -4], [1, 0, 4, 0, 6, -6, 7, -3, -5, -3, -2], [4, -2, 7, 1, 4, -4, 5, -7, -5, 2, -5], [-13, 8, -4, 2, -2, 6, -7, 3, -3, 5, 2], [4, 10, -2, -6, -3, 4, 9, 2, 4, -1, -1], [-1, -4, 4, 1, 4, -4, -1, -5, -5, 1, 1], [-11, 1, -3, -2, 9, -2, 2, -3, -6, 2, 1], [-13, 2, -4, -4, -2, -2, 3, 1, 9, -3, -4], [-12, 8, -6, -3, -5, 8, -3, 3, 9, 0, 0], [12, -4, 0, 4, 3, 1, -3, 1, -2, 1, 1], [7, -4, 0, 3, 5, -10, -6, -11, -9, 2, -3], [-2, -4, 7, 1, 4, 2, 3, -3, -6, 1, 8], [-1, 4, 0, 1, -2, 0, -5, -2, -6, -2, 5], [3, 6, 6, 5, 1, -6, 2, -1, 0, 1, 4], [-1, 2, -1, 1, 8, 0, 4, -4, -6, 3, -1], [5, 7, 2, 5, -1, -2, -3, -5, 2, 2, -3], [8, 2, 0, -6, -2, 4, 0, -4, 2, 0, -2], [7, -4, 2, -2, 0, -8, 1, -2, -2, 2, 3], [1, 0, -3, -1, 8, -10, -2, -3, -5, 1, -1], [-11, -6, 1, -1, 0, -4, 2, -3, 0, -8, -3], [-10, 4, 8, -6, 1, 2, 9, -1, 1, -4, 1], [2, 4, -4, 2, -2, 6, 4, 3, 6, 4, -2], [23, 0, -8, 0, 2, -2, -11, -4, -9, 7, -6], [-3, -4, 3, -3, -6, 4, -4, 1, 2, -3, 1], [-11, 0, 1, 6, 1, 2, 1, 5, 4, -7, 0], [-10, 4, -1, -3, -6, 8, 5, 11, 3, 1, 3], [-20, 4, 0, -4, -2, 6, -2, 5, -1, -3, 2], [5, 2, 4, -2, 5, -2, 0, -5, -5, 4, -1], [-2, 2, -1, 5, -5, 10, 4, 5, 6, -3, 0], [-15, 8, -2, -7, -5, 8, 2, -1, 6, 1, 3], [18, 0, 2, 0, 7, -4, -7, -9, -7, 2, -2], [1, 6, 0, 1, -4, 2, 5, 3, 2, -6, 0], [12, 4, -2, 2, 6, 1, -2, -6, -3, 1, 3], [5, 2, -1, -2, -6, 2, -2, -3, -6, 10, -1], [-2, 2, 1, -7, -4, 4, 3, 3, 5, 0, 3], [-7, -2, 7, 1, 8, -2, 6, 5, -8, -6, 3], [2, 0, 6, 0, 1, -4, 5, -5, -6, -4, -3], [11, -8, 5, -3, 4, -14, 0, -5, -4, -4, 1], [-10, 4, 3, 6, 0, 3, 1, -2, 6, 1, 2], [-4, 0, -6, -1, -3, 4, 3, 6, 2, 3, -8], [-3, 1, -2, 0, -2, 0, -6, 4, -6, 5, -4], [-16, -2, 5, 2, 3, 0, 10, 3, 5, -4, 2], [13, -2, 2, 4, 3, -6, -8, -8, 3, -4, -2], [-2, 0, 4, 6, 6, -4, 2, 3, -1, -3, 7], [8, 1, -9, -3, 0, -6, -8, -3, 1, 6, -2], [-18, -2, -5, 1, 1, 6, -4, 1, -3, 7, -2], [0, 2, -6, 6, -2, 6, -6, 6, 4, 0, 3], [-12, -4, -1, 3, 1, 4, 2, 2, 10, -7, 2], [-31, -2, 2, 8, 3, -1, -6, 3, -2, -2, 6], [-10, 2, 2, -11, -3, 8, 13, 7, 4, -1, 1], [-17, 2, 7, -7, 1, -2, 15, 1, 6, -3, 4], [-8, 2, 2, -3, 2, 3, 8, 5, -1, -7, -1], [-1, 2, 2, 5, -4, -2, -5, 1, -2, 1, -1], [5, 6, -8, 1, 2, 0, -5, 2, -6, 4, -7], [8, 7, -1, 0, -1, -4, -1, -3, -2, 0, 2], [3, -2, 1, -8, 5, 2, 1, 1, -6, -3, -2], [-14, 0, 6, 0, 9, 2, 5, -1, -2, -4, -9], [-14, -2, 5, -2, 5, -2, 0, 