
// Make newform 3936.2.a.v in Magma, downloaded from the LMFDB on 02 August 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_3936_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_3936_2_a_v();".
// 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_3936_2_a_v();".
// 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 := [-58, -174, -58, 81, 29, -14, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [-5, -1, 1, 0, 0, 0, 0], [-32, -74, 22, 35, -8, -4, 1], [38, 66, -26, -33, 8, 4, -1], [40, 80, -24, -35, 8, 4, -1], [41, 98, -15, -44, 6, 5, -1], [100, 250, -42, -117, 18, 14, -3]];
    Rf_basisdens := [1, 1, 2, 2, 2, 1, 2];
    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_3936_a();"
function MakeCharacter_3936_a()
    N := 3936;
    order := 1;
    char_gens := [3199, 1477, 1313, 1441];
    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_3936_a_Hecke( : Kf := Kf)
    return MakeCharacter_3936_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], [1, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, -1, 1], [1, 0, 0, 0, 0, 0, 1], [1, 1, 0, 0, 0, 1, 0], [0, 0, 0, -1, 1, -1, 0], [1, -1, -1, 0, 0, 0, -1], [2, -1, -2, 0, -1, -1, 0], [1, -1, 1, 1, -1, 0, 1], [1, 1, -1, -2, 1, -1, 0], [1, 0, 0, 0, 0, 0, 0], [1, -1, -1, 0, -1, 1, 0], [2, 1, 0, -2, 1, 1, -2], [2, 1, 0, 2, 0, 0, 1], [-1, 1, 0, 0, -1, 0, 1], [3, -1, 0, 1, -1, 0, 0], [-1, -1, 0, 2, 1, 0, -1], [0, -1, -1, 0, 1, 1, -1], [4, 0, 2, -1, 2, 0, -1], [4, -1, 1, -1, 0, 1, -1], [-2, 2, -1, 0, 1, 0, -2], [1, -1, 0, -2, 1, 0, -1], [-1, -1, 0, 3, 0, -1, 1], [6, 1, -1, 1, 1, 2, 0], [3, -3, -2, 0, -1, 1, -1], [-3, 1, -2, -2, 3, 0, -1], [1, 0, 3, 2, 2, 2, 0], [-1, 1, 1, 0, 0, 0, 3], [6, 0, -2, -3, 0, -1, -1], [-3, -1, -3, 0, -2, 0, -1], [0, -2, -1, -1, 1, 2, -1], [-4, -2, -3, 1, -2, -1, 0], [6, 1, -1, -3, 2, -1, -1], [-2, -1, -1, -1, 0, -3, 3], [4, -1, 1, 1, -2, 1, 1], [-3, 1, 3, 0, -1, 1, -2], [-2, 1, 3, -2, 2, 0, 0], [3, -3, 1, 0, 2, -2, 1], [-8, 0, 1, 1, -3, -2, 3], [2, -5, -2, 3, -2, -3, 2], [2, -1, -5, -1, -2, -3, 1], [1, 1, 2, 1, 0, 3, 1], [5, 3, -2, -2, 2, 0, 0], [1, -2, -1, 0, -4, -4, 4], [-3, 1, 2, 2, 1, 0, 1], [10, 2, 1, 0, 0, 0, 1], [-8, -2, 0, 2, 1, 