
// Make newform 4235.2.a.bl in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4235_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_4235_2_a_bl();".
// 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_4235_2_a_bl();".
// 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 := [-11, -38, 38, 140, -96, -106, 63, 26, -14, -2, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-44, -97, 67, 188, 6, -95, -9, 17, 1, -1], [121, 157, -511, -33, 381, -65, -92, 23, 7, -2], [44, -85, -372, 390, 329, -261, -87, 57, 7, -4], [-143, -225, 640, 62, -468, 62, 109, -23, -8, 2], [66, -72, -501, 372, 416, -258, -104, 57, 8, -4], [-198, -40, 1245, -570, -1022, 492, 257, -120, -20, 9], [-286, -184, 1629, -684, -1334, 647, 335, -160, -26, 12]];
    Rf_basisdens := [1, 1, 1, 11, 11, 11, 11, 11, 11, 11];
    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_4235_a();"
function MakeCharacter_4235_a()
    N := 4235;
    order := 1;
    char_gens := [2542, 1816, 2906];
    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_4235_a_Hecke(Kf)
    return MakeCharacter_4235_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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 0], [2, -1, 0, 0, 0, -1, 1, 1, 0, -1], [0, 0, 1, 1, 0, -1, 0, -1, -1, 0], [0, 0, 0, 0, 1, 0, -1, 1, 1, 0], [-1, 1, -1, 1, 0, -1, -1, 0, -1, 1], [0, -1, -1, 1, -1, 0, -1, -1, -1, 0], [1, 0, -1, 0, 1, -1, -2, 1, 0, 0], [-1, 0, -2, -1, 0, 0, -1, 1, 0, 1], [-3, 0, -1, 1, 0, -1, -1, -1, 0, 1], [2, 0, 1, -2, 0, 0, -1, 0, 1, 0], [1, 0, 1, 1, 0, 0, 1, 0, 1, 0], [-2, 4, 0, 0, 1, 0, -2, -1, 0, 0], [1, 0, 1, 1, 1, -1, 0, 1, 1, 0], [1, 2, -1, -1, 2, -2, 1, 0, 0, 1], [-2, 0, 0, 0, -3, -1, 0, -1, 0, -1], [-1, 0, 2, -1, 2, 2, 1, 2, 1, -1], [5, 1, -1, -1, -1, 2, -1, -2, -2, 1], [0, 1, -2, 1, 0, -4, -1, 0, -3, 1], [-3, 0, 2, 2, -1, 1, -1, 1, 2, -1], [0, 2, -1, 2, -3, 2, -1, -1, -2, 0], [2, 2, 1, -1, 3, 3, 2, 0, 2, -2], [3, 1, 2, -2, -2, 4, 1, -2, 3, 0], [4, -2, -1, -3, -3, 1, -2, 1, -1, 0], [6, -1, 2, 2, 2, 0, 0, -2, 0, 2], [0, -2, 3, 1, -1, -1, 1, 0, 1, -2], [-2, -1, 1, 1, -2, 2, -1, -4, 0, 0], [4, 2, 2, -2, -1, 3, -1, -1, 1, -1], [-3, 4, -3, -3, 0, 0, 1, 4, 0, 1], [-6, -6, -2, 0, 1, -5, 2, 5, 0, -1], [4, 4, 0, 0, -3, -1, -1, -2, -1, 2], [-4, 3, -1, -2, 3, -1, 0, 4, 