
// Make newform 6936.2.a.bp in Magma, downloaded from the LMFDB on 28 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6936_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_6936_2_a_bp();".
// 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_6936_2_a_bp();".
// 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 := [-8, -468, -1344, -1141, -45, 285, 47, -27, -3, 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, 1, 0, 0, 0, 0, 0, 0, 0], [-19352, -125538, -39319, 103525, 16119, -23799, 823, 1475, -195], [22304, 75244, 101941, 61625, -21085, -17747, 3167, 1091, -201], [780, 55306, 193809, 130907, -47375, -35847, 7317, 2221, -445], [-1584, -91188, -291259, -194607, 73665, 53947, -11467, -3351, 689], [-42324, -658395, -1085242, -347750, 256472, 100520, -29092, -5998, 1389], [19704, -1182714, -2419641, -880833, 575949, 235711, -65879, -13755, 3143], [-88936, -1405824, -2549737, -838933, 613153, 229659, -68223, -13371, 3149]];
    Rf_basisdens := [1, 1, 11164, 11164, 5582, 5582, 5582, 11164, 11164];
    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_6936_a();"
function MakeCharacter_6936_a()
    N := 6936;
    order := 1;
    char_gens := [1735, 3469, 4625, 6361];
    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_6936_a_Hecke( : Kf := Kf)
    return MakeCharacter_6936_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, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, -1, 0, -1, 0, 0, -1], [0, 0, 0, 0, 1, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, -1, -1, -1, 0, 0], [-2, 0, 0, 1, 0, -1, 1, -1, 0], [-2, -1, -1, -1, 1, 1, -1, 0, 0], [0, 0, 2, -1, 0, -1, 0, -1, 0], [0, -1, -1, 1, 1, 0, -1, 0, 1], [2, 0, -1, 0, -1, 0, 1, 0, -1], [-2, -1, -1, -3, 0, 1, 0, 0, 1], [0, -1, 2, 1, 1, 0, 0, 2, -1], [4, -2, -1, -2, 1, 0, 0, 1, 0], [0, 0, -3, -1, -2, -2, 1, 0, -2], [2, -1, -1, -1, -1, -1, 0, 0, 0], [4, -2, -1, -1, 0, 1, 0, -1, 0], [-4, 1, -1, -2, 0, 1, 0, -1, 0], [3, 1, 1, 0, -1, -1, 1, 0, 1], [1, 1, -5, -4, 0, 0, 0, -1, 0], [-4, -1, -1, -2, 0, 0, -1, 2, 1], [2, -1, 3, 5, 0, 0, 1, -2, 0], [2, 0, -2, 2, 1, 0, 1, -1, 