3, -1], [1, -6, 1, 0, 0, -2, 0], [3, -3, 4, 4, -3, 2, 1], [0, 3, 2, -1, 0, 1, -4], [2, 2, 1, 0, 0, -3, -1], [-8, -2, 2, -2, 2, 0, 0], [-2, -2, -4, -2, -1, 1, 1], [-2, 1, 0, -1, -1, 0, 3], [4, 0, 3, 2, 2, 2, -3], [7, -3, -6, 1, 1, -2, -4], [-1, 3, 1, 0, 3, 3, -2], [-3, -1, -1, -3, 4, 2, -4], [-1, -3, 2, 1, -1, 2, -2], [4, 1, 1, 1, -3, 0, 0], [-1, 3, 0, -1, -1, 0, 0], [-6, -1, 0, 3, 0, 1, 0], [7, -1, -2, 1, 0, 3, 1], [2, -3, 0, 2, -3, -1, 4], [-5, 1, 0, -3, 4, -3, -1], [-2, 2, 1, 4, -2, 3, 1], [-2, -4, 5, 1, 3, 4, -1], [-3, 7, 0, 1, -3, 2, 0], [0, -2, -4, -4, 2, 0, -6], [-13, -1, 1, 0, -2, 0, 1], [11, 3, 1, 4, -1, 1, 0], [1, 3, -2, -1, -1, 0, -2], [2, -4, -3, 1, 2, -1, 0], [10, -1, -1, -2, 1, -1, -1], [9, -1, -2, 0, 0, 0, 0], [-7, 2, 0, -4, 6, 2, -3], [2, -4, 0, 2, 0, -4, 2], [10, 2, 2, 5, 0, 0, 3], [-20, -2, -1, 0, -1, 0, -2], [-10, 3, 0, 1, -4, 1, 4], [-5, -5, -1, 2, -2, 0, 1], [4, 0, 3, 0, 4, 1, -3], [4, 1, -3, 1, -4, -1, 3], [-11, 1, 0, 2, -1, -2, 1], [-4, -2, 2, 0, 4, 0, -2], [6, -6, -4, 0, 0, -2, -2], [2, -4, -4, 0, 4, 2, -4], [11, 6, -3, -2, 2, 0, 4], [-9, 1, -2, 0, 1, 4, -3], [6, 7, 0, -1, -1, 0, -3], [7, 1, -1, -1, -1, 2, 1], [-10, -4, -1, 6, -1, 2, 0], [-7, 5, 2, 2, 1, 0, 1], [-4, -3, -4, -1, 1, 2, -5], [-6, -1, -2, -4, 1, -5, 2], [-5, -5, -3, -5, 2, 0, -4], [4, 4, -1, -1, 2, 1, -6], [-12, 6, 0, -2, 4, -2, 2], [7, 1, 6, -2, 5, 2, -5], [-8, -7, 2, -4, 3, 1, -4], [-8, 6, 1, 1, 3, 2, -1], [-3, 3, -5, -4, 0, -6, 1], [4, 2, -2, -6, 0, -2, 2], [-6, -8, -2, 4, -4, 4, -2], [-8, 0, -3, -3, -5, -6, 5], [13, -1, -4, -4, -2, -1, 0], [-11, -1, 1, -4, 4, 2, -3], [-4, 2, -2, -6, 4, -2, -4], [2, -2, -1, 5, 0, -1, -2], [-18, 0, 3, 3, 0, 5, 0], [-13, 1, -5, -2, -4, -2, -1], [9, 7, 2, -1, 1, -2, 2], [1, 5, -3, -1, -5, 2, -3], [-10, 0, 5, 7, -3, 6, 1], [-11, 1, 4, -4, -1, -4, 3], [-3, -1, 5, 0, -2, -2, 3], [-10, -1, 0, 2, 1, 3, -6], [0, -2, -6, 2, -2, 2, 0], [0, -6, 7, 1, -2, 3, 4], [-10, -2, 0, 0, 0, -2, 6], [-7, 5, 2, -2, 2, -4, -2], [2, 6, 5, 3, 2, -1, 4], [11, -1, 2, -1, -4, -1, 1], [-2, 0, -2, 6, 2, 6, -6], [-2, 11, -2, -1, 4, 3, -4], [-22, -1, 2, -1, 3, 4, -1], [-1, 0, 7, 6, 2, 4, 0], [-5, -1, 4, 5, -1, -2, 0], [9, -3, -3, 4, 1, 3, -6], [-2, 0, 6, 0, 