2, -1], [3, -2, -2, -2, 3, -3, 3, 3, 3, -1], [0, -4, 4, 5, 0, 3, 2, -5, -1, 1], [-1, 1, -3, -1, 5, 1, -1, 1, 1, 1], [-1, -1, -3, -2, 3, -3, 0, 4, 3, 0], [3, -2, -1, -2, -3, -1, 2, 2, -2, -2], [2, -6, 1, 3, -6, 2, -1, -2, -1, 3], [-1, -1, -3, 1, -4, -3, 0, -1, -6, 0], [0, 1, 0, -3, -2, 3, 3, -1, 3, -2], [4, -3, -2, 0, 5, 3, 3, 3, 2, 0], [0, -1, -1, -4, 1, 1, -2, 1, 2, -3], [2, 5, -1, 4, -2, -2, 1, -3, 1, 2], [6, -1, -2, 3, -1, 0, -3, -2, -3, 1], [10, 0, 2, -3, 4, 6, 4, 3, 5, -1], [5, 0, -5, -1, -1, -8, -3, 2, -3, -1], [4, 0, 3, 0, 3, 4, 3, 1, 0, -2], [-1, 1, -5, -2, -3, -5, -3, 2, 1, -1], [10, -2, 2, 2, -4, 4, 0, -2, -2, 2], [4, -1, 1, -3, -1, 1, 1, -2, 2, -3], [1, 1, -3, -3, 1, 3, -1, -1, -2, 0], [1, 0, 3, -1, -8, 2, -3, -5, -4, 2], [4, -3, -1, 2, -5, -1, 0, -3, -1, 0], [-2, -3, -2, 2, -3, -4, 0, 0, -3, -2], [4, 2, 0, 2, -3, -1, -1, -8, -1, 2], [5, -3, 2, 2, -1, -3, 6, 0, 2, -1], [-5, 3, -2, -3, 4, -3, 4, 5, 2, -2], [5, 0, -3, 3, -5, 1, -2, -1, -4, 0], [1, -2, -3, -1, -3, 1, 2, 1, 0, -2], [2, -4, 2, 1, -4, 9, -3, 0, -1, 1], [-5, 0, -4, -3, -1, 1, -6, 0, -3, 2], [8, 2, -3, -1, -1, -3, 4, 1, 3, -2], [-4, -1, 1, -1, -1, 0, -1, -5, 0, 4], [4, 3, 4, 2, 0, 7, -2, -5, 4, 3], [6, 1, 3, 6, -1, 5, -2, -7, 0, 3], [3, -6, 0, 0, 2, 2, -3, 5, -1, 4], [11, 2, -4, 3, 2, 1, -3, -1, -3, 4], [-2, 4, -1, -8, 5, 1, 2, 5, 5, -5], [-1, -4, 7, -4, 4, 5, 1, 2, 7, 0], [0, -2, 5, 0, 1, 0, -4, 0, 2, -1], [16, 7, 0, 2, -1, 5, -3, -5, -2, 2], [-7, 2, 2, 1, -2, -2, -1, -2, -5, 1], [-4, 4, -1, -2, 5, -4, 2, 8, 2, -3], [2, -3, -1, 1, -2, -1, -3, -3, -7, 0], [9, -6, 2, -1, 1, 6, 5, 1, 0, -3], [8, 0, -3, -6, 6, 4, -4, 2, 2, 4], [-3, 3, 3, 4, -4, 4, -3, -6, -6, 5], [8, -3, -1, 5, -11, 0, -2, -5, -4, 2], [-8, -2, 8, -3, 4, 3, 2, 0, 5, -4], [-6, -7, -1, 0, -4, -5, 0, 2, -2, 1], [0, -2, -4, -4, 2, -1, 2, 10, 2, 2], [-3, -2, 5, -1, -9, 2, -6, -3, -3, 1], [-2, 2, -2, -2, 1, -2, 2, 3, 2, -6], [11, 1, 1, -1, 0, 3, 1, -3, -5, -2], [8, 2, 3, 0, -7, 3, 0, 3, -1, -2], [11, -1, 0, 6, 1, 2, 0, -3, -6, 2], [-9, -2, 0, 4, 0, 1, -2, -2, -3, -1], [7, -8, 1, -5, 2, 0, 5, 3, 0, 0], [0, 1, 3, -1, 4, 3, 2, 3, 9, 1], [2, -5, 3, 2, 2, 6, 0, -3, 5, -3], [-13, 2, 1, 0, -3, -5, -1, 0, 3, -2], [-6, -5, -3, -2, 10, -6, 1, 3, 4, -4], [12, 2, 1, -1, 2, 0, -1, -4, -6, -1], [4, 2, 5, 1, 4, 4, -2, -3, 