
// Make newform 8030.2.a.ba in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8030_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_8030_2_a_ba();".
// 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_8030_2_a_ba();".
// 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 := [2, 0, -15, 9, 17, -11, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0], [-2, 8, 7, -10, 0, 1, 0], [-8, 0, 24, 5, -12, 0, 1], [-2, -6, 23, 22, -22, -1, 2], [8, 28, -42, -39, 34, 2, -3], [-5, -13, 25, 23, -22, -1, 2]];
    Rf_basisdens := [1, 1, 2, 2, 2, 2, 1];
    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_8030_a();"
function MakeCharacter_8030_a()
    N := 8030;
    order := 1;
    char_gens := [1607, 2191, 881];
    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_8030_a_Hecke(Kf)
    return MakeCharacter_8030_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, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, -1, -1], [-1, 0, 0, 0, 0, 0, 0], [-1, -1, 1, 0, 0, 0, -1], [-1, 0, 1, -1, -1, 0, 0], [0, 2, 0, 2, -1, 1, 1], [-1, 1, -1, 0, -1, 0, 0], [1, -1, 1, 1, 0, 0, 2], [0, -1, -1, -1, 3, 1, -1], [-4, -1, -1, -2, 0, 0, 1], [-1, -2, 1, -1, -1, 1, 0], [-3, -1, 1, -2, 0, 1, 0], [2, -3, 3, 2, 0, 0, 2], [-1, -1, 0, -4, 3, 1, -1], [4, 3, 2, 1, 1, -2, 0], [-4, 3, 0, 1, 0, 0, 2], [5, -4, 3, 0, 0, -1, 3], [1, 1, -3, 3, 1, 2, 1], [1, 0, 0, 0, 0, 0, 0], [3, 0, -2, -4, -2, 2, -1], [-5, -1, -2, -4, 3, 0, -3], [-4, 1, -2, -4, 1, 2, -3], [-4, 7, -2, 5, -1, 0, 2], [1, 6, 0, 4, 0, 1, -2], [3, 1, 7, 3, -2, -2, 3], [-6, 4, 0, 3, 0, -1, -1], [-8, 3, -1, 0, -1, 1, 0], [-1, -2, 0, 0, 5, -1, -1], [-7, 3, -1, 1, 3, 0, 0], [3, -2, 3, -1, 0, -4, -1], [-10, 3, -5, 0, 1, 1, -2], [1, -2, 2, 0, 1, -1, -1], [7, 4, -3, 1, -3, 1, -3], [-4, -1, 4, 4, -4, -1, 2], [-6, 1, -5, 2, 0, 5, -2], [-4, -2, 4, -2, 3, -2, 2], [1, -1, 2, 8, 0, 0, 3], [2, -5, 3, -1, -1, -4, -1], [-2, 6, -3, 3, -6, 3, 0], [-5, 3, -8, 3, 0, 3, -1], [-5, 5, -2, -3, -2, 3, -1], [-2, -1, 1, 5, -2, 0, 5], [-2, -6, 6, 2, 0, -4, 2], [4, -2, 2, -1, 5, -3, -1], [6, -1, -1, 1, -4, 4, -3], [3, -1, -1, -7, 2, 1, -2], [-3, 1, -2, -4, 1, 0, -5], [-7, 7, -2, 3, 0, 0, -4], [-4, 5, -2, 3, 4, -2, 2], [-3, -2, 0, 3, -4, 7, 2], [8, 0, 3, 6, 1, -6, -2], [4, -1, -1, -5, 