0, 1, 2, -3], [-7, 2, 4, -2, 5, -10, 12, 1, 1, -2, 5], [5, 4, -8, 7, -3, 2, -12, -5, -1, 2, -3], [-8, -5, 6, -2, 0, 4, 1, 4, 2, -3, -1], [-5, 4, 0, 3, -7, 10, 2, 5, 9, -6, 3], [5, 2, 5, 8, 7, -4, 3, 0, 0, -2, 0], [-15, 3, 7, -1, -1, 8, 8, 2, 3, 2, -2], [11, -2, 9, 1, -5, 1, -1, 3, -1, -5, -1], [-4, 6, 6, -6, 5, 1, 15, -1, 6, -2, 1], [-6, 0, 3, 4, 7, -4, 4, -5, -5, 6, -2], [-12, -10, 0, -9, 3, -4, 5, -5, 0, 3, -5], [13, -4, -2, 0, -4, 4, -5, 2, -2, 6, -3], [11, -6, -3, 1, -4, 8, -6, -5, -2, 2, -7], [0, 0, 4, 0, -4, 2, 8, 2, 1, -2, -3], [11, -2, 0, -6, 2, -2, 5, 0, 3, -5, -3], [-1, 8, 6, 9, 2, -2, 1, -2, 0, -3, -3], [2, 0, -2, 1, 7, -2, -1, -1, -9, 2, -6], [-9, -1, 1, -2, 3, 2, 12, 3, 3, -3, 4], [-16, -2, 0, 2, -2, 4, 6, 3, 6, -9, 4], [-1, -8, 11, 3, -2, 2, 4, 2, -2, 2, -1], [-5, 2, 9, 3, 4, 0, 8, -3, -4, 2, 8], [-15, -6, 4, 2, 5, 0, 2, 3, 10, -13, 4], [-1, 2, 6, -1, 15, -14, 8, -6, -5, -1, -1], [-5, 4, -4, 3, -9, 4, -8, -5, -2, 5, -2], [19, -8, 0, -1, 7, -8, 4, 5, -6, -3, -4], [-4, 6, 1, -3, -2, 2, -5, -4, -1, 1, -2], [1, -2, -2, -1, 1, 6, -2, -4, 9, -4, -3], [-7, 2, 1, -9, 2, 0, 12, 4, 6, -4, -5], [-6, 6, 5, 8, 11, -6, -2, -2, -7, -2, -2], [-6, 5, -10, -5, -16, 4, -5, -1, 7, 7, 2], [3, 8, -2, 7, -3, 6, -4, 2, -4, 3, -2], [2, -2, 2, 6, -2, 2, -2, 2, -2, 3, -2], [-28, -4, 5, -5, -8, 8, 9, 12, 8, -4, 2], [-1, -5, 0, -6, -4, 2, 2, 1, 2, -4, 2], [8, 0, 5, 4, 2, 0, 1, 2, -7, 1, -3], [3, -4, 9, -2, 8, -5, 10, 3, 5, -8, 6], [-22, -4, 1, -2, -7, 4, 4, 8, 5, -7, 2], [-10, 8, 3, 0, 1, 2, 0, 1, -5, 1, -4], [-9, 4, -4, -2, 4, 4, 7, 3, 6, -5, -5], [5, 6, 3, -1, 8, -10, 6, -6, 2, 4, 3], [-8, 10, -2, -1, 7, -6, 3, -2, -5, 2, -4], [-7, -7, 5, -3, 9, -4, 8, 1, 2, -15, 1], [-18, 7, 3, -5, 2, 4, 0, -1, 5, -10, -2], [-9, -1, 5, -5, 6, 2, 3, -4, -1, -3, 9], [4, 2, -6, -4, -2, 8, 0, -2, 1, 3, -3], [-1, -2, 5, 5, 3, -4, 9, 1, 4, -10, 0], [-15, -12, 5, -2, 3, 2, 3, -2, 0, 1, -4], [-2, -4, -3, -4, -3, -8, 2, -3, 9, 0, 0], [12, -6, 0, 8, -1, 0, -11, -1, -8, 5, -7], [0, 6, 1, -6, -3, 2, 6, -3, 3, 0, 5], [-4, -5, -3, 6, -17, 8, -5, 5, 7, 5, 3], [1, -6, 5, 5, 2, -6, -8, 1, 0, -1, 1], [-21, -4, -8, -3, -7, 4, -12, 3, 0, 2, 0], [-18, 10, -1, -5, 8, -6, 13, -2, -2, -2, 2], [-18, 8, -4, -2, -8, 10, -2, 0, 1, -1, 4], [15, 0, 5, 1, 10, -10, 0, -8, -9, 5, 0], [0, -3, 4, 11, 11, -8, 4, 0, 2, -6, 