5, 2], [4, -9, -1, 0, -5, -6, 2, 1, 1, 2], [12, -6, 6, 2, 0, 4, 2, 0, 6, 2], [6, 2, -1, -1, -1, -1, -3, 0, -3, -2], [4, 3, -5, 2, -2, -6, -5, -2, -1, 6], [5, -7, -1, 0, 6, 0, 6, -1, -6, 2], [1, 4, -2, -5, 5, 1, 9, 4, 2, -1], [-5, -2, 1, 1, -6, 0, -2, -7, -6, 1], [8, -3, 0, -1, 0, -4, -1, 2, 1, -3], [1, -4, 4, -1, -9, 7, -1, -2, -1, -1], [-1, 0, 2, -1, 7, 8, 7, 3, 6, -5], [14, -9, -3, 4, -6, -3, -4, 0, -4, 2], [-5, 2, 4, -6, 6, 2, 6, 8, 10, -8], [3, -5, -1, 2, -1, 4, 0, 3, -1, 1], [-3, 1, 6, -4, 3, 8, 0, -4, 4, -4], [16, -8, 1, 2, -1, 5, 4, -1, 0, -1], [-10, -2, 0, -5, 9, 0, 3, 12, 9, -1], [2, -4, 3, 4, -4, -8, 2, 3, -4, -2], [-7, -5, 0, 3, -2, 0, -4, -5, -3, 3], [0, 0, 3, -3, 10, 1, 4, 2, 7, -8], [17, 4, 1, 0, -4, 7, -1, 2, -3, 3], [7, -2, 6, 3, 3, -1, -7, -4, 1, 1], [-6, 5, 3, 9, -4, 1, 3, -4, -4, 3], [-13, 0, -1, -1, 10, 0, 6, 3, -2, -3], [-8, 4, -4, -3, 2, -6, 2, 6, -3, -2], [9, 4, -4, 2, -2, -6, 3, 5, -3, 2], [5, -8, 0, -5, -3, 11, -5, 0, 4, -3], [-4, 1, -1, -3, -4, -3, -1, 2, 0, -1], [3, -4, 7, -5, -6, 7, 1, -7, 7, -2], [3, -6, 4, 6, -8, 5, -1, -3, -1, 1], [2, 6, 2, 3, -4, 3, 4, 2, 5, 0], [3, 5, -1, 0, 9, -3, 1, -3, -3, -4], [4, 0, -6, -6, 8, 1, -4, 5, 5, 2], [11, 1, 2, -5, 9, -3, 7, 4, 7, -7], [-8, 3, 10, 1, -7, 8, 3, -4, 1, -5], [5, -1, 0, -6, 6, -2, -1, 5, 3, 2], [4, -1, -2, 1, 0, -5, 1, -5, -7, 4], [-2, 0, -6, 6, -9, -2, 2, -3, 0, 2], [-3, 9, -1, -8, 11, 9, 2, 4, 1, -1], [1, -1, 5, -4, 0, 3, -1, -3, 2, 5], [-16, 4, 6, 1, 12, -10, 8, 0, -2, -3], [12, 10, -11, 1, -4, 1, -4, -5, -5, 0], [11, -6, -5, -1, -6, -5, -3, 3, -2, 2], [-7, 2, -6, -11, 4, -2, 6, 6, 2, -6], [0, 2, 5, 2, 11, -7, 5, 6, 10, -7], [-11, -2, -1, 1, 2, 2, -2, -5, -4, -1], [-14, 7, 1, -1, -4, -3, 3, 0, -2, 3], [-14, 2, 2, 6, -2, 8, 2, -2, 1, 3], [-8, 3, -2, -14, 6, 0, 3, 2, 3, -4], [10, -5, -5, 2, -7, 8, -6, -3, -9, 6], [6, 14, -3, -6, 0, 5, 0, 2, 7, -2], [5, 0, 3, 0, 15, 13, 4, -2, 2, 4], [6, 7, -8, -1, 2, -4, 5, 6, -2, 2], [-7, -4, -2, -4, -1, -5, 4, 5, 3, 1], [10, 4, -5, 2, 15, -3, 6, 3, 2, -1], [16, -8, -6, -4, -7, 0, -3, 4, 0, -2], [-6, 1, 5, 3, 7, 4, -5, 1, 4, 0], [-1, 7, 0, -3, -1, 3, -3, 0, -1, 3], [4, 0, -4, 1, -5, 11, 1, 4, -1, 0], [19, 5, 5, -1, 0, 7, 3, -5, -1, 4], [17, 5, 1, 5, 0, -1, 8, -4, 0, 1], [-1, -2, -7, 6, -8, -3, -11, -9, -8, 2], [0, 