6, 3, 1], [3, 7, -1, 0, -3, 1, -7], [-2, 1, 4, 1, -3, -1, 2], [-7, 0, 5, 2, -4, -3, -1], [1, -2, -2, -4, 3, 3, 1], [5, -3, -1, 2, 2, -4, 1], [2, 2, 0, -7, 3, -3, -2], [-5, -1, 3, 5, 1, -3, -1], [3, 2, 5, -6, -3, -3, 0], [1, 5, 1, 1, -1, -2, -4], [9, 2, 3, 6, -4, -4, 5], [4, 3, -1, -2, 5, -2, 3], [10, -1, 1, 2, -6, -4, 5], [0, 3, 1, 3, -4, 6, 6], [-5, -7, -2, -1, 1, -4, 0], [4, -4, 0, -1, 5, 1, -4], [1, -2, 1, 4, -2, 7, 3], [5, -9, 6, -3, 1, -2, 1], [-6, -2, -5, 2, 3, 4, 0], [-1, -6, -5, -6, 3, 4, -3], [3, -7, 4, 0, 3, -8, 1], [-5, -8, 2, -1, -4, 2, -5], [1, 8, 4, 10, -1, 1, 1], [6, 2, 6, -2, 1, 1, 4], [-7, 11, -7, 0, -5, 1, -1], [-9, -6, -5, -6, 0, 3, 8], [5, -6, 4, 0, 4, -2, -3], [7, -10, 3, 2, 0, -8, 0], [-1, -3, -3, -2, -1, -6, 2], [14, -3, 0, 1, -1, -5, -1], [-2, 2, 2, -3, 1, 3, -6], [1, 1, 2, 4, 1, -10, -1], [2, 3, -3, 6, 2, -1, 5], [12, -12, 5, -1, 6, -5, 2], [-18, 2, -1, 6, -2, 0, 4], [7, -4, -7, -4, 7, 9, 1], [-3, -3, -5, -6, -1, 11, 1], [-2, -12, -1, -4, -1, -6, 1], [8, 8, -3, 3, -5, 2, -6], [0, 6, -9, 7, -5, 11, -1], [-6, 1, -1, -9, 2, -7, -2], [-9, -11, 4, 0, 10, 0, 4], [6, 7, -2, -2, -9, -7, -1], [-4, 5, -1, -7, -4, -2, 2], [11, -16, 8, 1, 1, -1, 5], [-1, 4, -2, 11, 0, -2, 5], [-11, -14, -8, -9, 8, 2, -4], [-15, 0, -4, 4, -2, 8, 7], [-8, -7, -6, -1, 1, -1, 7], [-2, -3, -14, -5, 3, -2, -6], [-5, -6, -2, -7, -4, 6, -5], [4, -2, -2, 0, -3, 0, 2], [-1, 0, -6, -3, 1, -8, 0], [7, 1, 0, -3, 5, 3, 1], [4, 8, -1, 12, -3, 2, -2], [23, 2, -1, 7, 0, 5, -2], [-21, 2, -9, -9, 6, 1, -9], [-1, 0, 4, -3, -1, 6, 5], [13, 7, -4, 2, -3, 0, -1], [-5, -11, 6, -3, 9, -9, 0], [-8, -2, -5, -11, 4, 4, -1], [-2, -6, 6, -9, 3, 3, 1], [-16, -4, -8, -9, 8, -2, -3], [-2, -9, -2, -1, -2, 7, -7], [-9, -8, -3, 2, -7, 0, 1], [1, 6, -7, 5, -9, 3, 5], [-5, -9, 6, 5, 5, -2, 4], [-3, 8, 10, 10, -16, 0, 9], [-9, 3, 2, 6, -6, -4, 5], [-11, -7, 1, -8, 8, 2, -1], [2, -6, 1, -11, 4, 1, -1], [-4, 5, -8, 5, -12, 7, 2], [-16, 8, -5, 5, -1, -6, -5], [5, -5, 0, -6, 3, -2, 7], [5, 11, -6, 4, 3, 8, 0], [-24, 10, -6, 1, -5, 2, 3], [-8, -2, 1, 13, 3, -1, 9], [4, 2, 9, -6, -6, -11, 6], [-14, -9, -1, -10, 5, -6, 6], [0, 0, -5, -10, 3, -3, 0], [-5, -9, 7, -5, 4, -10, 1], [-2, 7, -6, -11, 0, 3, -4], [10, -7, 5, 4, 2, 4, 2], [-6, 4, -1, -5, 