0], [2, -1, 0, 1, -2, -3, -1, 1, 0], [0, 0, -4, 2, 1, 2, 1, 0, 1], [-2, 2, -2, 0, 1, 1, 0, -2, 0], [1, -3, -1, 0, 2, 2, -1, 2, 1], [-2, -1, -4, -3, -1, -2, 0, 0, 1], [-6, -2, -3, -4, 0, 1, -2, 1, 1], [2, 2, -1, -5, 1, 2, 1, 2, 0], [12, -2, -2, -1, 0, 1, -1, 1, 0], [-6, 1, -5, -7, -1, 1, 0, 0, 0], [4, 1, 2, -2, -1, -1, 0, -2, 0], [0, 0, -5, -2, 1, -1, 1, -3, 1], [2, 0, 5, 9, -2, -2, 1, 1, -1], [3, 3, 2, 2, 1, -2, 2, 0, -2], [-4, 0, -1, -4, 1, 2, 1, 2, 1], [-2, -1, 1, 4, 0, -1, 3, 0, -3], [2, 0, -2, -6, -2, 1, -1, 5, -1], [2, -1, -1, -6, 0, 2, 1, 3, -1], [0, -1, 5, 2, -2, -1, 0, 3, 0], [-2, 2, -5, -1, 2, 1, 0, 0, -1], [0, 0, -5, 3, -1, 2, -1, 2, 2], [7, -1, 5, 3, -1, 0, 1, -2, 0], [-4, 1, 2, 5, 0, 0, 2, 0, -2], [7, -2, 1, 2, 1, 1, 1, 2, 1], [-2, 0, -1, 10, -1, -1, 0, 1, 0], [-1, 1, 5, 7, 2, -1, 0, -1, -1], [0, 3, -8, 0, -3, -1, -1, 1, -1], [6, -1, -6, -6, -1, -3, 1, 3, -3], [5, 1, -5, -6, -1, -1, 0, 1, 0], [2, -4, 4, 4, 1, 1, -2, 0, 2], [12, 1, -4, -3, -1, 2, 2, 2, -1], [-2, -1, -10, -1, 3, 2, 0, -2, 5], [4, 1, 11, 2, 0, -3, 1, 0, -3], [4, -3, 2, -4, 3, 3, -2, 1, 1], [6, -2, -2, -9, -1, 2, -2, 3, -2], [10, 0, 2, 0, 2, 0, 0, 0, 2], [0, -2, -9, -12, 0, 0, 1, 2, 0], [-2, -3, 1, 7, 1, 3, -4, -1, 3], [-3, 3, -3, 6, -2, -2, -1, 0, -1], [-6, -1, 3, -1, 4, 2, -1, 0, 2], [6, 1, -3, 6, -1, -6, 1, -3, -2], [-4, 0, 5, -4, 1, 1, -2, -3, 0], [6, -1, 1, 2, 0, 2, 3, -1, -1], [4, 3, 11, 6, 0, 0, -1, 2, -1], [-2, 0, 12, 12, 1, -1, 1, 1, -1], [1, -1, 2, -4, -1, 2, 0, 2, 2], [2, -4, -5, -5, 1, 1, 4, 1, -1], [-4, 2, 2, 5, 0, 1, -1, 3, 0], [4, -1, -1, 3, 2, 3, -2, -1, 4], [1, 1, 1, -7, -2, -1, 0, 3, 1], [2, 0, 3, 5, 4, -2, 1, -3, 3], [6, 2, 7, 4, 3, 2, 1, -4, 1], [8, 1, 3, -2, -1, -2, 2, 1, -4], [2, 3, -9, -2, -4, -3, 2, -3, -2], [2, 5, 8, 6, 3, -1, 3, 1, -3], [-2, -3, -9, -15, -1, 0, -2, 3, -2], [8, -3, -3, 1, 1, 1, 2, 3, -3], [10, 3, 3, 2, -1, 2, 1, 3, 2], [4, -3, -4, 1, -3, -2, 2, 2, -5], [10, 2, 12, 7, 1, -1, 1, 0, -4], [8, 1, -2, -4, -4, -1, -1, 3, 2], [-4, 0, -7, -3, 1, 4, 1, 0, 2], [-2, -1, -2, -12, -1, -1, -3, 3, 1], [17, -3, -1, -1, 1, -2, 0, -1, 1], [6, 0, -1, -1, 2, 