2, -4, -2], [-12, 1, -1, 1, -4, -1, 5], [7, 0, 5, -2, 2, 2, 0], [-4, -6, 0, 0, 4, 4, 0], [-2, 2, 4, -1, -4, -2, 3], [-8, -7, 0, 2, 1, 5, -2], [7, -1, -3, 2, -5, -5, 2], [-6, -6, 1, 4, 0, 2, -5], [-11, 1, 4, 4, -1, 2, 1], [13, -7, 2, 1, 1, -2, 2], [13, 7, 2, -2, 0, -2, 2], [-14, 3, 7, -3, 2, 1, -5], [6, -4, 1, 3, -5, 6, 1], [-11, -5, 8, 3, -3, 2, 2], [-2, 1, 4, 3, 0, 1, -2], [0, 8, -7, -1, 6, 1, -6], [-22, 0, 0, 0, 2, 4, -2], [1, 5, -3, -8, -1, -3, 0], [15, -1, 1, 2, -4, 6, -3], [1, -1, 4, 5, -3, 6, 4], [2, 4, -10, -2, -4, -4, 0], [10, 0, 2, 1, 3, 1, 4], [-16, -5, 3, -2, 1, 3, 1], [8, 4, 7, -1, -2, 1, 2], [3, 3, 1, 4, 8, 6, -9], [-3, 0, -9, 2, -6, -8, 6], [-15, -3, -3, 5, -2, -2, 6], [2, -8, 2, -2, -2, 2, -2], [-12, -2, 1, -4, 4, 6, -7], [-1, 9, 4, -2, 1, 2, -3], [1, -13, -5, 4, -4, -2, 1], [-8, 5, -1, -7, 4, 7, -3], [-6, -2, 1, 5, -1, 2, 5], [-5, 11, 2, 0, 0, 5, -2], [3, 5, 1, -2, 4, 2, -1], [14, -5, -9, 3, -2, -3, -3], [7, 10, 4, 0, 0, -4, 5], [3, -3, 5, 2, 3, 7, -2], [-4, -3, 1, -1, -8, -7, 7], [16, 6, -1, -1, 0, -1, -2], [-8, 0, 1, 1, -6, -9, 4], [-10, -5, 4, 8, -4, 6, 3], [3, -3, -6, -5, 5, 0, -8], [-4, -9, 5, -1, -2, 1, -3], [-4, 0, -6, -2, 1, 7, -3], [8, 2, -2, 3, -1, -5, 2], [0, 6, -1, 6, 2, 6, -5], [1, -2, 5, 2, -4, -2, 8], [0, 3, 2, 5, -4, 5, 0], [-1, 1, 11, 2, 7, 3, -6], [-10, 6, -10, 4, -6, -2, 8], [-14, 1, 6, 1, -4, 1, 8], [-10, 4, -2, 4, 4, -2, -2], [-2, -2, -6, -4, -2, 0, -4], [17, 3, -3, -2, 8, 2, -11], [-16, -6, 1, 5, 0, -5, 0], [-18, 4, 5, -1, 4, 1, -4], [5, -3, 0, 1, -2, -5, 1], [10, -3, -2, 1, -5, -2, 3], [7, -7, -9, 4, -3, -1, -4], [12, -2, 4, -2, -4, -2, 4], [-14, 2, 3, -1, 6, 5, -2], [6, 2, -8, -4, -2, -4, 8], [-4, -10, 3, 8, -3, -2, 4], [0, 10, 4, -10, 6, 6, -8], [4, 2, 0, 0, -8, -4, 4], [7, 2, 6, -4, 8, 0, -5], [-20, 0, -2, -4, -4, 0, -4], [-12, 3, -7, 2, 3, -3, -3], [14, -8, -3, 2, 2, -6, 5], [-5, -5, 0, 6, -7, 5, 3], [1, 0, 6, -2, 4, -2, -3], [20, 6, 10, 0, 2, -4, 8], [-2, -4, -1, 3, 1, 6, -15], [-8, -9, 3, 9, -1, -2, 6], [-2, 0, 6, 1, 4, 3, 7], [-12, -12, -8, 2, -5, 3, -5], [7, -3, -1, -2, -4, -8, 3], [-7, -5, 1, 12, -1, -1, 2], [-9, -1, -4, 3, 2, 7, -5], [-10, -3, -4, 4, -1, 3, 0], [18, 8, 2, 1, -4, -1, 7], [-4, -10, 2, 