
// Make newform 4030.2.a.l in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4030_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_4030_2_a_l();".
// 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_4030_2_a_l();".
// 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, -43, -32, 46, 54, -6, -14, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [7, 30, 2, -47, -7, 13, 1, -1], [-11, -31, -1, 47, 7, -13, -1, 1], [-118, -318, 50, 448, 41, -120, -7, 9], [176, 316, -155, -426, -7, 110, 4, -8], [-221, -501, 145, 666, 42, -175, -9, 13], [388, 933, -230, -1243, -86, 325, 17, -24]];
    Rf_basisdens := [1, 1, 1, 1, 5, 5, 5, 5];
    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_4030_a();"
function MakeCharacter_4030_a()
    N := 4030;
    order := 1;
    char_gens := [807, 1861, 2731];
    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_4030_a_Hecke(Kf)
    return MakeCharacter_4030_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, 0, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 1, -1, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, -1, -1], [2, -1, 0, 2, -1, 0, 1, -1], [-1, -1, 0, 0, -2, -1, -1, 1], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 2, -1, 1, 0, 1, -1], [2, -3, 0, -1, 1, -1, 0, 1], [0, -2, -3, -1, 1, 1, 0, -1], [2, -1, -3, -1, -2, 0, -1, 0], [2, -1, -1, -2, 0, 0, 1, 0], [-1, 0, -1, 2, -2, -1, -3, 0], [0, 2, 3, 0, -2, 0, -1, 0], [5, -1, 1, 2, 1, 1, 2, -1], [0, 2, 1, -3, 0, -2, 0, 1], [5, 2, 1, 1, 2, 0, -1, 2], [-2, 2, 0, 0, -2, -1, -2, -1], [2, 1, 1, 1, -2, 2, 1, 0], [3, -1, 1, -1, 3, 0, 0, 1], [3, 3, 0, 0, -2, -2, 2, 2], [0, -3, -3, -2, -4, -1, 0, -1], [-2, 1, -1, -2, 0, -2, -4, 2], [2, -4, 0, 1, -2, -3, -2, 1], [1, -2, 0, 1, 3, -3, 0, 0], [6, -6, -2, 0, 0, -3, 2, -1], [0, 2, -1, 1, -2, 1, 2, -2], [2, 2, 2, 0, 0, 0, 3, 2], [-2, 1, 0, -3, 1, 2, 1, 2], [-4, 2, 2, -1, 3, 4, -1, -1], [5, 0, -1, -2, 3, -2, 2, 0], [-1, -5, -1, -1, 1, -1, 0, 0], [0, 1, 1, -5, 2, -1, 0, -1], [-5, -3, 0, 5, -2, 0, -2, 0], [5, 6, 4, 4, 1, 0, -4, 0], [3, 3, 2, 6, -1, 1, -2, 1], [2, 2, 0, 5, -1, -1, -3, -1], [0, -4, 1, -3, 