0, -4, -4], [12, 2, 7, 7, -5, -4, 4], [-11, -8, -3, -16, 8, 2, -2], [-24, -5, 5, 9, 0, 1, 7], [-8, -4, -1, 0, 1, -7, 2], [-2, 6, -3, -1, 0, -3, 2], [-1, 11, 3, 2, -5, 1, 1], [-14, 0, -4, -9, -2, 4, -2], [-16, 0, 10, 2, 6, 4, 4], [5, 2, 2, -4, -1, -17, -3], [-6, -7, -6, -5, -1, -1, -5], [-11, 0, 4, 2, -10, 7, -1], [4, -3, 3, -10, -1, -7, 6], [-1, 11, -6, 10, -3, -2, 9], [-17, -12, -6, -1, 8, 0, 5], [-16, -5, -4, -6, 3, 8, 0], [9, -12, -2, -8, 0, -5, -7], [15, -12, 0, 2, 3, 1, 5], [-10, 0, 5, 7, 0, -2, 12], [-1, -1, -12, 1, -4, 8, -5], [13, -5, -1, -3, -4, -1, 6], [-2, -3, -8, -6, 10, 4, -12], [0, 4, 7, 8, 1, -8, 0], [7, 5, -3, 6, -5, -6, -9], [-12, 10, 1, 9, -8, -6, -6], [-2, 8, 0, 2, 3, 8, -2], [0, 6, 12, 5, -7, 1, 5], [-1, -5, -7, -12, 10, 4, -9], [11, -6, 5, 9, 3, 5, 6], [1, -3, 9, 1, -1, -1, 17], [-17, -2, 11, 1, 2, -5, 7], [-19, -6, 1, 12, -3, 3, 6], [-13, 6, 1, 6, -7, -4, -5], [10, 1, -1, 5, -2, -3, -10], [16, 13, 3, 20, -5, -6, 6], [0, -8, -2, -10, 5, -11, -4], [12, 11, 0, 1, 7, -10, 4], [10, 4, -1, -5, 4, 7, -5], [5, -10, -6, -9, 13, 0, -10], [-2, -13, 10, -6, 5, -4, 9], [-21, 1, 1, -16, 2, -2, 7], [30, 10, 8, 24, -12, -5, 11], [-16, 6, 2, -2, -10, -2, 5], [-4, 1, 5, -1, 5, 1, -11], [-8, 8, -2, 0, -6, -12, -14], [-1, -6, -5, 1, -1, -3, -3], [8, 14, -10, 6, -2, 8, -3], [-22, 7, -7, 4, -4, 1, 2], [2, -12, 4, -13, -3, 4, -11], [19, -8, -4, 1, 16, -3, -4], [-23, 18, -12, -3, -3, 7, -1], [24, 19, 4, 16, -6, 4, -9], [5, -25, 3, -6, -3, -11, -1], [33, 5, 5, 9, -6, 3, 8], [6, -19, 18, -8, 11, -13, -1], [-14, -8, 3, 2, 5, 5, 9], [5, 9, 1, -8, -7, -9, -7], [27, 10, 7, -4, 3, -5, -8], [7, -12, 3, 5, 1, 4, -5], [-7, 6, -5, 8, -1, 0, -3], [-9, 9, 5, 7, -6, 4, -1], [-1, -7, -2, 2, 11, 8, -1], [3, 9, -11, 2, -3, 3, -5], [8, 4, 19, 9, -5, -2, 1], [-11, -4, -1, -7, 3, -7, -3], [-4, 0, 3, -7, -3, -5, -11], [-3, -5, -14, -1, 4, 7, -4], [-24, -2, -2, -2, -9, 6, 11], [16, 7, -1, 9, -1, 11, 6], [-22, -7, 0, -20, 10, -1, -5], [0, 18, 2, 24, -6, -2, 4], [13, 14, -1, 5, 8, 5, -3], [4, -6, -6, -3, 0, 9, -3], [12, 2, 5, -2, 5, -9, -7], [-23, -2, -7, -8, 0, 12, 3], [27, -6, 4, -4, 5, -4, 0], [6, 10, 6, -2, 1, -3, -6], [-7, -2, 14, 4, -4, 6, -2], [3, -2, 5, -8, -3, -2, -17], [-5, 16, -4, 8, -3, -4, -17], [-17, -6, -9, 