
// Make newform 8034.2.a.u in Magma, downloaded from the LMFDB on 25 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_u();".
// 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_u();".
// 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 := [-4, -32, -53, 97, 230, -109, -249, 85, 64, -18, -4, 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], [-57834, 840961, 1771243, -4857162, -3375623, 7344639, -475719, -1973096, 347054, 123880, -25115], [326464, 2744613, 274806, -11137378, -266083, 12312604, -2408751, -3133089, 703885, 194155, -43620], [355594, -3554529, -7530379, 22005140, 12149715, -32593715, 3266695, 8763870, -1667268, -551264, 115421], [2166734, 10501909, -2052417, -43024636, 637757, 51186895, -10877367, -13311542, 3083592, 831576, -189461], [1342942, 7257953, -665823, -31274154, -344375, 37899765, -7853363, -9891326, 2271966, 618522, -140379], [3395542, 20040201, 1059979, -84025608, -4436099, 100542707, -19548755, -26121810, 5868288, 1630196, -366281], [-5761410, -28669817, 3928915, 119432904, -1255417, -139385317, 29219587, 36042260, -8303822, -2246898, 510619], [6971166, 36919609, -2584379, -153749580, -2609191, 182903005, -37292383, -47518076, 10849434, 2966642, -671471], [4665936, 24837390, -2225443, -105229792, -53294, 124845689, -26090460, -32427051, 7470473, 2024699, -460175]];
    Rf_basisdens := [1, 1, 57098, 57098, 114196, 114196, 57098, 114196, 114196, 114196, 57098];
    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], [-1, 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, -1], [-1, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0], [0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1], [-1, 1, 0, 0, 0, 0, 1, 0, -1, -1, 1], [-3, 1, 0, 1, 0, 1, 1, 0, 0, -1, 1], [0, -1, 0, 0, -1, 1, -1, 0, 0, -1, 1], [-1, -1, -1, 0, -2, 0, 0, 1, 1, -1, 2], [1, 0, -2, 0, -3, -1, 0, 1, -1, 0, -1], [-1, 2, 0, -1, 0, 0, 0, 1, 0, 1, 0], [2, 0, 1, 0, -1, -1, -1, 1, 0, 0, -1], [-3, 0, 1, 1, 2, 1, 1, 1, 2, -2, 2], [0, 2, -1, 0, 0, 0, -1, 0, 0, 0, 0], [-4, 0, 2, 2, 2, 0, 1, 0, 2, -2, 2], [-3, 0, 1, 0, 2, -1, 0, -3, -1, 0, -1], [-1, -2, 0, -2, 0, 1, 0, 0, -1, 1, -1], [-3, 2, -2, -1, 0, 1, 1, -2, -2, 2, -3], [-2, -2, 1, 1, 1, 0, 0, -1, -1, -1, 1], [-2, 2, -2, 1, 1, -1, 1, 2, 1, -1, 1], [1, -3, -1, -3, -1, -2, -1, -1, -2, 1, -2], [-2, -3, 1, -2, 1, 2, 0, -2, 0, -1, 2], [-2, 2, -2, 0, 1, -1, -3, -1, 0, 1, -2], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -2, 0, 4, 1, 