5, -5, -6, 2, 1, -4, 0, -6, 6], [7, -8, 1, -1, -3, 1, -6, 0, -5, 7], [1, -1, -1, 7, -7, 5, -9, -7, -7, 5], [-3, 0, 1, -4, -5, -9, -5, 0, -1, -5], [-10, -4, -4, -2, 9, -5, 1, 13, 8, -6], [5, 3, -3, -12, 14, 1, -2, 3, 3, 0], [-6, 9, -7, -7, -3, 1, 3, 1, 4, -7], [2, 12, -4, 6, 0, 2, -4, -8, 2, 6], [-15, 4, -3, 1, -8, -15, -6, -2, -7, -2], [5, 5, -1, 2, -3, -1, -4, 2, -1, -4], [7, -3, 2, 6, 0, 6, 5, 6, 5, -1], [-7, 2, 5, -1, -6, 13, -5, -7, 4, -4], [-7, -1, 4, -3, 0, -5, 6, 5, -2, -6], [15, 2, 1, 5, -3, -1, 2, -5, -11, 8], [-7, -7, 3, -1, -8, -5, 2, 4, 0, -9], [14, 4, -7, -2, -4, 6, -1, -2, -4, 3], [4, 5, 2, 8, 6, -2, -5, -4, -1, 4], [0, 1, -2, -1, -5, 4, -10, 4, 4, 2], [-3, 7, -2, 0, -3, 0, -3, -2, -13, 1], [1, -4, 1, 4, -2, -11, -3, -8, -2, 1], [-3, -9, -1, 2, -3, -6, 3, -3, -6, 3], [-1, -11, 6, -4, 0, -4, 9, 4, 3, -12], [2, 3, 9, 3, -8, 11, -7, -10, -6, 3], [-12, -5, 4, 5, 0, -1, -4, -7, 0, 0], [17, 3, 1, 3, -3, -7, -3, -7, -3, 1], [4, 5, -2, -2, 4, 0, 8, 0, 3, -6], [-14, 1, 7, -5, -2, 0, -1, 2, 4, -2], [-2, 12, -7, -2, 15, 8, 7, 7, 8, 2], [14, -8, -7, 0, 7, 5, 2, 1, -6, 7], [3, -4, -1, -6, 4, -5, 7, 3, 10, -7], [5, -6, 5, -4, 0, 1, 9, 2, 1, 0], [-22, 7, -6, -1, 5, -2, 1, 7, -3, 0], [-7, 8, 1, -3, 2, 3, 9, 3, 4, 2], [3, 1, -2, 8, -8, 8, -7, -13, -8, 11], [-19, 10, 0, -2, 8, 2, -8, 2, 10, 4], [-14, 3, -2, 2, -12, -6, -5, 0, -9, 6], [-17, 3, -2, 4, 4, -2, 3, 3, 0, -5], [-17, -8, 9, 3, -6, 17, 1, -12, -1, -4], [6, -5, -9, -2, -7, 17, -4, -4, -10, 7], [11, 3, 1, -4, 1, 6, -6, -6, -8, 1], [13, -7, 10, 4, -7, 4, -2, -2, 0, 4], [-11, 5, -5, 1, 3, -6, -3, 2, -2, 1], [17, -3, 0, -2, 0, 8, 13, 0, 3, -3], [-9, 12, 4, -5, 1, -1, 4, 6, 10, 2], [-1, 10, -3, -2, -14, 4, -1, -6, -1, 3], [7, -8, 1, 3, -8, -8, 3, 0, -4, -7], [5, -3, -7, -12, 10, -1, 2, 9, 9, -4], [-1, 10, 12, 1, -1, 4, -7, -11, -4, 7], [3, 2, -4, -4, 2, 3, 1, 10, 6, 0], [6, 9, 2, 4, -5, 11, -5, -13, 0, 4], [-1, -5, -1, -1, 6, 7, -4, 6, 4, 5], [0, 4, -4, 1, 4, 2, -6, -2, -3, 8], [10, 3, -2, 6, 6, -2, 4, 4, 1, -1], [-5, 1, 0, -6, 21, 9, 7, 8, 5, -2], [-7, -4, 4, -3, -4, -7, 5, 7, 0, 0], [-10, 4, 9, 4, 7, -5, 6, 7, 11, -11], [-17, -10, -8, -9, 10, -7, -7, 13, 5, -6], [-2, 5, -10, -3, 7, -12, -8, 6, 2, 2], [-4, -3, -3, 0, 17, -1, 2, 7, 1, -2], [-12, 1, 12, -1, 4, 19, 2, -1, 0, 