2, -2, 2, -7], [10, 1, 2, 8, -7, -4, 4], [-6, -10, -7, 4, 3, 2, 12], [-14, -11, -2, -2, -4, -4, -6], [-1, -7, -6, -4, 8, 1, 7], [-6, -9, 5, -10, 0, -2, 0], [17, 1, 6, -5, -3, -17, 1], [-6, 6, 1, 7, 11, -10, 6], [13, -7, 3, -2, -2, -5, 8], [-11, -7, -3, 4, 0, 15, -5], [6, -3, 10, -10, 5, -16, -8], [-16, 1, -6, 9, -4, 6, 14], [-5, -6, 13, 12, -2, 7, 7], [6, 8, -4, -3, 3, 0, -11], [4, 3, 19, 12, -5, -1, -6], [-13, -18, -10, -9, 5, 10, -3], [-18, -16, 1, -5, 3, -9, 3], [17, -7, 1, 9, 6, -4, -5], [-8, -14, -6, -19, -3, 9, -2], [21, 1, 17, 4, 5, -3, -6], [1, 4, 0, -4, -3, 6, -1], [-6, 19, -19, -4, -2, 10, -2], [6, -16, 11, -8, 2, 5, 4], [-2, -1, 3, 7, 3, -3, -7], [10, 3, -13, -3, 14, -2, -13], [19, -14, 6, -10, 9, -12, -10], [-9, 1, 17, 6, 8, -2, 1], [8, 9, -9, -3, -5, -15, 0], [-12, -3, 9, -15, -3, -6, -7], [-2, -13, -7, -8, 0, 16, 2], [-25, 1, -2, -6, -2, -5, 5], [17, -2, -4, 5, -10, 8, -7], [5, -12, -8, -15, 7, 5, -2], [-3, -16, 9, -6, 5, 2, 8], [8, 7, -4, 2, -4, -10, 9], [14, -4, 14, -1, 2, -6, 14], [-15, -14, 4, -19, 16, -1, -2], [-9, 18, -25, 5, -5, 24, -8], [28, 8, 1, 27, -4, 9, 0], [-1, 2, -3, -1, -8, -2, 3], [3, 9, -10, 10, -15, 9, -4], [-14, 9, -23, -3, 2, 11, -5], [6, -7, 6, 5, 2, 3, 4], [7, 5, 17, 7, 5, -4, -1], [1, 7, -8, -7, 5, 7, 6], [-4, -8, 7, -4, 7, 6, -2], [-22, -15, 12, -5, 13, -4, 1], [-43, -4, 2, -19, 3, 1, -11], [14, 10, 10, 11, -5, -5, 8], [-4, 4, 12, 0, -18, 4, 6], [39, 16, 1, 12, -11, 6, 1], [-6, 19, -6, 18, -4, 14, 14], [16, -13, 31, -9, 4, -19, 11], [1, -5, 8, 1, 0, -20, -2], [-15, 1, 15, 3, 0, -9, -2], [-20, -14, 7, -27, 18, -1, -9], [21, 6, 6, 10, -5, 5, 3], [2, 7, -18, -4, 4, 4, -9], [-42, 7, -8, -10, 8, 10, -10], [-1, -9, 11, 1, 0, 12, -1], [0, 23, 2, 2, -4, -7, 0], [6, -24, 2, -15, 4, -6, -6], [16, 11, -16, 13, -2, 6, 9], [15, 11, -15, 16, -2, 7, 1], [-20, 6, -16, -20, 13, 4, -20], [-3, 0, -19, -9, 18, -3, -9], [27, -11, 26, -17, 5, -8, 5], [3, 5, 12, -11, 1, -5, -3], [-20, 21, -16, -2, -2, 5, -18], [-1, -20, -16, -9, 17, 2, -7], [4, 1, 22, -6, 2, -1, 4], [-2, 7, -11, 26, -5, 7, 10], [9, -7, 6, -5, -2, -14, -7], [-9, 3, -17, -7, -7, 10, 8], [29, -6, -11, 4, 15, 12, 5], [5, -1, -11, -1, 9, -3, -15], [-1, 5, 0, 7, -21, -4, 6], [-15, -8, 6, -10, 8, 7, 1], [12, -5, 23, 