0, -3, -1, 1, 1, -1], [-6, 2, 1, 1, 0, -1, 2, 0, 1, 1, 0], [-4, -1, -1, 2, -1, 3, 1, 1, 2, -1, 2], [-4, 0, 0, -2, 0, -2, 3, 0, -2, 0, -1], [-2, 0, 4, 1, 2, 1, 3, 0, 3, -2, 2], [3, 0, -2, 2, -2, -2, 0, 3, -1, 2, -1], [-2, -3, -1, -1, 1, -1, -1, -1, -1, 2, -2], [0, 1, 0, -1, 0, -1, -4, -2, 1, 4, -4], [-1, -1, -2, 3, 0, 0, 1, 4, 0, 1, 0], [-3, -2, 1, -2, -1, 1, 0, 0, 1, -3, 3], [0, 1, 1, 0, -2, -1, -2, 2, 2, 3, 0], [2, -3, 0, -1, -1, -1, 1, -3, -3, 2, -2], [-1, 0, 1, -1, 2, 0, -2, -4, -2, 0, -2], [-8, 0, 3, 1, 2, 2, 0, 1, 1, -2, 3], [-6, 1, 0, -3, 2, -1, 2, -3, -3, 3, -3], [-2, 6, -1, 2, 2, -1, -2, 1, 0, 2, -3], [1, 0, -1, 1, -5, -1, -2, 3, 0, 4, 0], [3, -2, -3, -2, -1, 0, -5, 0, 1, 1, -3], [7, -4, -2, 1, 0, 0, 0, 3, 2, -1, 5], [-1, 2, 1, -1, 0, 0, -4, -2, 0, 4, -3], [2, -4, -2, -3, -3, 2, -2, -3, 1, -1, 0], [4, -1, -2, 4, -5, -1, 2, 2, 0, 7, -2], [-2, 0, -3, 0, -1, 1, -3, 0, 0, -2, -2], [2, -2, 0, 3, -1, -3, -1, 4, 1, 1, -1], [-3, 5, -3, -1, -2, 0, -3, 0, 0, 4, -5], [-4, -2, 6, 2, 3, 1, 2, 1, 2, -4, 2], [-4, 1, 2, 2, 5, 1, 1, 2, 0, -6, 3], [-8, 3, 0, 2, 1, 0, 1, -1, -2, 0, 3], [1, -2, -2, -1, -1, 2, 0, 0, -2, -1, 1], [-7, 1, -2, -2, 1, 1, 4, -1, -3, -2, -2], [-3, 5, -5, 4, 2, -2, -2, -2, -3, 6, -1], [1, -1, 1, 1, -1, -3, 4, 2, 0, 0, 0], [-1, -2, 5, 0, 2, 3, -2, -3, 3, -2, 0], [-6, -2, -1, -2, -2, 0, -1, -2, -4, 2, 0], [-3, -2, 0, -2, -1, -2, 3, -3, -5, 0, -2], [0, 2, -1, 3, -2, -1, 1, 5, 3, 0, 2], [-2, 2, -2, 1, 1, 1, 1, 0, -3, 1, -1], [-1, 3, 0, 4, 2, 1, -2, -1, 3, -3, 5], [1, -4, 5, -1, 1, 1, -5, 2, 4, -4, 2], [-2, -2, 2, -1, 0, -3, 2, 2, 1, 3, 3], [-8, -1, -5, 1, -1, 1, 1, 1, -5, 6, -2], [2, -2, -3, 3, -3, 2, 3, 0, -1, 0, 2], [-2, -1, 1, 0, 3, -1, -1, 1, 4, -2, 4], [-3, 0, -3, -1, 0, 2, -1, -1, -2, 0, 3], [3, 2, 5, -1, 3, -2, -3, -3, 0, 4, -6], [0, 2, 1, -1, 3, -1, -3, -2, 3, 1, -1], [-7, 4, 2, 1, 4, -3, 3, -2, -4, 5, -4], [-4, -2, 4, 0, -1, 1, 9, 1, -2, 0, 4], [4, 0, 1, -3, -4, 0, -1, 4, 5, 2, 5], [0, 0, 1, 2, -2, -5, 5, 4, 0, 1, -2], [-7, 2, 1, 1, 0, 1, -2, 4, 4, 0, 4], [-5, 0, 5, -2, 4, 2, 3, -1, 3, -6, 5], [-1, 5, 2, 0, 3, -1, 5, 2, 3, 1, 1], [3, -2, 0, 0, -2, -1, -2, 4, -1, -2, 2], [-3, 0, 4, 2, -3, 2, 3, 5, 1, -5, 3], [4, -1, -1, 6, 0, -2, -4, -3, 0, 5, -6], [2, 0, -2, -1, -2, 1, 0, -4, -3, 2, -4], [-10, -1, 4, -3, 6, 5, -2, -8, -1, -4, 2], [6, 1, -1, 6, 4, 0, 0, 2, 6, 2, 3], [-3, 2, 5, -1, 3, -2, -3, -4, 2, -4, -5], [2, 2, -3, 4, 1, 1, -7, 3, 6, 0, 3], [-1, -1, -2, 5, 4, -3, -5, -1, -2, 4, -4], [-6, -1, 