0], [-5, -9, -1, 4, -12, -10, 0, 3, -2, -3], [1, 1, -9, -4, -8, -19, -8, 7, -2, -6], [32, -8, 4, 8, -21, 10, 0, -12, -12, 6], [-16, 15, 1, 9, 12, -4, -6, -4, -1, 1], [5, 1, -2, 5, 6, 8, 6, -1, -1, 5], [-6, 0, 8, 13, 5, -5, 3, -6, 1, -2], [16, 1, 0, 0, 2, 10, -4, -4, -3, -4], [1, 5, 5, -7, 9, 1, -2, -8, 2, -1], [17, 6, -13, -2, 2, -2, -11, 3, -2, 9], [-9, -8, 5, -5, 10, 4, 7, 14, 3, -3], [4, -6, -1, -5, 6, -6, 1, 4, 0, -5], [-5, 1, -1, 7, -4, -1, 2, 0, -8, -3], [-3, -10, 2, -2, 10, -7, 4, 8, 3, -9], [10, 5, 7, 5, 4, 2, 8, -12, 1, 3], [-4, 6, -3, 10, -8, 2, -6, -3, 4, 5], [-27, 8, 9, 6, 14, 13, 5, -2, 11, -2], [16, -2, 3, 4, -7, 13, -6, -9, -15, 3], [10, -3, 5, -4, -12, 1, 2, 0, -6, -8], [-4, -1, 0, 2, 4, -4, 13, -4, 1, -2], [11, 7, -3, -4, -5, -5, -9, -6, -9, -1], [-10, -7, 5, 9, -4, -3, 9, -3, -3, 0], [-4, 9, -1, -1, 2, 4, 9, 1, 2, -2], [2, -4, -4, -3, 6, 13, -9, 7, -7, 8], [-7, 1, -18, -6, 2, -5, -3, 5, -7, 2], [-21, 8, -2, -1, -5, -15, -7, 2, -1, -3], [-1, -19, -8, 6, -14, -12, -13, -3, -18, 4], [20, -17, 9, -1, 8, 1, 12, 5, 6, -2], [9, 10, -5, 1, 0, 13, -7, -5, 0, 3], [11, 8, 1, 9, 0, 4, -3, -7, 2, 8], [3, -1, -12, 0, 4, -11, 2, 6, -3, -2], [29, -10, -1, -3, 0, 2, -4, 3, -4, 7], [-23, 2, -2, 4, 5, -5, -10, -10, -2, 3], [24, -17, -6, 4, -10, 6, -6, -2, -13, 7], [-5, 4, 4, -7, 8, 3, -9, -1, 3, -2], [-14, -1, -3, -6, 0, 12, 14, 11, 5, -5], [11, 3, 5, 6, -24, -1, -2, -13, -13, 0], [12, -5, 1, 5, -6, -2, -3, 0, -5, 11], [-8, -13, 7, 1, -8, 4, 9, -5, 7, -4], [1, -7, -12, -7, -4, 6, 10, 7, -3, -1], [-7, -8, 4, 0, 13, 9, 7, 1, 2, 5], [10, 2, -11, 3, 9, -9, -12, -7, -9, 10], [5, 1, -12, -12, 1, 2, -2, 14, 4, 2], [-12, -6, -5, -4, 7, 1, 4, 13, 16, -5], [-8, -9, 11, -4, 4, 4, 9, 2, 1, -2], [22, 5, -2, 3, 9, 16, -5, 1, 1, -2], [27, 2, -4, 9, 3, 1, -3, -10, -16, 11], [0, 6, -6, 6, 2, -1, 2, -10, -2, 4], [18, -16, -6, 7, -10, 2, -8, -10, -13, 12], [-13, 15, 11, 5, -2, -3, 7, -3, 7, -6], [16, -2, 10, -1, 8, 10, 14, 4, 9, -6], [-12, 8, -17, -10, 15, 5, -6, 11, 10, 5], [12, 2, 4, 4, 12, -8, -6, -4, -4, 0], [-9, 4, 6, 4, -12, 5, 3, -6, 2, -2], [-2, 6, -6, -4, 2, 16, 8, 2, 4, -2], [-4, 4, 3, 5, 7, -14, 12, 7, -1, -8], [1, 3, 0, -4, 1, -10, 1, 16, 5, -3], [1, -6, 2, 5, -7, 0, -4, -9, 1, -2], [3, -2, 1, -4, 10, -5, 5, 4, 7, -3], [-2, 11, -11, -12, 