8], [-7, 6, 1, -1, 0, 0, -2, -4, -4, 6, 1], [-16, -2, 3, 2, 3, -2, -2, 2, 3, -6, -1], [-15, 4, 0, -6, 3, -3, 6, 8, 5, -4, 1], [23, -8, 4, -4, 6, -8, 3, -11, 3, -3, -2], [16, 0, 0, -4, 1, -2, 3, 6, -1, -2, 1], [-1, -5, 0, 3, 2, -2, 0, 4, 5, 1, 7], [11, -2, -5, 1, 1, -6, -1, -4, 1, 6, 5], [-13, 8, -4, -4, 1, 5, 4, 3, 6, -6, 2], [13, -10, 5, 10, 3, -8, -1, -11, -11, 4, 5], [-20, 2, 3, 2, -5, 2, -12, -6, 5, 1, 4], [-9, 8, -5, 4, -8, 6, -10, -4, 7, 1, 4], [-6, 10, -8, -10, 0, 4, 8, 2, 7, -5, -6], [-20, 4, -2, -1, -11, 4, -3, -1, 5, 0, 7], [5, 6, 6, 6, -4, 6, -3, 2, 0, 4, 7], [22, -2, 9, 5, -3, 0, 6, -1, -1, 0, -2], [4, 3, 5, 2, 15, -6, 5, -15, 1, -2, 5], [-3, -4, 4, -1, 9, -12, 4, -3, 4, -2, -4], [-22, -2, 2, 2, -4, -6, 2, -4, -2, 0, 2], [28, 6, -3, 6, 3, -6, -12, -12, -2, 3, -1], [-4, 1, 5, 1, -8, 6, 4, 3, -4, 4, -1], [-18, -4, 0, -2, -13, 6, -7, -4, 12, -3, 3], [12, 4, -8, 4, -1, -2, -9, -6, -4, 8, -1], [-8, 0, -4, -5, -1, 2, -1, 3, 9, -8, 0], [17, 0, -5, 1, -6, 4, -2, 1, -2, -4, -3], [-24, 2, -2, -1, 1, 4, 1, -1, 5, 3, -2], [5, 4, -7, -2, 5, -8, -5, -4, -7, 4, 1], [-3, 6, -4, -3, 1, 6, 6, 11, -4, 3, -2], [5, -2, 4, 5, 7, 4, 2, 1, -6, 1, 6], [-2, 6, 2, -7, -5, 2, -3, 1, 8, -7, 1], [12, -8, 7, -2, 1, -6, 10, -4, 1, -6, 0], [-12, -10, 2, -7, 7, -4, -3, 1, -2, 1, 3], [-10, -11, 9, -2, 6, 2, -2, -6, -6, 0, 1], [7, -8, 2, 9, -1, 0, -6, -3, 2, -1, -6], [5, 0, 4, 5, 5, 0, -2, 2, 2, -7, 4], [-3, -4, -2, 7, -5, -2, -8, -4, 7, 5, 1], [-3, 4, -6, 1, -4, -4, 1, -2, 5, -1, 2], [9, 14, -8, 1, -11, 8, -12, 2, -1, 6, -3], [-6, 0, -6, -4, -9, -1, -7, 6, 3, 3, -4], [-16, -4, -15, -1, -8, 2, -3, 3, 2, 0, -8], [6, 10, -2, 4, -1, 8, -13, -3, -3, 0, 4], [-9, -2, 2, 11, 1, -6, -12, -2, -6, -1, 4], [-9, -7, 4, -4, 4, 0, 10, 4, 6, -3, -4], [-5, -12, -3, 0, -10, 3, -10, 3, 7, 2, 2], [28, -4, -3, 2, -8, 0, -1, 3, 8, 0, 0], [-7, -8, -2, -6, 2, -8, 7, -1, 1, -3, 2], [10, -8, 1, -1, 2, 6, -11, 0, -8, -2, -4], [-6, 6, -3, 5, 1, -4, -14, -9, -10, -1, -2], [-3, -10, 5, 7, 10, -10, -6, -2, -4, 2, -9], [-28, -2, -4, 6, -10, 10, -2, 7, 1, 1, 2], [-15, -2, -11, -3, -16, 14, -4, 7, 18, 2, 1], [-13, 1, -6, 10, 4, -8, -12, -4, -3, -2, -5], [-8, 20, 0, 2, -4, 8, 0, 5, -2, 0, 3], [10, -3, -3, 6, -2, 0, -10, 0, -3, 1, -2], [3, -8, -2, 5, -1, -4, -6, 2, -2, 7, -4], [-11, 2, -7, -7, -10, 6, -2, -2, 2, 