3, -2, 1, -2, -4, -2, -4, -4, 2], [2, 0, -4, 2, 2, 2, -3, -1, 0, -1, -6], [4, 2, -1, 1, 1, 0, 4, 3, 5, 4, 5], [-1, 0, -2, 3, 2, 0, 3, -4, 2, 1, 5], [0, -2, 0, 0, -1, -1, -6, 2, 4, -2, 3], [0, 0, -2, -4, -2, 3, 2, 6, 2, -2, 1], [4, 3, 4, -3, -1, -3, 0, 2, -1, -9, 0], [6, -8, 0, -5, 0, -1, -2, -3, 1, -1, -2], [3, -3, -3, -1, -9, -5, 0, 4, -4, 2, -6], [2, 1, 4, -3, 0, -2, 2, -1, 1, 4, 0], [-5, 0, 3, 3, 2, -2, -4, -1, 2, 5, -9], [-3, -5, 0, -1, -1, 4, -3, -4, -2, -5, 1], [0, 2, -4, -7, -6, 1, 3, 1, -1, -2, 3], [6, 1, -1, -1, -5, -1, 4, -2, -3, 1, -2], [-15, -2, -4, -1, 2, -4, 0, -2, -8, 4, -1], [-5, 3, 3, -4, 1, -2, 6, 2, 1, -4, 6], [-9, 0, 3, -1, 6, 4, 3, -3, 0, -6, 3], [4, 4, 5, -1, 6, 1, 5, 0, 5, -2, 0], [5, -1, -1, 2, -1, 0, 8, 1, 1, 2, 4], [0, -8, -4, -7, -6, 3, -1, 1, -1, 1, 0], [4, -2, -6, 0, -7, 2, -2, 6, 2, 9, 3], [2, -8, 0, 6, -2, 1, 2, 3, 0, -6, 2], [-9, 4, -2, -1, -2, -4, -2, 4, -2, 0, -6], [-12, 7, -3, -5, 0, 4, 6, -1, -1, -1, 5], [3, -8, -4, -3, -6, 5, -5, -5, -2, 0, -3], [-8, -4, -5, -2, -1, 2, -3, 0, -2, -1, 4], [0, 4, 0, -2, -4, 3, -2, -1, 0, -6, 3], [-11, -2, -2, -6, -1, 2, -1, -7, -9, 5, -10], [9, 2, 3, 4, -2, 1, 5, 6, 1, 4, 1], [-8, 6, 5, -4, 7, -1, -1, -9, -2, 3, -3], [2, -4, 1, -3, -4, -1, 0, 1, -1, -3, 0], [-5, -3, -5, 1, -2, 3, 2, -3, -4, 13, -2], [-18, -1, 1, -4, -2, 3, 2, -7, -6, -2, -8], [10, 1, -1, -4, -4, -3, 1, 2, -2, 7, -4], [-3, -1, -4, -2, -2, 3, 3, -3, -1, 4, 2], [3, 0, -4, -3, -5, -2, 6, -2, -8, 2, -2], [-3, -8, 6, -2, 2, 4, -6, 1, 7, -12, 3], [-7, 0, -5, -2, -2, -3, -8, -3, -7, 7, -12], [11, -8, -3, -3, -8, 5, 2, 1, 2, -8, 6], [0, 5, -2, 0, -1, -4, 0, -4, 2, 7, -9], [-8, -5, 8, -3, 4, -1, -5, -3, 1, -4, -8], [0, 4, 3, 4, 4, 0, 11, -1, 0, -1, 5], [3, 0, -4, 1, -1, 0, 3, 1, 4, -1, 13], [10, -2, 0, 3, -2, -5, -3, 6, 5, 4, -2], [5, -2, -1, -3, 2, -3, -4, -5, -2, 4, -3], [6, 5, 4, -1, -1, 0, 9, -1, 1, -4, 8], [-2, 2, 2, 0, 5, -6, -4, -8, -6, 0, -10], [0, -7, -3, 2, 4, 6, 2, -1, 2, -7, 9], [-10, 6, 3, -2, -8, 1, 2, 0, 0, -1, 1], [12, -9, -1, -2, 1, 8, 1, -7, 2, -6, 9], [-4, 1, -5, 3, -5, 1, 2, 7, 1, -2, 9], [-7, 0, 2, -3, 3, -1, 0, -3, -2, -6, -6], [-1, -2, -2, -3, 4, 2, 2, -8, -4, -2, -2], [6, 7, 0, 8, 8, 2, 1, -3, 4, 1, 4], [-4, 4, -7, 3, -3, 3, -2, -1, -1, 5, -2], [-1, -6, 0, -4, 1, 0, 6, 3, 1, -5, 2], [8, 5, -1, -3, -3, -4, 5, 8, 1, 8, -1], [-1, -2, 4, -2, 7, 3, -5, -2, 7, -1, 5], [12, 8, -3, 7, -4, 2, -3, 7, 5, 2, 1], [2, -6, 1, -4, -1, 0, 8, 4, -2, 1, 1], [1, 4, -4, 2, -9, 