4, -9, 2, 13, 3, -4], [21, 12, -2, -5, -7, 6, -8, -6, -2, -6], [-16, -4, -4, -6, 3, -4, 4, 12, 6, -2], [-7, 13, 4, -10, 12, 10, 7, 8, 11, 4], [10, -18, -15, -1, -4, -7, -4, 5, -11, 8], [19, 8, 1, 1, 8, -3, 4, -12, -1, 2], [-15, 5, -5, -6, -9, -19, -2, 4, -5, 0], [30, 9, 6, -1, 7, 11, 3, -4, 3, 0], [11, -3, 8, 4, -1, 7, -13, -5, 5, 11], [-1, -17, 3, 5, -16, -5, 13, 1, -1, -4], [-21, -9, -7, 7, -3, -2, 3, 3, -3, 6], [28, -2, -3, 2, 18, 10, 9, 8, 14, -3], [-17, -7, -12, -13, 2, 1, 4, 18, 14, -2], [-11, 15, 3, 4, 7, 13, -4, -8, 1, -5], [-14, -17, 3, -1, -6, 7, 7, -3, -1, -4], [14, -8, -1, -6, 3, 7, 0, 1, -3, -1], [1, 10, 3, 2, -7, -9, -1, 2, 6, 0], [-7, -4, -6, 1, -8, -14, 11, 4, -11, -10], [19, 18, 12, 0, -2, 8, -1, -15, 1, 2], [7, 0, -8, -1, -2, -10, 1, -8, -3, 2], [-22, 0, 5, -10, 0, 0, 2, -5, 10, -4], [19, 10, 3, 3, -2, 7, -3, -9, -10, 8], [-7, 6, 0, 1, 5, 5, 6, -2, 10, 2], [11, 2, -3, -2, 11, 7, 10, 12, 10, 2], [27, -4, -4, -3, 7, 4, -8, 5, 9, 8], [2, 16, -2, -4, -2, -8, 0, 8, 8, 2], [14, 11, -3, 0, -12, 15, -8, -8, 4, 7], [-9, 14, 0, -7, 2, 6, -7, -7, 2, 7], [-9, -1, 10, 1, -10, -1, 9, -11, -1, -6], [1, -12, 2, -9, -1, 8, 3, 8, 2, -9], [18, -1, 5, -1, -14, 13, -1, -6, 2, 3], [-29, -7, 3, 4, 0, -11, 6, 9, 9, 0], [-9, -5, 2, 0, 6, -10, -1, 5, 10, -4], [27, 2, 0, 8, -10, -4, -7, -5, 2, 9], [-17, -6, -3, 8, -27, -7, -4, -6, -8, 8], [-7, -21, -3, -5, 12, 2, 4, 13, 3, 1], [10, 10, 12, 8, -12, 10, -4, -20, -14, 12], [-28, 15, -9, -7, 24, -5, 1, 12, 2, -1], [24, -10, -1, 6, -12, -10, 1, -6, -4, 1], [-2, 6, -3, 2, -1, -6, -1, 3, -8, -3], [12, -11, 3, -5, 18, 2, 3, 7, 16, 4], [-19, -3, -1, -6, 5, 10, 7, 9, 9, -6], [-16, -14, 4, 2, 4, -7, 0, 4, 6, 0], [-21, -11, -16, 4, 6, -12, 5, 2, -13, 1], [-10, -12, -9, -6, 7, -7, 11, 10, 2, -1], [14, 21, -3, 8, -9, 4, -10, -14, -1, 9], [-3, -15, 1, -8, 9, 5, 8, 20, 11, -1], [-2, -5, 9, -5, 8, 4, 7, 8, 4, 6], [-21, 9, 13, -2, -8, -6, -6, -1, -2, 2], [14, 5, 1, -5, 20, 6, -9, -3, -4, 2], [-15, -4, -6, 2, -2, -15, 3, 2, -6, -5], [38, 14, -14, 9, -4, -3, 0, -5, -7, 3], [2, -3, -5, 6, 2, -12, 1, 2, 5, -8], [21, 12, 1, -12, -5, 3, -13, 0, 2, 4], [5, 12, -11, 1, 15, 0, -12, 5, 2, 5], [-6, -9, 3, -2, -5, 2, 10, 2, 5, -5], [2, -15, -4, 17, -20, -14, -8, -8, -10, 4], [-22, 0, 13, -10, 8, 2, 13, 13, 