2, -2, -5, 3], [-2, -4, 9, 6, 1, 0, 3, -1, -2], [-4, -1, -6, 1, 0, 1, -4, 2, 1], [-2, -3, 5, -12, 0, 3, -2, 1, 2], [10, -2, -6, -2, -4, 0, -2, -3, -1], [-6, -2, -6, -11, -1, 0, -2, 4, 3], [10, -3, -2, 7, 1, 3, 1, 2, 0], [0, 3, 3, 11, -2, 0, -1, -4, 0], [-6, -4, -3, 7, 2, 0, 0, -5, 1], [-2, 3, 0, 6, -3, 1, 1, 1, 3], [3, 2, -4, 3, 2, -2, 0, 6, 1], [13, -3, 12, 2, -1, 2, -2, 2, -2], [2, -3, -9, -6, 0, 4, 1, 1, 1], [-4, 4, 6, -6, -2, -1, -1, -3, -1], [-16, -2, -1, 11, -4, -2, 0, -1, -3], [10, 3, -11, 0, -4, -3, 0, -1, 0], [2, -3, 2, -8, 3, 0, 0, 1, 2], [4, -4, -6, -6, 8, 4, -4, 1, 4], [2, 3, -8, 7, 0, -3, -1, -3, 0], [2, 0, 12, 12, 6, -2, 4, -6, 0], [8, 5, -7, 6, 0, -3, 4, -5, 0], [10, -6, -9, -2, 1, -1, -1, 4, 4], [14, 1, 8, 3, 1, 1, 0, -2, 2], [-10, 3, -3, 1, -5, -7, -2, 2, -2], [-8, 5, -2, 2, -3, 3, 1, -1, 1], [-5, -1, -3, 3, 2, -1, 0, -1, -1], [2, 0, 0, 0, 2, -5, 2, -2, -3], [-10, 2, 0, -1, -2, -1, -3, 7, -2], [4, -5, 9, 4, 3, 1, 2, -3, 1], [-2, -5, 1, 1, 4, 4, -1, 0, 0], [-6, -4, -3, -4, 3, 4, -1, -2, 1], [0, 3, -3, -3, 0, -2, 6, -1, -5], [6, -3, -6, -6, 1, 2, -4, 5, 2], [6, -1, 9, -10, 0, 2, -3, 7, 3], [-2, -1, 3, -12, 0, 2, -5, 2, 7], [-4, 3, 1, -9, 1, -1, 0, -3, -3], [-6, -2, -12, -8, -2, -3, 1, 0, 1], [2, 6, -2, 0, 2, 0, -3, -1, 1], [-10, 2, 7, 5, 3, 6, -3, 5, 2], [4, -2, 10, 8, 0, -2, 2, -2, -2], [2, -9, 10, 13, 2, 0, -2, 5, -1], [-2, -7, -10, -1, -1, 1, 1, 6, 2], [-2, 8, -4, -15, 1, 4, 2, -1, 0], [12, -2, 6, -3, 2, 1, 1, 3, -2], [10, -8, -1, -5, 1, -3, 0, 2, 0], [-10, 5, -10, 1, -1, 1, 1, -6, 4], [-2, -3, -2, 2, 5, 3, -3, 1, 5], [9, -1, -1, -11, 1, 2, -3, -4, 6], [-4, 12, 0, 1, -2, -3, 2, -6, -3], [4, -10, 10, 2, 3, 2, -2, 6, -3], [-2, 4, -11, -5, 3, -2, 1, -4, -2], [-10, 3, 0, -2, 1, -5, -3, -1, -1], [2, 0, -1, 4, 5, 1, 2, 0, 6], [-14, -3, -10, -8, -3, 1, -7, 7, 3], [-6, -6, -16, -11, 1, 1, -2, 2, 2], [-4, 4, 6, 6, -7, -5, -1, 3, -1], [6, -4, 5, 10, 0, -3, -2, -2, 1], [4, 0, -5, -8, 5, 6, -5, -2, 7], [4, -6, 1, 5, 2, 5, -2, 1, 2], [2, 5, -9, -9, -4, -2, 5, -4, 0], [-20, -2, -5, -11, 2, 3, -4, 3, 6], [-2, 5, -14, -8, 3, 5, 1, -7, 3], [10, 