2, 7, 7, 1, -4, 4], [-9, -3, 4, 3, 4, 2, 7, 1, 6, -2, 3], [-6, 2, 6, -4, 5, 0, -7, -7, 0, -4, -4], [-1, -3, 5, 2, 6, 2, 9, 3, 1, -12, 11], [2, 0, 10, -5, 9, -1, 2, -7, -5, -6, -4], [-1, -4, 0, 4, -5, 4, 3, 0, 3, -4, 0], [0, -6, -2, 10, 6, 2, 3, -5, -2, 3, 3], [0, 1, -3, -7, 0, 3, 4, -1, -3, -5, 3], [16, -4, 10, 0, 3, -2, 0, 2, 6, -9, -1], [6, -2, -4, -5, -4, -4, -2, -2, -7, -3, -3], [-2, 9, -1, 6, 2, -3, -6, 0, 0, 5, -7], [13, -9, 7, -3, 7, -1, -11, -4, 0, 1, -6], [0, -2, 5, -3, 2, 5, -2, -6, 3, -2, -3], [5, 6, -7, 7, 1, -1, -3, 4, 4, 3, -9], [3, 0, 2, -4, -3, 4, 1, -4, -1, 4, -3], [16, 2, -2, 3, -4, -2, 0, 5, -1, 10, -4], [20, 7, 2, -8, -2, -6, 0, 4, 2, 0, 0], [3, -2, 1, -6, 6, 1, 0, -3, -1, -9, 2], [2, 3, -5, 2, 2, 3, 5, 2, 0, -1, 6], [4, 3, -8, 9, -2, -5, -2, 0, -5, 6, -11], [-2, 0, -3, 0, -1, -2, 2, 6, -4, 9, -2], [-8, 6, -2, 6, 3, 1, 4, -2, -6, -8, -2], [2, -7, 1, -6, -5, 1, -4, 3, -6, 2, -8], [15, 0, 0, 1, -7, 4, 4, 8, 8, -7, 4], [2, -8, 1, -8, -9, -1, -9, 3, -6, 2, -6], [4, 2, 0, 3, -9, 6, 3, 3, -5, -5, 3], [3, -6, 6, -2, 4, 6, -5, -4, 3, -5, -1], [13, 2, 1, -3, -5, -2, 5, 2, 2, -6, -4], [-1, 0, -1, -3, 1, 6, 1, -6, -6, 4, -10], [3, 8, -5, 0, 6, -2, -2, -9, -3, 6, -9], [4, 5, -2, 6, 2, 4, 0, 0, 2, -4, 10], [10, -5, 2, -5, -3, -1, -1, -2, -1, 8, -2], [-12, 4, -3, -4, 1, 2, 0, -5, -4, -2, -7], [-16, 10, 3, -7, 9, -2, 0, -8, -9, -6, -8], [4, -3, 1, -2, 3, -4, -6, -1, 4, 1, -5], [12, -8, 6, 0, 0, 1, -5, -3, -2, -1, 6], [19, 4, -4, -1, -6, -9, -8, 0, -4, 5, -8], [-11, -2, 5, -7, 5, 3, 0, -1, 2, -9, 1], [-2, 5, 1, 4, 5, 4, 6, 2, 4, 1, -2], [-8, 2, -8, 6, -5, 4, -1, -2, -6, 12, -8], [8, 3, 0, 12, 4, 4, 3, 7, 10, -9, 11], [2, -6, 3, 5, -1, 4, -2, 5, 7, 3, 5], [3, 4, 2, -2, 4, -5, 1, -9, -7, 9, -12], [12, -1, -11, 0, -3, -3, -7, -1, -2, 7, -4], [1, 0, 3, -3, -5, -2, -4, 4, 0, 8, 0], [8, -4, 7, -3, 0, 0, 3, 0, 3, -11, 3], [13, 2, -3, 6, -2, -4, -2, 9, 5, 8, 4], [-5, 5, -8, 2, -4, -4, 6, 9, -3, -4, 2], [12, -5, 7, -1, 3, 4, -3, 0, 9, -12, 10], [9, 7, -10, 4, -4, 3, 8, 5, -5, 12, -3], [-2, 0, 6, -3, 0, -3, 2, -2, 1, 6, -3], [-4, 2, 1, -4, -1, -3, 7, -2, 0, 1, -6], [-14, 8, -10, 2, -2, 0, 10, 2, -6, 0, 4], [-6, 8, 0, -3, -5, 1, 1, 1, 3, -2, -1], [9, 5, -10, 1, -5, -2, 5, 4, -6, 7, 2], [7, -4, -1, 4, -1, -7, -3, -5, -5, 2, -9], [13, 6, -7, 0, -6, -1, -1, -1, -3, 5, 2], [-3, -10, 1, 1, 1, 6, -3, -8, 2, -8, 2], [14, 3, 0, 7, -3, -2, 4, 3, 5, 5, 2], [-4, 4, 8, 0, 0, 11, 5, 5, 12, -11, 10], [13, 9, 4, 2, 4, 3, 1, 5, 