6, 2, -4, 4], [8, 13, 1, 1, 3, 0, 0], [-12, -8, -1, 6, -4, 5, 3], [-23, 2, 5, 4, 2, 0, -4], [7, 5, -5, -4, -4, -8, 9], [-2, 7, 1, -6, 7, 1, 3], [28, 0, 8, 2, 4, 4, 0], [3, -8, 2, 2, 2, 4, -3], [-7, 1, 4, 2, 3, 11, -3], [22, 8, 6, 0, 3, -1, 3], [17, 10, 3, 0, 10, 4, -2], [-8, 2, -1, 2, -5, 4, 0], [15, -8, 0, 2, 0, -6, 1], [-16, -10, 8, 4, 6, 6, -6], [-9, -5, 3, 2, 0, 4, -1], [8, 5, 1, 5, -2, 5, 1], [20, -8, 1, 4, -7, -8, 8], [-10, -6, -6, -8, 6, 0, -6], [9, 11, 2, -3, 2, 7, 3], [5, 3, -2, 3, -1, -2, 4], [3, -3, -5, 2, 0, 6, -5], [-20, -18, 2, -5, 1, -3, -4], [-10, 3, 1, -6, 6, 0, -4], [25, 1, -6, -1, 4, -3, -1], [0, 11, 7, -5, 4, 5, -5], [6, -2, -10, 8, 2, 2, -4], [2, 13, 1, -9, 3, 2, -8], [6, 2, 6, -4, 6, -4, 0], [21, 1, 0, 2, -3, -6, 3], [-16, -1, 3, 1, 6, -1, -3], [11, 3, -7, 0, 3, 1, -6], [26, 4, -6, -4, 0, -2, 2], [11, 5, 5, 2, -6, 4, 1], [16, 4, -2, -3, -8, -5, 9], [8, -10, 4, 2, 1, -3, 3], [11, 3, 0, -1, 3, -6, 8], [8, 0, 0, 6, -2, 4, -4], [-39, -11, 3, 2, -1, 1, -2], [-11, 7, 5, 6, 2, 2, 3], [-4, -6, -2, 4, -6, -10, 10], [35, -11, -8, -3, -3, -2, 4], [12, -5, 11, 3, 10, -3, -3], [0, -8, 0, -4, 8, 0, 0], [-13, -2, -6, 0, 4, 4, -1], [-12, -12, -2, 3, -5, -1, 0], [-24, 1, 5, 3, -5, 2, 8], [-3, 1, -8, -11, 2, -1, -11], [5, -7, -7, -7, 3, -4, -1], [16, 8, 0, -2, -3, -1, 5], [-10, -11, -7, 7, 0, -3, 1], [-8, 11, -2, 1, -2, -5, 6], [15, -7, -7, -2, -8, 4, -3], [-3, -11, 2, 9, -1, 4, 8], [4, -1, 3, 5, -5, 8, 2], [-12, -2, -4, 4, -4, 8, -4], [2, 12, 5, 3, -4, 5, 4], [-24, -5, 9, 5, -3, 4, -10], [14, -7, -6, -9, 6, -11, 0], [13, 7, -5, 2, -7, -7, 4], [14, -2, 10, -5, 6, 6, -9], [-15, 15, 5, -1, 1, 0, -3], [-28, 10, -2, -4, 4, -2, 2], [24, -4, -1, 0, -4, -4, 3], [4, 6, 2, -6, 7, -1, -1], [-12, 1, 6, -1, 4, 3, -8], [-23, -5, -2, 3, 0, -11, 9], [4, -4, -7, 1, 1, 0, -1], [42, 3, 4, -1, 5, -2, 3], [-15, -14, -5, 4, 4, -2, 0], [14, 6, 4, 8, -2, -2, 2], [18, -6, -13, -9, 6, -7, -8], [-3, 9, 6, 2, -5, -10, 11], [-2, -15, 6, -7, 5, 0, -7], [-1, -5, 16, 3, 6, 5, -9], [2, 2, -1, -6, -4, 0, -1], [-1, -4, -9, -10, 0, -4, -2], [24, 0, -3, 1, -7, 0, 1], [6, -1, -7, -3, -5, -6, 6], [14, -10, -2, 0, -2, 2, 0], [-11, -1, 7, -1, 0, -2, -2], [16, 8, -3, 4, 1, 4, 