1, -14, -12, -1, -1, -3, 3, -1], [-10, 1, 1, -1, 0, 0, 1, 8, 0], [4, -2, 0, -7, 2, -1, -1, 5, -2], [0, 10, 8, 1, -2, -3, 5, -5, -6], [18, 9, 4, 6, -6, -6, 8, -7, -3], [10, 4, -10, -1, 0, -7, 7, 3, -6], [-1, 0, 1, 5, -3, -7, 4, -1, -5], [10, 0, 2, -1, -2, -5, -1, 1, -6], [14, 5, -7, 0, 0, -5, 8, -4, -3], [-14, 5, 3, -4, 0, 1, 0, 1, 0], [-2, -6, -14, -11, 0, -3, 0, -2, 5], [12, 6, 3, -3, 1, 1, 2, -3, -3], [-4, -6, 9, -1, 0, -4, 3, -4, 0], [-14, -4, 15, 11, 3, -3, 1, -2, -4], [4, 0, -1, 2, 5, 2, -5, -2, 1], [-12, 9, -15, -9, 2, 0, -1, -2, 0], [5, 3, 7, 18, 0, -7, -2, -3, 2], [-6, 6, 17, 11, -2, 0, 7, -3, -3], [16, -7, -5, -10, 3, -1, 0, 8, -1], [-30, 4, -4, -10, -2, -2, -3, 3, 3], [0, -5, -1, 3, 7, 9, -7, 2, 4], [-14, -1, 12, 10, 1, 2, -2, -3, 0], [-6, 1, -6, 2, -3, -5, -1, 1, 3], [0, 0, -13, 1, -3, -1, 0, -6, 1], [14, -9, -5, -6, -1, 3, -2, -2, 2], [-16, -2, 11, 12, -1, -4, -1, 0, 1], [2, 5, 10, 7, 0, 3, -5, 4, -1], [2, -3, -15, -7, 5, 3, -3, 4, 6], [-6, 8, -4, -7, -4, -2, 2, 0, 5], [4, -7, -5, -3, -1, 0, 3, 0, -5], [-3, 4, -20, -14, 4, 4, 2, -2, 0], [6, -6, 4, -3, -1, 4, 4, 0, 3], [-8, 2, -10, -6, 5, 2, -2, 0, 3], [-10, 0, 3, 0, 1, -8, 3, 0, 1], [-14, -4, -8, -10, 2, -2, -6, 2, 0], [-6, -4, 6, 11, 0, -7, 5, -7, 0], [-6, 0, 19, 2, 1, -3, 2, -8, -6], [-12, 0, 19, 19, -5, -1, -2, 7, -5], [-12, 2, -15, -21, 3, 2, 0, -2, 1], [-4, 4, -1, -1, -7, -5, -4, 7, -1], [-4, 5, -3, 9, -7, 0, -1, 2, -1], [12, 0, -1, -5, 5, 2, -3, 6, -2], [2, 12, 7, 11, -3, -2, 3, -3, -4], [-16, -2, 4, -13, 0, -1, 2, 8, -1], [4, -3, -6, -4, 3, 4, 4, 8, -6], [-16, -6, -5, 0, -1, 2, -5, 8, 1], [10, -3, -2, -4, 2, -1, 2, 4, -5], [13, 4, 0, -1, 2, 2, -6, -4, -1], [0, -1, -15, -6, 4, 7, 0, -1, 2], [-20, 4, 8, 5, 0, -7, -1, -7, 4], [2, 2, -9, 13, -1, -6, 3, -6, -2], [6, 2, -21, -14, -11, -9, 2, 1, -8], [0, -4, 13, -3, 0, 3, 4, -1, 0], [2, -4, -2, -5, 6, 6, -3, -9, 5], [-6, 3, 0, -9, 1, -2, -6, -4, 5], [-10, 3, -6, -22, -5, 4, -2, 6, 0], [0, 13, -4, -7, 3, -2, 2, -5, -1], [2, -7, 8, 22, 1, -5, 2, 0, -6], [-12, 1, -13, 8, 2, 5, 4, -9, 4], [2, 6, -11, -15, -5, -4, -3, -1, -4], [4, 6, -1, 11, -1, -4, 