1, -1, 7, 1], [0, -1, 0, 6, -5, 4, 1, -2], [-1, 2, 1, 2, 0, 2, 4, 2], [0, 4, -2, 5, -2, 1, -2, 1], [-4, 4, 3, 1, 4, 0, 0, 3], [-8, -3, -3, -2, -4, 2, 2, -4], [-4, 3, 0, 1, -5, 4, -3, -2], [6, 7, -2, 1, -2, 1, -2, 2], [5, 1, 2, 5, -3, 2, -5, -1], [10, 1, 2, 1, 4, 0, 7, -1], [10, 5, 1, -3, 1, 1, 3, -2], [-2, 2, -4, -1, 4, 0, 0, -1], [-6, -1, 1, 3, -1, -2, 2, 1], [7, -6, -2, -4, 2, -5, 1, 0], [1, -5, 4, -4, 2, -1, 3, 1], [1, -2, -5, -4, 0, 2, 1, -3], [3, -5, -7, 1, 0, 0, 5, -1], [0, 3, -2, 1, -2, -3, 1, 0], [8, 5, 6, -3, 3, -2, 0, 0], [-1, 2, -1, -4, 8, 1, 4, -1], [9, 2, 4, -2, 6, 3, -1, 0], [10, 1, -3, -2, 1, -2, -3, -1], [-4, 5, 2, 2, 0, 3, 6, -5], [6, 3, -2, 5, -1, 5, 6, -3], [8, 0, -3, -3, 1, 2, 6, 2], [-4, 1, 3, 4, 3, 2, 2, -1], [4, 3, 1, 7, -5, 4, 3, -2], [6, -5, -2, -1, -2, 0, 3, -3], [0, -8, -2, 7, -6, 3, -2, -1], [-2, -7, -5, -6, 2, -3, 4, -3], [12, -10, -3, 3, -6, 2, 1, -5], [3, 4, -1, -2, 0, -3, -5, 5], [-2, 3, -1, -5, 0, -4, 2, 1], [-9, 7, 4, -4, 3, -2, 0, 4], [-2, 0, -1, -1, -7, -1, -12, -1], [1, 3, 2, -5, 0, -2, -1, 2], [7, -7, -6, 3, -8, -6, -2, 2], [4, 2, 5, 0, 8, -2, 4, 3], [11, -6, -2, -2, -1, -7, 1, 1], [-3, 8, 3, -3, 7, 2, 2, 1], [-7, 8, 6, -3, 0, 3, 3, 2], [-10, 8, 3, -1, 7, 5, 2, 3], [-2, -4, -4, -7, 7, -2, 3, 0], [-2, -1, -4, -6, 6, 0, -2, 2], [5, -3, -4, 7, -3, 4, 2, -5], [9, 2, 0, 3, 5, 4, -2, -4], [9, -2, 2, 6, -2, 8, 7, -5], [5, 3, -7, -6, 6, 3, 4, 0], [-11, -3, -4, -3, -9, -8, -3, 1], [14, 7, 4, 0, 6, 3, 1, -5], [13, 4, 5, 1, -4, 1, 3, 1], [9, 5, 7, 2, 2, -4, -1, -1], [13, 1, 4, 8, 6, 7, 3, -2], [0, -7, -2, -3, -2, 1, 8, 0], [1, -5, -8, 1, 1, -2, -3, -1], [-9, -6, -5, 5, -5, 1, 0, 0], [5, 3, -2, 2, -2, -1, 3, 5], [-1, -1, 0, 7, 1, 1, 3, 2], [10, 1, 2, 8, 2, 1, -1, 3], [9, -7, -4, 4, -2, -3, -5, 0], [11, 7, 5, -3, 1, -10, -4, 7], [0, 2, 1, -1, 1, 4, 8, -2], [7, 1, -2, 5, 2, 0, -4, -2], [-16, 8, 4, 2, 2, 5, 0, 3], [9, 5, 3, -3, 10, 0, 2, 2], [-7, -5, -4, -2, -6, 0, -1, 5], [-1, 4, 3, 0, -3, -1, -9, 3], [-12, -4, -9, -10, 4, -1, 3, -3], [-10, -7, 0, 1, -9, 0, 2, -2], [1, -1, 4, 1, -1, 4, -3, 1], [-5, -5, 6, -2, 0, -5, 5, 