1, -7, -17, 10], [-20, 12, 12, 11, -2, 3, 13], [20, -5, 19, 1, -3, -11, 18], [25, -10, 2, 2, 13, -5, 6], [-6, 18, 2, 14, -10, -7, 9], [-1, 3, -14, 18, 12, -1, 7], [-23, 28, -2, 11, -11, 1, 4], [32, 3, 9, 17, 0, -14, -2], [15, 14, -1, -4, -1, 3, 4], [29, 7, 17, 17, -13, 7, 9], [7, -26, -7, -16, 18, 1, -13], [34, -4, -8, 10, -1, -7, -4], [31, 11, -7, 3, -2, -3, -2], [-12, 2, 2, -17, -11, 5, -2], [31, -15, -3, 3, 4, -8, -4], [0, -12, 1, -6, -12, 2, 3], [9, 11, 1, 2, -2, 3, 4], [20, -7, -2, -7, 12, -1, -4], [-12, -12, 10, -19, 8, -21, -9], [-11, 1, 11, -16, -2, 10, -8], [-19, -7, -11, 0, -6, -9, -7], [8, -29, 13, 0, 12, -12, 4], [-16, -7, 13, -11, 3, -11, 2], [25, 6, -6, -9, -10, -12, 1], [-10, -11, 0, -8, 10, -5, 0], [-21, 17, -7, 2, 4, 13, -8], [-1, -9, -4, -12, 4, -2, -15], [-16, 25, -12, 2, -1, 19, -7], [-33, 16, -6, 7, -9, 1, 3], [6, 1, 2, 3, 6, 3, -4], [-19, -9, 7, 0, -7, -8, -1], [-15, -18, 6, -29, 13, -6, -3], [-16, -3, -26, -2, 14, 13, -14], [18, 23, -7, 4, -10, 9, -9], [15, 3, 8, 11, -7, 3, 19], [-30, -10, 1, -17, 11, -1, -5], [-14, -12, -10, 6, -2, 10, 9], [9, -1, -12, -10, -4, 5, -15], [14, 10, -6, 8, -3, -5, -4], [-6, 21, 1, -6, -3, -4, -5], [17, 17, -2, -6, -8, -2, -19], [-11, 3, 8, -7, -15, 11, -17], [-22, -6, 17, -6, -7, -12, -2], [22, -7, -7, -10, 9, 11, 13], [25, 8, -12, -4, 8, 10, -6], [36, -15, 4, -18, 9, -5, -12], [29, -19, 9, -18, 0, 4, -5], [-19, 6, -20, -10, 6, 19, -6], [0, 19, 2, 17, -15, -7, 6], [-11, -11, 6, 10, 9, 5, -1], [7, 8, 9, 18, 3, -12, -10], [3, -21, 13, 10, 8, -8, 12], [-17, 22, -1, 15, -10, 3, -1], [4, 9, -8, -3, 10, 2, 4], [-21, 10, -10, 18, -13, 11, -1], [-11, -2, -6, -28, 9, 8, 3], [-24, 4, 21, 5, -8, -7, 13], [-40, -10, -10, -23, 10, -6, -8], [12, -20, 7, -3, -9, -12, 14], [10, 6, -17, 1, -11, -7, -2], [31, -16, 6, -1, 11, -10, -3], [25, -1, 24, 9, 0, -8, 7], [-19, -20, -1, -12, 9, 4, 2], [27, -17, -4, -5, -9, -17, 1], [-3, 14, 0, 40, -6, -2, 13], [10, -10, 15, 16, -8, 5, 16], [5, 20, -16, -9, -5, 3, -19], [-7, -27, 4, -16, 24, 4, 4], [0, -2, 4, 12, -6, 17, 9], [3, -10, 1, -13, 13, 2, -16], [-53, -5, -1, 1, -7, -1, 8], [-37, -7, -14, -1, -1, 18, 5], [16, 18, -14, 19, -11, 15, 10], [-26, 10, -17, -4, -5, 13, 0], [-8, -3, -23, 11, -2, 4, -8], [33, -30, 18, -12, 