1, 0, -5], [-6, 10, -9, 1, 1, 1, 5, -2, -2], [12, 8, 6, 8, 2, 0, 2, 4, -2], [18, 4, -4, 12, 2, -6, 9, -3, -7], [-28, 4, 8, 3, 0, -3, 1, 1, 0], [4, -5, 8, 21, -3, -7, 0, -1, -3], [-14, 8, 5, 15, -8, -1, 3, 0, 1], [6, 3, 10, 2, 1, -3, 3, 1, 3], [-2, 10, 9, 13, 3, -3, 2, 1, 1], [-8, -1, 0, 12, 3, -2, -2, 0, 4], [-6, -8, -3, 0, 1, -3, -6, 0, 4], [10, 6, 9, 20, 0, 3, 7, -2, 5], [-6, 6, 13, 20, -3, -2, 7, 0, 1], [-10, -7, -4, -10, 0, 6, -2, 8, -3], [-12, -5, 11, 5, 2, 0, -4, 1, -3], [-18, 1, 2, 1, 4, 5, 1, 6, 1], [-10, -1, 16, 9, 9, 0, 2, -6, -1], [4, -9, -1, 2, 4, 3, 4, 3, 0], [-12, -6, 1, -2, 6, 2, -3, -1, 0], [14, 8, 24, 2, 2, 0, 1, 1, 3], [-6, -3, 12, -5, 3, -3, -1, 2, -4], [-21, -1, -15, -14, 8, 6, -3, 2, 9], [-6, -1, -1, 3, -3, -4, -3, 3, -4], [4, -2, 3, -3, -3, 5, -8, 3, 3], [2, 1, -10, -16, 1, -4, 6, -1, 4], [2, 1, 14, -3, 5, 4, 2, 2, 1], [0, -4, 13, 2, -2, -9, 5, -4, -3], [0, 1, -1, -2, -1, 4, -3, 2, -7], [10, 1, 17, -3, -3, -1, -3, -4, 0], [-2, 3, -2, -7, -3, -2, -2, 2, 1], [12, 4, 14, -4, 9, 5, 0, -2, 2], [-8, -4, -3, 13, 0, -3, 2, -1, 6], [-6, -4, -5, -10, 5, 6, 4, 3, 0], [-11, 1, 3, 4, 4, -2, -5, -6, 9], [2, -8, -4, -6, 0, 6, 6, 4, 2], [12, 6, -4, 28, -4, -2, 0, 0, 0], [0, -1, 19, 16, -6, 0, -3, 8, -4], [-8, 6, -15, 1, -4, 2, 2, -3, 3], [10, 2, -6, 11, 1, 4, 4, -3, -2], [-8, 1, 3, 1, 2, 2, 3, -2, -4], [-20, 5, 13, -4, -3, 1, -4, -3, 3], [-8, 2, -14, 0, 6, 2, -2, -4, 12], [6, -7, -3, 1, -4, 2, -1, 10, -4], [30, -3, -12, 0, 3, 9, -1, -2, 0], [-8, 4, 2, 12, -9, -5, -1, 3, -1], [6, -6, -8, -17, -2, 3, 2, 2, -3], [24, -3, -1, 1, 7, 0, 1, 2, -3], [-22, 1, 7, 3, -2, 3, 2, -4, -3], [-4, -3, -7, 3, 0, -6, -2, -5, -1], [0, 1, 10, 2, -5, -1, 5, -2, -5], [4, -8, -2, -3, -1, -6, -4, 10, -1], [-10, -8, 9, -6, 1, 2, -3, 18, -1], [2, -8, -6, -14, -2, 3, -1, 4, 3], [22, 4, -4, -15, 9, 2, 2, -1, -2], [0, -6, -7, -7, -3, -9, 2, 5, -9], [-7, 1, -19, 8, -2, 6, -5, 4, 5], [-4, 6, 1, 7, 6, -2, 4, -1, 1], [-12, -6, 9, -17, 1, 9, 2, 7, 1], [14, 7, 9, 10, 5, 2, -1, -9, -6], [-8, -4, 2, 8, 2, 11, -5, -1, 9], [6, -3, 15, 18, 2, 1, -2, -3, -2], [-22, 4, 6, -7, 1, -8, 