6], [9, -8, -5, -4, 3, -2, -2, -6], [5, -7, -4, 1, 1, 7, 6, -4], [0, 4, 12, 1, -3, -2, 0, 5], [-2, 0, 2, 6, -3, -1, -4, 4], [5, 3, 6, 10, 1, 5, -4, 6], [10, 7, 5, 2, 10, 5, -5, -1], [4, -3, -4, 3, -11, 0, -2, 0], [-3, 4, 8, 6, 4, 1, -5, 2], [-10, 8, 5, 0, 0, 1, 4, -5], [-8, -7, -1, 3, 2, -4, -2, 5], [10, -4, 0, 6, 3, -1, 0, 2], [-1, -6, 0, 0, -4, 3, -5, -8], [-8, -11, -6, -8, 6, -8, 3, -2], [-1, -6, -11, -3, -4, -3, -2, -1], [-13, 1, -3, -2, -1, -5, 0, 1], [-9, 5, -3, 0, -2, 0, 0, 5], [9, -11, -4, -5, -5, -3, -3, 2], [-2, 3, 5, 2, 9, 5, 3, -4], [8, 6, -4, 4, -7, 4, 2, 1], [-5, -1, -2, 13, -8, 6, 2, -2], [-18, -5, 1, -3, -5, -4, 3, 7], [-10, -2, -1, -5, 5, -2, 5, -6], [10, -1, -2, -6, -2, -9, 1, 3], [-3, 2, -1, -9, 7, -6, 1, 9], [0, -10, -8, 3, -12, -8, -7, 2], [-1, -4, 1, -10, 5, -9, -5, 4], [-5, 12, 7, -3, 6, 10, 4, -2], [0, 2, 4, 4, 8, 9, 5, 1], [-12, -8, -9, 7, -9, 5, 7, -3], [-14, -3, 6, 9, -5, 2, 0, 0], [-17, -2, 6, -5, 1, -3, 0, 0], [11, -6, 2, -1, 6, -3, 4, 5], [-18, -2, 0, -3, -5, 1, 1, 10], [10, -11, -9, 2, -6, -5, -1, 1], [-9, 9, -3, -4, -8, 1, -5, -2], [-5, 16, 4, 4, 1, 8, 4, -2], [0, 11, 0, 2, -6, -6, 4, 6], [2, -4, -4, -8, 7, -8, 3, -1], [-12, 3, -2, -5, 0, -1, 6, -6], [-19, -2, -1, -15, 2, -2, 5, -2], [-16, 3, -4, 3, -9, 0, -8, 2], [10, 1, 0, 7, 7, 2, -2, 4], [-16, -6, -4, 7, -5, -5, -13, -1], [-16, 7, -3, 0, 5, -4, 4, 1], [-5, 7, 2, -2, 4, 4, 1, 7], [-4, 7, 10, 10, 1, 2, -6, 0], [-2, -9, -1, 2, 2, 0, 5, -6], [7, 1, -7, -2, 3, -1, 6, -1], [-10, 2, -11, 1, -11, -5, -8, -1], [4, -11, -2, 6, -2, 5, 2, -4], [-9, 2, -3, 1, -7, -4, 2, -7], [4, 2, -4, 2, 2, -1, 5, 1], [-2, -9, -14, 0, 0, -1, 12, -3], [-8, 1, 0, -5, 1, -5, 0, 5], [-15, -5, 5, 0, 0, -3, -4, -4], [10, 2, 2, 3, -5, 4, 11, -1], [4, -7, 0, -6, 1, -1, -5, -3], [-6, 7, 6, -3, -2, 3, 8, 2], [5, -5, -8, 1, -6, -2, 7, -2], [-3, 7, -3, 0, 9, 6, 2, 0], [9, -10, -7, 8, -1, -2, -7, -2], [17, 0, 3, 4, 3, 7, -4, -8], [-10, 7, -1, 1, -1, 1, 9, -6], [9, 4, 8, 6, 8, 6, 3, 3], [-17, -8, -5, 3, -12, -3, 2, 5], [18, 1, -8, -2, -4, -2, -2, 5], [-20, -2, -2, -8, 5, 5, -6, 0], [-3, -14, 2, 9, 3, 7, 3, -6], [15, 10, 