9, -1, -5], [-20, -3, 1, -2, 2, 6, 2, 1, 8, -11, 7], [-3, -6, 1, 8, 3, -4, 2, 1, -1, 5, -6], [9, 1, 1, -5, 3, -5, -8, 1, 0, -5, -3], [10, -9, 6, -1, 10, -1, -5, -6, -1, 1, -3], [-6, -1, 9, -3, 10, 6, -5, -15, -5, 0, -4], [4, -12, 3, -4, -3, -5, 6, 1, -4, -4, 1], [12, -2, -3, 7, -2, 6, -7, 5, 13, 6, 4], [-9, 7, -3, 4, 5, 4, 5, 0, -5, 0, 4], [4, 4, 1, -5, 4, -3, -1, -2, 1, 6, -2], [6, -5, 8, -6, 1, 4, 9, -1, 0, -4, 9], [-18, -3, 3, 3, 2, -2, 1, -4, -1, 3, 9], [13, 0, -2, -8, -8, 3, 0, 11, 1, 5, 0], [-2, 1, 1, 1, 2, 0, -5, 6, 3, 4, 7], [2, -11, 4, -3, 0, -1, -2, 0, 5, -3, 1], [2, -5, -1, 9, 6, -2, 3, 8, 11, -7, 12], [-5, -7, 3, 7, 4, -3, -9, -4, 0, 5, -4], [13, 5, -6, 12, -2, 1, 8, 8, 5, 6, 0], [6, -3, -10, 6, -3, 4, -8, -2, -2, 9, -2], [26, -5, -1, -1, -5, 3, 3, -4, -7, 13, -7], [17, -2, -5, 5, -3, 3, -1, 11, 8, 1, 4], [-6, -2, 6, -4, 1, -5, 8, 5, 4, -1, 5], [3, -4, -2, -6, -9, 0, 12, -6, -7, 2, -4], [10, -6, 3, -1, 0, 7, -7, 3, 7, 1, 6], [-19, 5, -1, 4, 2, 8, 1, 1, 1, -2, 4], [-1, 5, 10, 4, 11, -4, 4, 0, 5, -8, 3], [24, 6, -3, 11, -2, -6, 0, 13, -1, 1, 0], [-9, 10, -9, 7, 6, -3, 2, 0, -4, 8, -3], [-16, 10, -1, -1, 2, 2, 9, 1, -7, 1, 6], [7, 8, -10, 2, 2, 1, 3, 0, 1, 2, 9], [-5, -6, 2, 4, -4, 3, 6, -3, 7, -5, -1], [-4, -6, 7, -4, -4, 0, 2, -6, -4, 2, 0], [-5, 8, 12, 12, 7, -4, 5, 1, 3, -9, 0], [5, -1, 4, 7, -8, -9, 2, 14, 0, -1, 6], [3, 6, -15, 8, -13, -7, 0, 14, -7, 9, 0], [-10, 8, 2, -4, 4, -2, 10, -4, -4, 10, 4], [-16, 2, 6, -8, 4, 0, 0, -6, -2, -6, 0], [-10, -3, 11, -2, 13, -1, -2, -11, 0, 0, 4], [9, -6, 3, 0, -6, 1, -4, 13, 9, 1, 2], [-7, 8, 0, -2, 3, 2, 2, -4, -1, 8, -3], [0, 7, 0, 4, 1, 2, -9, 0, -2, 7, -8], [10, 0, 3, -5, 0, -1, 3, 3, -3, 7, -6], [-1, -2, 4, 5, -9, 4, 3, 4, 8, 3, 8], [26, 4, -2, 12, 6, -3, -2, 4, 4, 8, 5], [5, 1, 5, 4, -7, 6, -2, 1, 3, 7, -3], [19, -13, 7, -5, -2, 7, -2, -11, 2, -4, 10], [-9, -7, -1, -9, -8, 6, -6, -2, 8, -9, 5], [-1, 10, 3, -4, 4, 2, 0, 2, 7, 1, 8], [9, -5, -3, 3, 6, 1, -4, 3, -4, 0, 0], [-15, 13, -1, 1, -1, 2, -8, -2, 10, 1, -3], [13, 3, 7, -3, 0, -4, 0, -7, -4, -1, -3], [0, 2, -6, -5, -15, 2, -2, 2, 1, 0, 2], [4, -2, 8, -4, 12, 8, 14, 0, 4, -12, 9], [8, 3, -9, 2, -8, 0, -3, 5, -4, 8, -10], [-6, 9, -3, 1, -8, 0, 3, 4, -3, -3, 5], [10, 10, 3, 3, 10, 4, 2, -1, 5, 0, 4], [6, -2, -8, 0, -9, 0, -1, 4, -6, -1, -3], [4, -11, 5, 4, -4, 6, 10, 7, 12, -11, 17], [3, 1, 4, -6, -7, 1, 1, 3, -5, -1, 1], [-21, -8, 0, -4, 3, -6, 0, -9, -7, 10, -2], [14, -2, -12, -8, -8, -2, -7, -12, -4, 