8, -11], [10, -2, -9, -4, 5, -14, 0, -1, -4, -15], [38, -4, -7, -2, 1, -7, 0, 7, 1, -3], [-17, 3, 2, 1, 12, -1, -3, -3, 7, 10], [-15, 14, -5, -12, 7, 5, 0, 4, 6, -4], [5, 4, 6, -1, -12, 7, -13, -8, -7, 4], [2, -5, 0, 11, -8, -7, -7, -12, -10, 9], [-20, -2, 6, 9, -20, 16, -11, -21, -9, 3], [15, -4, 0, 0, 17, 4, 4, -9, -5, 0], [-10, 4, -1, 4, 13, -3, 13, 12, 8, -5], [14, -7, 2, 7, -8, 15, -5, -7, -9, 6], [16, -1, 1, -10, -4, 8, 10, 3, 3, -9], [-3, -10, -1, 1, -18, -14, -2, -5, -10, -1], [25, 7, 6, -16, 9, 19, -1, 4, 9, -6], [-15, 18, 4, -5, 20, 11, 7, 1, 12, -2], [-26, -6, 8, 6, 5, -4, 3, -3, 5, -2], [14, -12, 13, 9, -1, 6, 14, -11, 5, 0], [-5, -5, -8, -6, 16, 0, 9, -4, -1, -10], [-8, 18, -4, -11, 16, -1, 15, 13, 3, -8], [-20, 8, -5, -12, 12, -10, 2, 16, 8, -6], [-7, -1, -1, 5, 3, 20, 5, -8, 2, 3], [20, 1, 1, -1, 21, -4, 19, 11, 11, -12], [9, 4, -1, 15, -7, 12, -1, -10, -7, 7], [-20, -3, -3, 8, 1, -6, 0, -9, 1, 6], [-11, -1, 5, 10, -8, -10, 9, -2, -8, -3], [-11, 6, -5, 4, -3, 11, -17, -8, -9, 9], [14, 0, -12, 5, -18, 13, -9, -13, -17, 14], [24, 4, -10, -8, -8, 0, -8, 0, 2, 4], [11, -6, 0, 1, -21, 16, -22, -15, 6, 12], [0, -10, -2, 16, -19, 5, -6, -21, -24, -1], [-1, 25, 0, 2, 9, 16, -23, -10, -3, 9], [-9, -6, 0, 7, -3, -14, -1, 6, 1, 3], [1, -16, -2, 1, -9, -4, 3, 4, -2, -3], [19, 2, -10, -3, -15, 4, 0, 1, -9, 0], [-32, 10, 6, -14, 9, 3, 5, 3, 16, -10], [10, 0, -2, 5, 5, -25, 5, 9, 5, -5], [-33, 7, 3, 1, 10, -7, 1, -3, 9, 2], [-6, 7, -5, 4, -20, -6, -7, -3, -7, 8], [-23, 13, -1, -4, 1, -9, -9, 10, -2, -2], [4, -25, 2, -12, -5, -7, 16, 7, -2, -4], [-16, -2, -6, -10, 10, -3, 6, 5, 10, -4], [4, -8, -4, 1, -16, -16, 11, 11, -5, -3], [-21, 7, 1, -7, 9, -13, 5, 7, 2, -2], [-2, -4, 1, -15, 18, 3, 9, 12, 9, -4], [-1, -6, -11, -4, 1, -10, 1, 3, -11, 1], [30, 1, 2, -13, 6, 18, -2, 6, 2, -2], [-1, 19, 1, 0, 15, 4, -7, -5, 0, 11], [11, -13, -8, 2, -14, -17, -2, 0, -7, -6], [19, 13, -7, 2, -1, 0, 10, 2, -2, 1], [14, -18, 11, 6, -16, -14, 11, -4, -10, 1], [-35, -20, 11, 8, -12, -16, 10, -3, -1, -13], [14, -14, 10, 0, 4, 2, 4, 6, 2, -4], [9, 2, -2, 11, -3, -4, 11, -7, -6, 3], [-29, 13, -11, -7, 2, -4, -18, 7, 1, 3], [4, 4, -3, 9, 3, -3, -16, -14, -14, 6], [-5, -10, -4, 12, -20, -12, 5, -9, -18, 1], [-18, -10, 3, 2, -11, 20, -4, -6, 4, 9], [9, -11, 