1, -2, -8, 5, 9, -1], [-26, 0, 2, -16, 5, -3, 5, 2], [-21, -8, -9, 2, -1, 2, -8, 4], [4, 15, 5, -4, 3, 6, 0, -1], [1, 0, -7, -12, 0, 4, 3, 2], [5, -6, -4, -7, -5, -7, -4, -10], [5, 5, -1, -5, 15, -4, 9, -4], [8, 6, 7, 5, 1, 1, 5, 3], [-1, -10, -1, 6, -1, 4, 15, -7], [12, 9, 2, 1, -7, 12, 3, -6], [7, 9, 4, -8, 2, -9, 3, 3], [0, -16, 1, -9, 0, 6, 0, -3], [3, 4, -4, -1, 15, -1, 3, 5], [-6, 0, -4, -12, 9, -2, 12, -3], [10, 2, 8, 7, -1, 0, -3, 3], [-7, -13, -5, 1, 6, -9, -9, 7], [-8, -14, -8, 0, -1, 0, 2, -1], [-17, 0, 3, 11, 6, 12, 1, 0], [6, 4, 8, -1, 14, -5, -4, 6], [9, 3, 3, -4, -7, -5, -8, -3], [12, 11, 6, 4, 6, -2, -12, -3], [1, 7, -6, -5, 9, 1, -3, 0], [-9, 1, 1, -4, 14, -4, -3, 3], [-13, -2, 3, -2, 3, 1, 7, 7], [-5, -1, 0, 6, 9, 0, 13, 4], [2, 4, -6, 4, 4, 5, 12, -5], [-4, -7, 4, -11, 5, -3, -4, 5], [-13, -2, 7, 1, -7, 1, 8, 8], [4, -10, -1, 5, 5, 7, 1, -7], [-11, -1, 0, -9, -1, -1, 2, 8], [-13, 3, -9, -1, 8, 4, -8, -3], [3, -6, -2, 9, 3, 0, -11, 8], [1, 14, 6, 0, 0, 8, -3, 1], [0, -5, 8, -7, 9, -1, -4, -1], [-19, 14, -3, -6, 8, 3, 4, -2], [20, 6, 6, 8, 2, -4, -1, 4], [21, 7, -2, 1, 7, 4, 0, 1], [-11, -2, 2, 3, -12, -11, -1, 8], [-12, 9, 4, -3, 8, 1, 12, 2], [22, -5, 5, -6, 8, 4, 6, 6], [1, -11, -12, 6, -2, 9, 4, -4], [-4, 1, 10, 7, -13, -2, -17, 0], [8, 1, -4, -5, -9, -12, -10, 2], [4, -6, 6, 11, 3, 8, -2, 5], [-1, -10, 3, 2, -13, -9, -1, 4], [-11, -4, -4, -2, 2, 1, -10, -6], [-7, -5, 5, -8, 9, 5, 3, -5], [-6, 6, -9, -7, 13, -1, 10, -1], [-15, -7, -2, -10, -2, -8, 1, 11], [5, -1, 9, -11, 11, -4, -4, 4], [-19, 13, 7, 6, 2, 4, -8, 3], [16, 12, 5, -3, 10, -7, -1, 0], [-6, 2, 4, -2, -5, 0, -7, 5], [15, -4, -7, -5, -1, 3, 10, 2], [6, -14, -2, -4, -8, -1, 3, -7], [1, -8, -8, -12, 9, -3, 9, 5], [-18, -6, -8, -10, -1, -6, -6, -1], [-4, 17, -3, 8, -2, -6, -5, 2], [1, 7, -6, -3, 3, -3, -4, -2], [-5, 0, 11, -7, -3, -9, -11, 2], [-10, -5, 7, -14, 8, 3, -3, -3], [-19, 10, 8, 7, -9, 0, 3, 3], [-6, 3, 3, 1, 4, -14, -8, 4], [-26, 9, -4, 5, 1, 10, -5, 0], [6, -10, 5, 1, -2, -4, -1, 1], [-11, 3, 4, 8, -10, 6, -2, 0], [-21, 8, -4, -6, -2, 7, -2, -2], [-2, 13, 3, -10, 