6, -11, 8], [1, 0, 9, 13, -25, -3, 10], [27, 9, -2, 7, -12, -3, -1], [10, 11, 28, 10, -20, -10, 16], [-9, 2, -11, 11, 16, -7, 7], [3, 1, -13, -12, 8, -15, -24], [-7, -3, -3, 1, -12, 24, 3], [6, 22, 1, 35, -14, -5, 6], [-2, -10, 17, 5, 6, -6, 24], [-21, -4, -4, -18, 0, -3, 4], [23, -4, 0, 9, -19, -4, 7], [1, 10, 3, 4, -3, 7, -4], [-8, -26, -9, -21, 19, 6, 0], [-21, -4, 4, -11, 3, -4, -10], [-24, 16, 6, 7, -10, -8, 1], [-11, 18, -16, 13, -9, 5, -21], [7, -4, 7, 6, 3, 1, 12], [16, 2, 9, -8, -1, -11, 1], [12, -2, 2, 20, -9, -6, -6], [-7, 16, -2, 2, -16, 10, 1], [32, 8, -5, 10, -11, -16, 4], [24, 0, 14, 2, 1, 9, 5], [-27, -10, -13, -8, -3, 24, 4], [38, -3, 10, 13, -14, -6, 21], [-2, -6, 32, -4, -7, -15, 8], [-11, 25, 2, 6, 2, 4, -8], [-6, -1, 26, 7, -5, -8, 26], [-9, -1, 16, 3, -10, 5, 7], [-17, -18, 1, -17, 2, -18, -7], [16, 22, 4, 25, -19, -1, 12], [51, 10, 1, 25, -3, 3, -11], [-25, 10, 3, -18, 1, 17, 5], [-24, -2, -1, -7, 0, -4, 5], [-8, -25, 4, 7, 3, 2, -2], [-11, 3, -17, -11, -1, 9, 1], [11, -12, 12, 17, 4, -6, 5], [-18, 9, -11, -13, -1, 13, -20], [17, -15, 16, 19, 0, -5, 20], [23, -4, -1, 14, 0, 0, 12], [44, -3, 3, 12, -7, 7, -1], [-6, 4, -7, 7, 9, 0, 2], [-15, 5, -5, -1, -2, -7, -6], [31, -18, 3, 17, 8, -1, 5], [4, -5, -7, -26, 13, 3, 4], [47, 11, -8, -13, -8, 4, -10], [-4, -10, -8, -3, -11, 20, 5], [-11, -3, 6, -13, 12, -4, 0], [-43, 5, -12, 3, -13, 14, 1], [-12, 18, -18, 0, -10, -6, 0], [-18, -8, -10, -13, 5, 7, -2], [3, -3, 6, -3, 6, -6, -3], [33, 1, 3, 0, 19, -22, -14], [-35, 1, 9, 9, 4, 21, 12], [-60, -7, 3, -21, 1, 1, 0], [7, -5, -21, 0, 4, 1, -3], [-28, 19, -3, 9, -10, 5, 5], [-33, 43, -8, 16, -20, -4, -5], [1, -15, -3, 1, -5, 16, 15], [24, 9, -5, -20, 4, -4, -14], [-35, 11, -7, 9, 7, 9, -17], [-10, 35, -17, 7, -12, 8, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8030_2_a_ba();".
// 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_8030_2_a_ba(:prec:=7)
    chi := MakeCharacter_8030_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_8030_2_a_ba();".
// 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_8030_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-34, -82, -13, 54, 14, -12, -2, 1]>,<7,R![-254, 308, 615, -27, -128, -10, 7, 1]>],Snew);
    return Vf;
end function;