2, -9, -2], [-14, -9, -2, 12, 2, -4, -4, -2, 3], [-2, 1, 15, -2, 1, -6, -1, -5, 4], [5, -6, -7, 9, 3, -1, 0, 7, 3], [-16, 2, 23, 22, -10, -4, -3, 2, -2], [-4, 4, -10, 1, 1, -2, 1, 5, 0], [2, -1, -7, -4, 1, 0, -7, -1, -6], [6, -2, 5, -22, 3, 2, 1, -3, -2], [-4, 0, -15, 8, -3, 8, -4, -1, 6], [10, -8, -26, -12, 0, -2, -6, 6, 2], [-8, 5, 16, 18, -2, 4, 0, -2, -4], [0, -4, 22, 3, 0, -3, -1, 1, 2], [14, 12, 13, 5, -2, -8, 5, -11, -5], [13, 7, 5, -1, 2, 5, 4, -1, -5], [-14, -1, -19, -15, 2, 2, -1, 0, 0], [19, -3, -11, -16, 6, 6, -2, -13, 6], [0, 4, -2, 9, -4, -11, -4, -6, 3], [10, -5, -3, 5, 4, -6, 5, -11, -3], [10, -1, 35, 8, -4, -3, 2, -1, -8], [14, -8, 5, -25, -3, 7, 0, 3, -3], [4, -14, -13, -20, 5, 5, -2, 11, -8], [26, -5, 7, -2, -2, 7, -2, 7, 0], [20, -7, 15, 6, 3, -4, 1, 2, -5], [15, 2, -8, -13, 4, 4, 2, 0, 7], [-4, -3, -24, -6, -8, -8, -8, 8, 4], [-5, 3, 1, -6, 1, -5, -2, -3, -2], [8, 6, -16, 11, -4, -4, 4, 0, -3], [12, -5, 0, -3, 1, -3, -3, 1, 8], [-4, -6, -2, -14, 5, 0, -4, 8, 1], [-5, 0, -4, -16, -2, 7, 2, 2, -4], [-4, 6, 11, 7, -4, -4, 4, -5, -3], [16, -15, 13, 13, 3, 4, 0, 6, -1], [8, -11, -3, 11, -5, -3, -6, 3, 3], [-26, -5, -2, 1, -1, 5, 3, 4, 4], [12, -8, -1, -1, -3, 3, 9, -2, 0], [4, -7, -2, -5, 4, 5, -2, 2, -3], [18, -5, 3, -1, 0, -5, 4, 4, -7], [-10, 4, -1, 7, -7, -9, -6, 1, 5], [26, 2, 15, 8, -3, -3, 4, -2, -6], [4, 0, 0, 7, -4, 5, -5, 5, 0], [-4, 6, -6, -1, 4, -3, 2, 4, 5], [8, 5, 3, -4, -4, -1, 0, 3, 0], [-1, 7, 2, 2, 7, 6, -6, -8, 10], [-4, 0, -12, -17, 10, 3, -5, -3, 0], [0, 6, 20, 25, 6, 0, 3, -2, -2], [-10, 6, 1, -4, 5, 3, -2, 1, 6], [-10, -8, 10, 7, -2, 5, -1, -1, 4], [4, -3, 3, -1, 2, 1, -3, 1, 8], [4, -1, -17, -29, -2, 2, -7, 2, 0], [-30, -13, -13, -18, 4, 8, -1, 9, 7], [13, 8, -11, -28, -3, 0, 3, -4, -1], [2, -4, 0, -11, 0, 6, -2, 6, -5], [4, -5, 4, -8, -4, 7, -7, -2, 12], [-6, -3, -6, 10, 9, 1, -3, -3, -1], [-19, 10, 4, -4, -3, -4, 5, -1, -1], [11, 4, 19, 14, 9, 5, 8, -9, -2], [5, 3, 1, -2, -10, -10, 1, 2, -7], [6, 4, 3, 1, 1, 3, 2, 1, -1], [-6, 9, 10, 1, -3, 3, -1, 5, -2], [-18, 9, -4, -5, -3, -1, 1, 6, 2], [0, 5, -16, 13, 1, 3, 