10, -2, -2, 2], [3, 0, 2, -6, 10, -4, -11, 3], [11, 15, 4, -1, 4, 14, 11, -2], [1, -12, 0, -8, 5, -5, 7, -1], [23, 1, 2, -1, -5, 0, 16, -5], [26, 16, 3, -1, 15, -1, 7, -9], [10, -7, -6, 6, 8, 0, 1, -2], [3, 0, -4, 2, -19, -8, -16, 6], [15, -3, 5, -7, 2, 3, 6, 2], [-5, 2, 2, 20, -12, 8, -11, -7], [-16, -16, -1, -8, -3, -8, -1, 11], [-3, 2, 0, -5, -2, 3, 2, -9], [14, 18, 7, 8, -10, 4, -11, 4], [24, -13, -8, -1, 0, -7, 5, 2], [11, -4, -3, 9, -4, -2, -14, 8], [12, 3, -8, -9, 3, -4, -15, -2], [-14, -16, -4, 3, -4, -2, -5, 9], [1, -11, -3, 12, -7, -11, -9, -3], [20, 13, 3, -6, 3, 5, 2, -6], [-22, 2, -4, -1, -17, 1, -11, -3], [4, 3, 7, -4, 3, 0, -10, 7], [-14, 3, -8, 3, -4, -1, -5, 10], [17, -27, -7, -7, 0, -12, 1, 7], [-4, 2, 1, 1, -2, 6, 3, -5], [9, 0, -3, 6, -5, 3, 1, -8], [4, -8, 4, -4, -2, -10, 2, 6], [12, 15, 0, -3, 4, 2, 10, -9], [28, -2, 4, -2, 12, 0, 7, 8], [3, 2, 2, 1, -11, -9, -7, -3], [3, -4, 0, -6, 2, 9, 15, -2], [14, -7, -3, -8, 1, -11, -12, 12], [18, 9, -1, -9, 6, -3, -1, -5], [8, 16, 5, 3, 9, 0, -3, 8], [5, 3, 7, 12, -14, -1, -8, 6], [-7, -6, 0, 14, -9, 6, 10, -6], [-26, -5, 5, 0, -19, -7, -12, -2], [-25, -7, -10, 0, 0, -2, 0, 12], [12, -7, -4, 9, -2, 4, 12, -11], [2, 3, 7, 6, -10, -11, -14, 1], [3, -10, 0, 6, -7, -6, 3, 2], [5, 6, 6, -2, 8, -4, -3, 11], [-12, 11, 4, -7, -9, 6, 6, -10], [4, 8, 4, 4, -9, -5, -19, 6], [20, 0, 10, -2, -3, 7, 9, -2], [4, 23, 19, 1, 3, 1, -21, 7], [0, 7, 13, 6, 16, 1, 6, 1], [9, 1, -2, 11, -13, 6, -5, -3], [11, 9, 5, -10, 26, 6, 4, -5], [-2, -11, -3, -1, 12, -1, 4, -1], [-19, 20, 8, 6, 5, 18, 2, -2], [-9, -15, -6, -2, 6, 7, 1, -6], [4, -24, -12, 1, 1, -4, 9, 1], [9, -6, -5, -2, -15, -13, 3, 3], [9, 20, -1, 5, 13, 7, -4, 4], [-11, -5, 2, -4, 4, -7, 8, 10], [-2, -9, 2, -7, -7, -5, 11, -5], [-5, -5, 1, -19, 0, -9, 11, 3], [-6, -16, -11, 4, -3, 3, 10, -11], [27, -8, 4, 2, -2, -2, -1, 9], [-24, 3, -3, 6, -8, 5, -14, -9], [-3, 12, -1, 11, -4, 7, -18, -5], [9, 6, 9, -1, 11, -3, -1, -6], [3, -4, 2, -15, 7, 2, -12, -4], [21, -3, -5, -7, 9, -3, 16, 0], [10, -5, -4, 9, -3, 8, -9, -6], [-6, -13, -4, 6, 4, 8, -4, -5], [-1, 16, 