12, -7], [-13, 6, -5, -1, 2, -9, -6, -7, -10, 8, -15], [4, 6, 2, -8, 9, -5, -11, 0, -2, -7, -11], [2, -9, -1, 9, 6, 1, 7, -8, 1, -10, 5], [2, 6, -2, -2, 1, 1, 4, -8, 0, 12, -6], [-3, -13, -8, -4, -3, 4, 4, 6, 3, -2, 13], [-1, -2, 6, 6, 7, 0, -6, 2, 9, 6, 0], [24, -4, -3, -7, -5, -3, 7, 5, -1, 8, 2], [2, 12, 9, -9, 8, -5, 4, 1, 5, 1, -4], [-1, -1, -9, -4, 1, -1, -6, 0, -1, 5, -9], [6, -6, 19, 1, 12, 2, 1, -5, 1, -10, 0], [9, 7, -15, 5, -2, -3, 5, 13, -2, 8, 0], [0, 5, -7, 3, 1, -8, 2, -4, -11, 15, -15], [-1, -2, -2, -1, 6, 1, 15, -1, -4, -8, 6], [-24, 12, 4, 9, -5, 9, 16, 6, 1, -9, 9], [-8, 5, 6, -6, -2, -3, -1, -9, -8, 4, -13], [-25, 1, 0, -8, 8, -4, 5, -5, 3, -4, -4], [9, 1, -3, 2, -10, 0, -4, 12, 1, 1, -8], [8, 2, 1, 1, -5, -7, -9, 5, 9, -4, 5], [-17, 2, 7, -2, 4, 6, 12, 10, 9, -15, 9], [-2, 1, -3, 1, 4, -4, -6, 1, 9, -2, 11], [12, -6, 1, 1, -1, -7, 6, 0, -7, 9, 1], [14, 4, -7, -6, 5, -2, -11, -11, -8, 8, -18], [-7, -2, 0, 3, 2, 8, 5, -4, 4, -4, -1], [-1, -9, 6, -5, 3, -5, -14, -3, 0, 4, -6], [-12, 10, -8, 4, -1, -3, -1, 8, 8, 0, 4], [-15, 8, -3, 1, -6, -1, 16, 1, -10, -6, 9], [-7, 8, 2, -6, 5, 2, -1, 1, -1, -13, 8], [18, 1, -1, -10, -7, 4, -7, 2, 0, -5, -2], [-9, -7, 18, -5, 0, 4, -2, 0, 6, -12, 2], [10, -2, 3, -9, 15, 4, -21, -11, 3, -1, -5], [4, -9, -1, 1, 3, -2, 10, -2, -1, -6, 6], [41, -10, 2, 3, -5, 0, -5, -5, -10, 3, -1], [-10, 3, 12, 3, 11, -1, 1, -6, -3, -11, -3], [-13, -2, 2, -14, 10, -4, -2, -24, -1, 11, -14], [-11, 0, -6, -8, 1, -5, -14, -9, -11, 6, -9], [-10, 3, 5, -8, 9, 7, -7, -22, -10, 0, -13], [-13, 9, 3, 10, 2, -2, -4, 0, 7, 1, 1], [18, -4, -12, 8, -12, 3, 3, 9, 2, 0, 0], [-3, 11, -5, -3, -2, 2, -8, -3, -2, -1, -8], [-14, 3, -7, -3, -1, -3, 0, 6, -7, 7, 6], [5, 2, 4, 6, -5, -6, 6, 8, 3, -6, 5], [22, 5, 0, 9, -3, -5, 7, 14, 5, 1, 11], [0, 14, 2, -3, -9, -5, -2, -2, -3, 7, -9], [4, -19, 3, -4, 1, -4, -1, -4, -4, -9, 0], [0, 4, 7, -14, 1, 4, -2, -2, -4, -4, -3], [19, 3, 5, 2, -5, -4, -1, 3, -3, 13, -10], [4, -5, 6, -4, 4, 2, 2, -9, 4, -10, 7], [-14, -4, 2, -1, -5, 2, 4, 6, -7, -5, 8], [-23, 12, -4, 1, 3, 1, -10, -7, -4, 6, -1], [-17, -1, 6, 10, 6, 1, 14, 9, 5, -7, 7], [-23, -8, 1, 1, -1, 6, 3, -8, 4, -3, 0], [-20, 2, -9, -2, 0, -3, -2, -2, -10, 20, -12], [15, -7, -11, 1, -12, -6, -8, 12, 10, 4, 2], [-5, 2, 5, 8, -3, -4, 7, 7, -1, -19, 7], [-6, 8, 4, 2, 7, -2, -1, 3, -2, -2, 6], [-4, 0, 6, -12, 3, 6, 2, -6, 6, -7, 9], [27, -7, -6, 4, -13, 2, -3, 3, 1, -1, 7], [-13, 4, -6, 3, 9, -1, 