0], [11, 8, 2, 0, -6, -2, -1], [20, -11, 3, -1, 7, 0, -2], [-16, -1, 5, 5, 2, -3, -1], [4, 2, 5, -2, 10, -2, -7], [8, -4, 8, 6, -2, 2, -4], [-5, 1, 2, -1, 10, 1, 1], [33, -7, 5, 1, -3, 1, 3], [0, -9, -3, 7, -14, -7, 11], [-21, -5, 8, 4, 3, 5, 3], [18, 4, -5, -5, 3, -4, -1], [9, -1, -2, 1, -1, 0, -10], [-19, -7, -4, 10, -5, -8, 7], [20, -2, -11, -6, 9, -4, -8], [17, -5, 1, 8, -11, -3, 2], [-2, -5, -6, -6, -4, 4, -9], [29, 0, -3, -4, 10, 0, -4], [-8, -7, -11, -7, -7, 2, 2], [16, 5, 1, 5, 2, -1, -5], [4, 10, 0, -4, -4, 0, -4], [-15, 7, 4, 7, 1, -4, 4], [-7, 11, 1, 4, -8, 2, 9], [23, -1, 15, 3, -5, -1, 9], [33, 13, -4, 0, 3, 2, 3], [37, -1, 5, 2, 4, 4, 3], [-4, 10, -6, 0, 6, 10, -8], [3, 1, 13, 7, 0, 10, 4], [-24, 8, 3, 3, 6, 1, -2], [5, 21, 1, -7, 9, -3, -5], [-1, 5, 8, 2, -4, -2, 0], [19, -4, -7, 2, -10, -14, 10], [23, 13, 3, -2, 6, 8, -3], [-4, -4, -6, -12, 4, 2, -12], [39, 9, -4, -3, -4, 1, -1], [-17, 3, -6, -5, 0, -9, 9], [-4, 11, -1, -5, -5, -6, 2], [8, -2, -3, 0, -2, -4, 3], [0, -3, -8, 7, 0, -5, 6], [22, 0, 1, -6, 10, 2, -7], [8, -2, -6, -2, 0, 6, 2], [-6, -20, 2, -2, 2, 2, 0], [5, 11, 12, -5, -3, 0, 4], [-16, -10, 13, 5, -5, 4, -1], [-11, 6, -5, -2, -8, 0, 4], [17, -13, 1, 2, 6, -8, 7], [-5, 5, -18, -9, 6, -9, -5], [35, 5, -2, 2, 9, 5, -5], [2, 9, 7, -7, 0, -1, 5], [20, 6, 4, -4, -2, 4, 2], [12, 1, -2, 6, -10, -8, 11], [-8, 11, 7, -7, 4, 5, -9], [36, -4, 1, 8, -8, 14, -5], [1, 9, 0, 0, 3, -6, 3], [-4, 4, 13, 9, 6, 7, 4], [1, -9, -6, 7, -11, -12, 6], [30, -3, 4, 7, 0, -3, 6], [6, 10, 10, 2, 2, 2, 8], [8, 10, 6, -5, 7, -5, 2], [-20, 4, -1, 4, 0, 1, 5], [8, 13, 11, -7, 9, 8, -2], [-7, -13, -13, 4, -3, -5, -2], [-16, -1, -4, 7, -8, -5, 8], [22, -2, -1, 3, 6, -1, 4], [-31, -13, 1, -2, 0, -8, -1], [17, 1, -6, 6, 3, 0, -1], [-15, 11, -11, -6, 6, 0, -5], [-3, -11, 7, 4, 4, 8, -3], [3, -3, 0, 7, -7, -12, 10], [-14, 2, -4, -8, -8, -8, 2], [-16, 7, 2, -3, 7, 2, -1], [-16, 0, 2, -2, -6, 0, -2], [-31, 3, 2, -4, -3, 4, -3], [23, 1, 8, -1, -4, -1, 7], [-16, -5, -7, -1, -6, 1, -5], [39, -9, -4, -1, -5, 2, -4], [32, -12, -7, 4, -4, -5, -1], [18, 10, 0, 0, 4, 4, -4], [19, 9, 2, 12, 1, 10, 1], [-8, 4, 2, 4, -6, -2, 0], [-2, -5, -2, -6, -1, -7, 6], [-14, 2, 