11, 15, 2, -3, -1, -9, -3, 6], [-14, 8, -5, 5, 7, -11, 2, 4, 6, 2], [19, -7, 0, -10, 4, -5, 10, 5, 0, -5], [-13, 4, -5, 6, -10, 3, -19, -16, -1, 1], [-2, -3, 5, 17, 10, -1, 3, -7, 7, 2], [-7, 10, -8, -4, 1, -13, -6, 10, 3, 4], [-9, 6, 0, -11, -8, -3, 3, 15, 13, 0], [11, 16, 7, 4, 13, -5, 2, -13, -4, -2], [-24, 10, 7, -3, -14, 6, 5, 4, 5, -9], [-2, -4, -13, -8, 0, 0, -2, 9, -14, 7], [6, 3, -15, -5, -3, -25, 9, 6, -4, -1], [2, -5, -5, 6, -4, -2, 0, -3, -11, 1], [16, 10, 1, 8, -7, -11, -10, -11, -11, 12], [3, 6, -1, 0, 0, -7, -1, 2, -15, 1], [10, -16, 2, 0, 12, 3, 10, 5, 2, -6], [-24, -2, -5, 0, 0, -1, 10, 9, 9, -9], [8, 22, -8, 3, 1, -4, 9, -4, -3, -3], [-27, -2, 0, -4, -3, 7, 3, 5, -5, 1], [10, 6, 2, 4, 0, -8, 0, -1, -1, 14], [-11, -7, 3, -4, -20, -7, 4, 5, 7, -4], [30, -16, 13, -11, -2, 15, 20, 7, 9, -8], [-3, 15, 5, -2, -2, 6, -6, -5, 5, 5], [7, -9, 19, 5, 7, 3, 15, 7, 19, -1], [-15, 13, 6, -1, -1, -3, 0, 5, 2, -4], [6, -12, 11, -4, 16, 2, 5, 4, 5, -12], [-13, 5, 0, 7, -2, 7, -6, -10, 0, 6], [23, 1, 1, -7, 7, 13, 4, 18, 8, 1], [30, 8, -7, -11, -9, 9, -10, -14, -2, 12], [-12, -10, -7, 8, -15, 8, 2, -5, -7, 7], [-35, 15, 13, -9, 29, -4, 7, 10, 14, -7], [0, -9, 6, -4, -26, -1, -6, -1, 2, -3], [25, 9, 10, 11, 5, 8, -6, -17, -4, 8], [-15, 9, -9, -6, 17, -23, 6, 14, 4, -4], [27, 8, -17, 7, -2, 7, -3, -7, -8, 1], [3, -2, 2, -3, 15, 4, -5, -10, -3, 5], [-10, 10, -7, -10, 5, -23, -4, 23, 1, -3], [-30, -2, 0, -7, 16, 6, 3, 19, 7, -5], [-8, 9, 6, -12, 2, 1, 12, 15, 12, -2], [11, 20, 0, -2, -4, 9, -7, 0, 10, 0], [13, -7, 15, 12, -7, -7, -2, -8, -3, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4235_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4235_2_a_bl();".
// 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_4235_2_a_bl(:prec:=10)
    chi := MakeCharacter_4235_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_4235_2_a_bl();".
// 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_4235_2_a_bl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4235_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-11, -38, 38, 140, -96, -106, 63, 26, -14, -2, 1]>,<3,R![1, -14, -3, 120, 45, -166, 13, 50, -10, -4, 1]>,<13,R![14641, 31944, 2904, -26840, -5473, 7894, -206, -896, 246, -26, 1]>],Snew);
    return Vf;
end function;