1, -1, -6], [2, -2, -4, -3, 4, -1, 2, 0, 11], [2, 10, -11, -4, -1, 3, 2, -5, 0], [-16, 5, 12, -5, 4, -6, -2, -6, 6], [-10, -12, -1, 1, 8, 4, -2, -3, 11], [-12, 0, 0, -1, 7, 3, -1, -3, 7], [16, 5, 15, 8, 0, 3, 6, -3, -8], [-8, -2, -6, -11, -3, 1, 7, 4, -3], [1, 10, 0, -3, 1, 3, -5, 7, 2], [-3, -2, -6, -13, 0, -6, 6, 2, -5], [0, -1, 7, -4, 8, 11, -2, 9, 2], [16, -6, 8, 25, 5, 0, -1, -1, 2], [2, -2, -3, 9, 1, 1, -3, 4, 2], [-6, 7, 12, 13, -6, 1, 4, -16, 1], [26, -1, 23, 23, 2, 6, -1, -6, 0], [-15, -5, -10, -9, 5, 9, -1, 3, 10], [8, -1, -19, -14, 7, 13, -4, 7, -5], [-14, -10, 0, -26, 2, -2, -4, 0, 0], [-14, -4, 6, 10, 1, -7, 0, 2, -2], [1, 4, 3, -2, 11, 7, 4, -5, 6], [4, -6, -3, -1, -2, 1, -5, 3, 4], [-2, 5, 33, 32, 1, -4, -1, -6, 1], [-16, 0, -25, -10, 2, 0, 6, -6, 6], [18, 0, 30, 8, 1, -11, 1, 4, -12], [2, 6, 5, 5, -13, -13, -2, 3, -9], [6, -8, -10, -23, 6, 3, -5, 3, -4], [14, 2, -7, 14, -6, 9, -1, 2, 5], [-26, 12, 12, -1, -4, -3, -5, -1, 2], [2, 8, -1, 18, -3, -10, 3, 8, -5], [-5, -4, 10, 12, 10, 7, -3, 11, 1], [0, -15, 18, -1, 7, 4, 0, 2, -1], [-4, -5, 17, -3, 6, -6, 5, 0, -10], [20, -8, 9, 6, -9, -2, -3, 4, 1], [-16, -7, -20, -3, 1, 2, -2, 8, 3], [19, -8, -19, -10, 1, 4, -9, 4, 5], [10, 1, -20, -38, -4, 0, -5, 13, -7], [-1, -8, -17, -16, 0, 0, -5, 12, -1], [14, 0, 27, 9, -7, -5, -2, 8, -2], [0, -5, 27, 8, 0, 6, -7, 2, 1], [-36, 4, -11, -21, 2, -7, 4, 1, 0], [-2, 0, 6, 12, -4, 0, -6, -4, 8], [-2, 8, 12, -8, 6, -10, 2, -4, -4], [-9, -11, 7, -19, 1, 10, -3, 8, -2], [18, 11, 9, 18, 1, -8, -3, -1, -4], [4, -4, 31, 18, -9, -4, -2, 1, -2], [-12, 5, 11, 2, -2, 5, 0, -3, -6], [8, 2, -17, -14, -3, -2, -2, 8, 5], [16, -3, -40, -28, 1, -5, -3, 1, -7], [2, -10, 18, -3, 11, 8, 0, -8, 4], [4, -1, -26, -11, -1, 8, -3, 3, 4], [16, 11, 12, -2, -1, 1, 9, 1, -13], [-36, 6, -13, -32, -2, 7, -7, 5, 8], [32, -10, -3, -8, 7, -4, 3, -4, 1], [-6, 5, 1, -11, -1, 5, -2, -4, 2], [24, -16, 11, 1, -1, -1, 0, 3, -7], [32, 0, 11, 12, 1, 11, 2, 1, 5], [-34, -2, -11, 13, -5, 0, 1, 5, 2], [-17, 11, 14, 21, -3, 3, 5, -3, -2], [-20, 16, 5, 17, -1, -3, 8, -7, -1], [-6, 5, 8, -19, 9, 9, -1, 8, 