17, -5, -2, -8, 11], [-5, 0, 3, -11, -8, 4, 4, -11, -14, -1, -5], [15, -7, 6, -9, -10, 6, 8, 5, 6, -9, 1], [10, -14, 2, -13, -9, -3, 3, -1, -3, 6, -5], [-17, -4, -8, -9, 2, 2, -5, -16, -6, -2, 5], [29, 4, -15, 5, -17, -3, -5, 7, 0, 5, 4], [23, -7, -14, -2, -11, 3, -8, 6, -7, 0, -3], [-2, 0, -12, 4, -4, 7, -10, 7, 12, 11, 0], [-13, -12, -2, 0, 0, -1, 0, -6, -5, 2, -3], [-4, -6, -11, -2, -15, -5, -9, 7, -10, 10, -1], [13, -6, -8, 3, -10, 10, -7, 6, 12, 4, 5], [-13, -3, -2, -6, -10, 0, 0, 0, 1, 11, 5], [5, -3, -5, -5, -15, 0, 11, 3, 0, -8, 1], [-15, 3, -4, 0, 3, 4, -13, -13, -5, 0, -11], [5, -11, 13, -2, 11, -1, 7, -4, -5, -15, 5], [25, -8, 11, -3, -9, -1, -9, 3, 8, -8, 8], [6, 7, -1, 10, 5, 1, 4, 21, 6, -1, 3], [4, 14, 2, -1, 7, -8, -15, 0, 5, 1, -6], [-18, -7, -4, 4, -5, -3, 7, 3, -8, -7, 6], [-15, -7, 3, -14, 10, 4, 15, -7, -1, -8, 6], [-13, 1, -9, -2, 1, -3, -19, -16, -9, 1, -15], [-16, 8, 7, 5, 3, 8, -9, -5, 1, -12, -1], [-1, -14, 0, -4, 5, -7, 1, -4, -1, -7, 4], [-20, 15, 11, -6, 8, -3, 1, 6, 12, -9, -6], [-3, -9, 10, -8, 1, 3, -4, -4, 9, 3, 1], [-21, -4, 6, -6, 11, 2, -4, -9, -3, -20, -6], [23, -6, -5, -6, -9, 8, -8, -1, 3, 3, 10], [-28, 5, -8, -3, 1, 4, 0, -9, 3, 3, 2], [-16, -20, -8, 1, -2, 12, 1, 7, 1, -14, 20], [21, -3, -10, 15, -12, -6, 3, 8, 0, 12, 2], [15, 4, -10, 0, -11, -1, -9, 3, -7, 1, -12], [-50, -1, -1, -7, 7, 3, 0, -5, -3, 0, 9], [12, 2, -16, 6, -19, -5, -7, 6, -8, 3, -8], [-8, 10, -1, -9, -9, 0, -7, 0, 3, 4, 2], [0, 0, 0, 9, -4, -7, 5, 3, -5, 11, -4], [-23, 10, -5, -1, 6, 4, 15, -1, 2, -1, 4], [-17, -3, -11, -6, -10, 4, 10, 9, 1, -1, 15], [11, 0, 0, -4, -12, -7, -5, 15, -7, -11, -4], [-20, -8, -1, 6, 5, 3, -1, 0, 10, -4, 9], [-4, -6, 7, 0, 12, 0, -4, -19, 0, -4, 3], [-8, 6, 3, 4, -6, -1, -9, 5, -2, -9, -4], [-20, -10, 3, -3, -6, 3, 2, 0, -5, 2, 2], [4, -6, 2, 5, 3, 4, 10, 16, 13, -4, 5], [16, -6, 1, 1, -13, 7, -1, 6, 3, -7, 5], [12, 8, 0, 10, 6, -5, -2, 8, 4, 4, 5], [1, 8, -1, 3, 1, -3, -3, 14, 8, 6, -9], [-10, -10, -2, 1, -9, -1, -2, -9, 7, 0, 0], [16, -11, 0, -4, -6, -1, 11, 0, -12, 11, -4], [-17, 0, 11, -1, 11, 8, 2, -3, 8, -4, 9], [9, 13, -4, 0, 11, -7, 7, -5, -1, 8, -12], [11, -16, 8, -9, 6, 3, 1, -13, -6, -4, 3], [-11, 0, 13, 12, 17, 9, 2, -4, 11, 1, 9], [6, -4, 3, -5, 14, -3, 5, -2, 9, 6, -8], [3, -14, 8, -16, -8, 1, 6, 3, -7, 1, -5], [-9, -12, 23, -1, 14, 4, -4, -18, 2, -2, -1], [4, 16, 9, 2, 15, 1, 4, 4, 10, -12, 5], [-25, -8, -4, 8, 0, 1, -1, 10, 9, 4, 9], [-7, -6, 4, -1, 7, 