10, -5], [23, -4, -4, 3, -4, -1, -9, -3, -11, 8, -2], [9, -10, 3, 0, 1, -4, -5, 2, -9, -2, 3], [-24, 10, -5, -2, 6, 0, 19, 3, 4, -4, 2], [15, -10, 2, 5, 15, -14, -2, -4, -9, 2, -9], [-10, 0, 3, -14, 3, 2, 18, 5, 5, -8, -2], [14, 6, -10, -8, 1, -11, -7, -8, 0, 3, -1], [32, -5, 0, -1, -1, 4, 2, -1, 2, 0, -2], [-26, 10, 0, 3, 5, -8, 7, 2, -3, -2, -10], [12, 2, 3, 7, 8, 6, 1, 5, -7, -1, 1], [3, 6, -5, 9, -2, 4, 6, 3, 7, 1, -4], [-8, -6, 1, -8, 1, -4, -4, -8, 4, 3, 1], [-2, 0, -6, -4, -18, 6, 2, 5, 8, 4, 2], [8, -12, 7, -6, 15, -12, 4, -6, 3, -6, -4], [-16, 4, 8, -6, 6, 2, 14, 1, -2, -8, -2], [26, -6, -2, 4, 4, -2, -4, -1, 3, -1, 9], [3, -15, -7, -3, -10, 2, -9, 5, 1, 1, 1], [-11, -10, 3, 2, -2, 7, 10, 8, 3, 4, 0], [-8, -2, -10, 5, 7, -6, -11, -3, -5, 5, -8], [0, -16, 4, -8, 5, 5, 3, 4, -3, 5, 2], [14, -6, 8, -7, 0, -8, 12, 3, 9, -5, 3], [-12, -6, 9, -4, 9, 0, 0, -2, -2, -8, 5], [-3, -12, -3, -11, -3, 9, 3, 0, 8, 6, 2], [6, -2, 6, -5, 2, -8, 12, 2, -4, 6, 1], [-10, -4, -2, 6, -2, 2, 0, -3, -1, 1, -1], [-18, 13, 5, 5, -6, 8, 16, 6, 11, -2, 10], [-13, -2, -2, -3, 9, -6, 2, 1, -12, -3, 0], [-10, -10, 3, 1, 0, 4, 3, 4, -1, 1, -4], [-22, -12, -7, -6, -2, -3, -15, -1, 1, 3, -5], [-12, -2, 8, 2, 8, -2, 12, 1, 0, -8, 12], [-5, 0, 7, 0, 6, -5, 4, 5, -2, -10, -7], [-2, -3, -2, -8, -8, 4, -3, 5, 4, 1, 1], [0, 4, -2, 9, 2, 0, -6, 3, -2, -2, -6], [1, 0, 0, -1, -3, 4, 4, 0, 6, 8, 0], [-4, -2, -3, 1, -2, 7, 5, 11, 10, 6, 6], [24, 6, -5, -3, -16, 10, -11, 7, 0, 4, 2], [9, 0, 7, 1, 2, -2, 6, 2, -5, 1, 4], [27, 0, 3, 2, 4, -7, 4, -14, -2, 3, -7], [-22, 2, 9, 3, 0, 6, 11, 0, 0, -3, 10], [3, 2, 4, 2, -3, 6, 8, 3, 9, -5, -7], [4, 0, -2, 0, 8, -2, 2, -3, -4, 0, 0], [17, -10, 1, 2, 6, -6, -2, 6, 1, -3, 7], [8, -5, 6, 11, 5, -8, -8, -5, -5, 1, 5], [-7, -10, 2, -11, 11, -6, 8, 5, -5, -12, 3], [-26, -8, 2, 11, 7, -6, -3, 0, -4, -5, 3], [-2, 4, 6, 2, 11, -12, 17, 2, -4, -7, 1], [-3, -7, -6, 4, -16, 10, -10, 6, 1, 0, 3], [-2, -16, -5, 1, -9, 8, -10, 10, 0, 1, -1], [-11, -6, -5, -10, -5, 4, -9, 2, -1, 2, 1], [-16, 22, -8, -1, -7, 4, -1, 0, 4, 1, 1], [6, 8, -1, -5, 7, 5, 0, 1, -4, -2, -9], [14, -4, 3, 7, 8, -20, -5, -3, -8, 1, 0], [0, 8, -7, -7, -4, -6, 9, 0, -2, 10, -2], [4, -4, -1, -1, 0, 8, -1, 7, 10, -2, 0], [15, -6, 3, -2, 13, -14, -3, -9, -10, 1, -8], [20, -8, -2, 12, 0, -6, -12, -7, 