10], [2, 6, 3, 24, 3, -6, 2, 1, 2], [8, 4, 24, 25, 10, -3, 4, -4, 3], [-7, 1, 5, 38, -2, -4, -5, -4, -1], [18, -2, 2, 5, -2, 1, -9, -3, 8], [-14, -2, 24, 24, 8, 3, -2, 8, -1], [16, -1, -10, -4, 4, 6, -9, 6, 12], [-16, 9, 7, -8, 4, -3, -2, -13, 2], [-8, -10, -4, -29, 7, 9, 1, 1, -1], [-12, 3, 9, 17, -12, -11, 10, 1, -9], [16, 8, -5, -2, -2, 0, -11, -4, 12], [48, -2, 1, -17, -3, -3, -4, -3, -1], [23, -6, 25, 8, 1, 3, 1, -2, 3], [-14, -9, -11, -9, -6, 2, 3, 8, -8], [-4, 5, -33, 6, -2, 5, -4, 1, 12], [-38, 4, -12, 1, 1, -1, 0, -4, 2], [-10, 1, -25, -8, 4, 8, -1, -5, -1], [-16, -5, -4, 15, -5, 2, -6, 4, 3], [0, -4, 17, 19, -1, -5, 6, -7, -5], [-6, 2, -16, -27, 4, 3, -7, 7, 2], [-3, 6, -24, 9, 2, 1, -5, 1, 2], [8, -9, -20, -15, 2, -3, -1, -3, -6], [8, -4, 0, -14, 1, 5, 3, -1, 9], [-14, -2, -14, 15, -13, -6, -4, -1, 0], [38, 10, -6, 10, 0, -3, -2, -10, 7], [16, -6, -13, -12, 4, -1, -3, 6, 1], [-10, 7, 10, -14, -11, -5, -1, -3, 1], [16, 12, 5, -24, -4, -5, -1, -5, -4], [18, 3, 15, 3, 6, -8, 13, -12, -4], [-12, 3, -17, -2, 0, 7, -2, 3, -6], [20, -11, 17, 7, 7, 3, 4, -1, -1], [0, 1, -5, 6, -13, -5, -2, 13, -5], [16, 1, 11, 37, -4, -8, 3, -8, -8], [18, 4, -12, -23, 7, 2, 2, -4, 5], [30, -5, -13, -2, 2, -11, 0, 0, -7], [-44, 16, 8, -1, -7, 0, 1, -1, 6], [16, -7, -9, -8, 7, 4, 6, -1, 6], [22, 4, -36, -25, -2, -1, -1, 3, -6], [14, -8, -13, -26, -2, 3, 0, 12, -5], [-8, 5, 14, 21, 1, 0, 4, -10, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6936_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6936_2_a_bp();".
// 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_6936_2_a_bp(:prec:=9)
    chi := MakeCharacter_6936_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_6936_2_a_bp();".
// 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_6936_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6936_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, -468, 1344, -1141, 45, 285, -47, -27, 3, 1]>,<7,R![629, 1578, -669, -1684, 144, 456, -8, -39, 0, 1]>,<11,R![204744, 174456, -60102, -36333, 5847, 2721, -227, -87, 3, 1]>],Snew);
    return Vf;
end function;