0, -3, -8, -5, 3], [-11, 14, 7, -2, -2, 6, 2], [12, 0, -1, -8, -5, -12, 8], [29, 7, 5, -4, 8, 8, -3], [43, 7, 1, -5, 1, -5, 1], [28, -14, -6, 6, -7, -5, 3], [-23, 5, 20, -9, 6, 5, -5], [-10, 1, -13, -1, -3, 2, -6], [-29, 1, -5, 2, 8, -4, -1], [9, 5, 2, 3, -5, -10, 14], [-28, 10, 10, 0, -2, -4, 0], [15, 5, 12, -2, 7, 14, -5], [20, -6, 8, -2, 8, 6, -12], [-5, 6, -3, -2, -8, -14, 14], [32, 1, -6, 1, -6, 7, 0], [17, -7, 9, 4, -5, 7, -8], [-2, 1, 3, -9, -2, -11, 3], [2, -4, -2, -10, 4, 0, -8], [-10, -1, -17, -1, -1, -4, -2], [14, 2, -4, -6, 2, -10, 4], [-14, -8, -14, 2, -4, -14, -2], [20, -6, 12, 10, 0, -4, 4], [10, 6, -4, 0, -6, 0, -4], [11, 4, -3, -6, 4, -6, 8], [4, 10, 8, 0, 8, 6, 4], [3, -7, -1, -8, 3, -5, -8], [-8, -13, -7, 3, 4, 13, -11], [-35, -5, 14, 1, 7, 10, -8], [35, -13, 0, 14, -13, -8, 7], [-1, 21, 8, -4, 9, 3, -1], [-43, -7, 0, 2, 5, 4, -7], [29, 1, 3, -4, 8, 4, -13], [-37, 10, -5, 2, 2, 6, -6], [-35, 3, 6, 7, 7, 6, -6], [-1, -1, -2, -2, 0, -11, 4], [-42, -20, -2, 3, 9, 7, -8], [12, 22, 7, -4, 3, -4, -2], [-1, 9, -3, -6, -3, 5, -4], [-16, -8, -15, 10, -4, 4, -3], [18, 6, -10, -10, -6, -18, 10], [6, 6, 4, -2, -10, 0, 8], [-19, 7, 10, -1, 0, -7, 9], [-11, 3, 1, -6, 12, -10, 5], [1, 9, -3, 4, -6, 6, 5], [-3, 4, 13, -2, -2, -6, 8], [-15, 9, 7, 7, -6, 16, 6], [4, 2, -8, -2, 0, 4, -4], [-9, -19, -2, 6, -6, -4, 4], [-8, -7, 9, 9, 0, 3, 9], [-9, -9, -11, -12, -2, 2, -11], [-30, -2, -6, -16, 9, 5, -7], [20, -8, -3, 4, 0, -5, -1], [-13, -7, 8, -2, -12, -2, 4], [-20, 2, 10, 2, -1, 11, 11], [-30, 2, -13, -7, 3, 6, -7], [0, -16, 0, -1, -11, -1, 4], [-8, -9, 4, -1, 11, 8, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3936_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3936_2_a_v();".
// 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_3936_2_a_v(:prec:=7)
    chi := MakeCharacter_3936_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_3936_2_a_v();".
// 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_3936_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3936_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 144, -170, -66, 84, -7, -6, 1]>,<7,R![64, 192, -120, -240, 146, -4, -8, 1]>],Snew);
    return Vf;
end function;