2, -6, 9, -5, -3, 1], [13, 6, 2, 0, -13, 6, 13, -2], [-1, -3, -3, -2, 11, -6, -15, -2], [-18, -4, -10, 12, -6, -1, -16, -1], [-10, 19, 3, 8, -6, 1, 9, 5], [7, 16, -5, 1, 8, 6, -4, 0], [22, 20, 1, 5, 18, 5, -10, 2], [13, -3, 4, 4, -2, -1, -12, 0], [10, -18, 9, 5, 0, -1, 9, -6], [4, 5, -7, 4, -2, -6, -19, -4], [5, -5, 7, 12, -2, 4, 8, -9], [45, 3, 4, 8, 4, 4, 7, -4], [18, 6, -8, -6, -6, -3, 2, 11], [11, -3, 0, 10, 11, 7, 4, -4], [20, 8, -3, -5, 13, -1, 12, -1], [-5, -11, -14, -4, 6, 2, 13, -2], [-16, -2, 9, -1, 5, 6, -13, 2], [0, 13, -7, -7, -1, 1, 4, -12], [-9, 16, 8, 11, 0, 17, -7, -5], [7, 3, 7, 17, -5, 4, 3, -2], [3, -3, -5, -11, -8, 10, 4, 0], [-13, -8, 2, 16, 4, 15, -2, 0], [-8, 9, -8, -5, -7, -6, -8, 8], [-1, -6, 21, 1, 5, -6, 4, 11], [27, 17, 8, -12, 5, 4, 3, -2], [-6, -24, -2, -10, 9, -12, -2, 1], [13, -20, -8, -5, -14, -12, -6, 6], [-8, 15, -4, -5, 3, 6, 3, -6], [9, 3, -9, -6, 2, 1, 0, -10], [-3, 3, -16, -1, -11, -7, -15, 2], [-20, 12, 18, -4, 5, 6, 14, 11], [-10, 19, 25, 0, 18, 3, -10, 11], [10, -18, 0, 10, 7, -9, 2, 8], [14, -16, 7, 9, -17, 1, -4, 3], [2, -17, -14, 6, 8, 11, 5, -4], [-3, 0, -8, 9, -6, -6, 2, 4], [7, 1, 13, 1, -3, 8, -2, 3], [-16, -11, -21, 2, -22, -8, -5, 4], [28, 6, 12, 6, 7, 7, -1, 0], [-10, 15, 6, 2, 6, 5, -13, 2], [-6, 5, -2, -8, 9, -2, 8, 0], [28, -4, 1, -9, -3, -8, 9, -6], [-12, -4, 0, -7, -2, -1, 13, -10], [-6, -23, -15, -9, -10, -9, 1, 12], [-10, 11, 0, -3, -6, -1, -11, 8], [-3, -2, 15, 6, 18, 8, 7, -1], [7, 8, -7, -14, 22, 1, 6, 4], [-16, 23, 20, -1, -5, -5, 1, 11], [11, -5, 3, 4, 2, -9, 4, 8], [7, -12, -14, 7, -4, 4, -4, -7], [-13, 15, 2, -2, -5, 4, 10, -8], [-14, 17, 5, -1, 0, -6, -4, -5], [4, -5, 1, 5, -10, 0, -1, 2], [-10, -8, 6, 22, -9, 14, 1, 1], [-7, -22, -4, -4, 6, -13, -2, 0], [27, 19, -1, 5, 2, 2, -6, -3], [1, 2, 12, 14, -3, -5, -9, 7], [13, 9, -2, -1, 1, -5, 5, 4], [37, -9, -5, 9, 9, -3, 3, -4], [-1, 8, 1, 9, 10, 6, -21, -4], [-12, 4, -5, -8, 17, 4, 0, -5], [-1, -3, 6, 15, -1, 9, 2, -4], [1, 6, 4, 8, 7, -4, -7, 4], [14, 3, -4, 0, 9, -3, -9, -2], [20, 10, -12, -12, 6, 3, -8, -7], [22, 5, -8, -12, -3, -3, -7, 