0, -4, -9, 16, -5, 1], [5, -2, 0, -8, -16, 5, 7, 0, -9, -11, -7], [1, 16, 0, 5, -2, -5, -18, -6, 2, 0, -8], [-31, -11, 17, -13, 6, 7, -10, -11, -2, -12, 2], [-17, 14, 5, -11, 1, -2, 7, -5, -8, -7, -8], [31, 5, 2, 19, 9, 1, -17, -1, 14, 7, 5], [-7, 4, -6, -1, 0, -6, 1, -7, -6, 2, -5], [11, 5, 8, 2, 2, -9, -20, 8, 9, 4, -9], [7, 12, -12, 6, 7, 6, -14, 1, 7, 6, 3], [7, 1, 8, -2, 4, -7, 3, -10, -7, -2, -13], [23, -15, 12, 4, -6, -2, 9, 11, 1, -22, 16], [-22, 6, -3, -16, 3, -4, -6, -17, -8, 10, -24], [1, 4, 9, 5, 14, 4, -5, -11, 0, 1, -10], [-27, 12, -4, -1, 1, 2, -11, -3, 0, 6, 1], [5, -2, -5, 5, -8, 1, 13, 25, 8, -6, 13], [7, -7, -14, -8, -13, -5, -15, 4, -5, 13, -15], [-24, 6, 3, -14, 5, -3, -8, -1, 6, -10, -3], [12, -3, 6, -12, 0, -5, -5, -9, -16, 4, -15], [-2, -3, 15, 3, 11, 2, -4, -9, -5, -18, -12], [-8, 5, 9, 8, 9, 0, 10, 7, -2, -21, 16], [5, 0, -19, 8, -19, -1, -2, 3, -5, 26, -4], [-18, 0, 6, -2, 16, -4, 7, -22, -8, 0, -12], [0, -10, 8, 5, 0, -2, 19, 7, 1, -4, 5], [-3, -5, -7, 1, -8, -8, 19, -5, -14, -4, 2], [-19, 2, -10, -7, -13, 2, 12, 6, -2, 6, 16], [-13, -12, -5, 1, 13, -4, -5, -8, -6, 17, -10], [-21, 4, 1, 0, 6, 2, -5, 0, -1, -5, -11], [-11, -1, 1, -1, 13, 1, 11, -6, 0, -4, 10], [9, 3, -6, -2, -1, 7, 3, 14, 5, -5, -5], [23, 0, -2, 1, -7, 1, -11, 6, 2, 10, 9], [5, -7, -4, -9, 1, -2, -4, -2, -8, 11, -16], [-6, 2, 7, -9, 11, 7, -16, -12, -3, -9, -8], [1, 4, 4, 2, -2, -2, -8, 6, 7, -7, -1], [0, 7, 5, 3, 10, 9, -13, 1, 11, -23, 3], [-16, -14, 0, -9, 8, 0, 1, -9, -9, -10, 2], [16, 11, -7, 0, -6, -6, -9, 5, -8, -3, -12], [5, -18, -2, -6, 4, 1, -12, -4, 7, 6, -7], [-9, 8, 6, 2, 10, 9, 14, -3, 7, -11, 15], [6, -2, 9, -1, 12, -7, -4, -7, -7, 0, -7], [17, -4, 0, -7, -1, -9, -8, 3, -16, -1, -19], [-4, -26, 6, 2, -8, -3, -1, 7, 2, -11, 10], [21, 0, 0, -8, -5, -4, 3, 3, 1, -6, 10], [-33, 8, 6, -2, 7, 4, 8, 2, -5, -6, -4], [-21, 13, 6, -5, 1, 5, -1, -8, 2, 3, 1], [-41, 12, 18, -3, 17, 1, 13, -8, 8, -7, 9], [-13, 2, 14, -6, 9, -3, 11, -7, 1, -8, -14]];
    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_u();".
// 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_u(:prec:=11)
    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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8034_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![32, -272, -190, 998, -45, -805, 141, 221, -33, -25, 2, 1]>,<7,R![-256, -1616, 1240, 4397, -1977, -2192, 566, 417, -58, -34, 2, 1]>],Snew);
    return Vf;
end function;