7, -1, -1], [11, -4, -12, 4, 0, -8, -7, 2, -9, 1, 2], [14, -12, -5, 13, -9, 3, -16, 3, 3, 2, 2], [9, 4, 7, -8, 5, 2, 15, 9, -1, -6, 3], [-3, 0, -5, 0, 5, 4, -3, 3, 8, -11, 4], [-21, 6, -4, 1, -4, 6, 3, 4, -2, -4, -1], [1, 4, -7, 3, -13, 14, -15, 1, 4, 7, 2], [20, 6, 1, 2, -10, 5, -7, 7, 7, 3, 7], [5, 6, -4, 2, 1, -1, 0, 0, -3, -2, -1], [-9, 2, -6, 1, -3, 2, -4, 0, -4, 9, 2], [-1, 8, -6, 9, -5, -2, 8, 3, 2, -1, -6], [-8, 1, 10, 3, -1, -2, 10, 0, 2, -2, 6], [-2, 7, -1, 9, 6, 6, 0, 5, -6, 1, 9], [3, 1, 9, -5, 4, 2, 13, -2, 1, -1, -1], [34, 5, 1, 6, 1, -4, -3, -4, -11, 5, -1], [3, 2, 2, -2, 8, -10, 3, -1, 1, 8, 0], [-28, 4, 10, 3, 10, -1, 14, -1, 0, -1, 4], [13, -6, 11, -4, 18, -9, 2, -9, -6, -2, 3], [4, 0, 4, 12, 0, -10, 4, -1, -7, -1, -9], [10, 10, -1, -8, 3, 6, 4, 5, 8, 3, 5], [-12, 6, -11, -10, -7, 11, 2, -1, 2, 1, -3], [24, -6, -2, -5, -15, 14, 1, 3, 4, 5, 2], [26, 6, -7, -7, 0, 0, -7, 1, -1, -1, -11], [7, -12, 5, -3, 6, -2, 6, 0, -1, -7, 0], [-14, 0, 0, 5, -3, -2, -3, -9, -2, -1, 5], [-1, 10, -1, -7, -4, 0, 6, 3, 8, 6, 1], [-18, 6, 8, -1, 2, 2, 2, 3, -4, -6, -4], [-10, 14, -1, 5, -1, -4, 8, 2, 4, -1, -5], [-3, 2, -10, -6, 5, 6, 0, 3, -2, -1, -5], [6, -12, -3, -4, 5, -3, -6, -3, 8, -5, -3], [7, -2, -5, -2, -5, 0, -1, -5, 3, 5, -1], [18, -2, -3, 3, 5, 0, 6, 5, -8, 5, 2], [-8, 6, -8, -3, -3, 0, -5, -4, -2, 1, -11], [22, 4, -10, 0, 3, 4, -7, -7, 2, 7, -2], [8, 0, -2, 10, 2, -11, -14, -6, -4, -6, 0], [21, -8, 2, -2, 16, -4, 5, -2, -9, -5, -1], [-24, 10, -3, -2, -12, 12, -3, 6, 4, -2, 4], [18, -6, 6, 3, 7, -18, 5, -10, 2, -5, -5], [21, -16, 6, 12, 6, -6, -17, -9, -12, -2, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8016_2_a_be();".
// 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_8016_2_a_be(:prec:=11)
    chi := MakeCharacter_8016_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_8016_2_a_be();".
// 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_8016_2_a_be( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![512, 5280, -2906, -6948, 2246, 2665, -864, -385, 155, 12, -10, 1]>,<7,R![256, -704, -2096, 3304, 3572, -4561, -692, 800, 46, -49, -1, 1]>,<11,R![86400, -69984, -101976, 76492, 26456, -19694, -2359, 1865, 84, -73, -1, 1]>,<13,R![10688, 16160, -19504, -22496, 16124, 8438, -5699, -408, 484, -27, -10, 1]>],Snew);
    return Vf;
end function;