6], [-42, -6, -20, 0, -6, 0, -13, 0], [14, 0, 13, 13, 20, 8, 6, -1], [-3, 1, 8, -3, -10, 0, 14, 2], [22, -10, 0, 14, -21, 9, -7, -2], [-3, 4, 9, -15, 9, 0, -11, -1], [-17, 3, 10, 5, 8, 10, -22, 0], [26, 0, 6, 8, 5, 15, 9, -12], [2, -16, -10, 2, -4, -7, -5, 5], [10, 14, 15, -9, 14, 7, 13, 2], [6, 4, -12, -5, 19, 4, -9, -6], [-13, -13, -1, 7, -18, -7, -20, 2], [10, 13, -2, -1, -1, -10, 3, -4], [-17, -12, -25, -2, -16, -11, 1, 0], [40, 11, 12, 0, 7, 0, -15, 9], [5, -1, 0, 4, 12, -10, -16, 2], [15, 8, 10, -4, 5, 5, 14, -11], [25, 6, 2, -8, -1, -4, 17, 0], [5, 3, 0, 0, -14, -17, -21, 4], [-9, -5, -18, -4, 2, 6, 1, 5], [-25, -15, -9, -5, 4, -13, 13, 9], [-13, 15, -4, -13, 7, -4, 1, 3], [27, 16, 8, 2, 15, -4, -13, 14], [-21, 6, 0, 0, 14, 8, 13, -5], [14, 10, -6, -13, 11, -1, -2, -3], [17, -1, -10, 2, -3, 14, 10, -11], [-26, -17, 2, -5, 1, -9, -5, 3], [-5, 5, -6, 0, 15, 0, 7, 9], [-20, -4, -15, 1, -9, -8, -4, -6], [31, -6, 0, 1, -3, -10, 8, -4], [4, -1, -7, -8, -8, -12, 7, 14], [-15, -2, -11, -1, -3, 7, 2, 2], [35, -17, -9, 5, 3, 7, -1, -9], [-23, -4, -10, -7, -16, -7, -2, 2], [0, 8, -8, -14, 10, 1, 12, 1], [-20, -16, -4, 3, -21, -7, -11, 5], [17, 18, 22, -4, 11, 6, -1, 6], [-10, 20, 16, -6, 28, 9, 8, -5], [-38, 29, 13, 5, -5, 6, -2, 0], [-9, -2, 9, -11, -6, -6, 0, 4], [-5, -1, -6, 13, -5, -4, -23, 1], [-1, 17, -2, 19, -5, 5, -25, -6], [1, -6, 4, 9, -9, -4, 3, -8], [-3, -17, 0, 8, 0, 5, -10, -6], [-41, 3, 2, 15, -11, 16, 8, 7], [-14, -1, -9, 8, -19, -5, -1, 10], [2, 2, 6, 21, -8, 10, 10, -13], [9, -4, 5, 4, 8, 0, 9, -4], [-20, -2, -8, -6, -11, -1, 2, -10], [19, -2, -1, 6, -5, -5, -24, 4], [39, 1, 2, 13, 3, 0, 2, 1], [-11, 17, 13, 14, -1, 2, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4030_2_a_l();".
// 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_4030_2_a_l(:prec:=8)
    chi := MakeCharacter_4030_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_4030_2_a_l();".
// 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_4030_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 34, -69, -30, 68, 12, -16, -1, 1]>],Snew);
    return Vf;
end function;