
// Make newform 2013.2.a.f in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2013_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_2013_2_a_f();".
// 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_2013_2_a_f();".
// 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, -16, 118, 98, -404, -141, 534, 13, -266, 32, 55, -11, -4, 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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, -4, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-47, 14, 228, -495, -255, 770, 46, -374, 29, 73, -11, -5, 1], [8, -184, 201, 441, -695, -320, 506, 108, -150, -17, 20, 1, -1], [66, 176, -386, -142, 749, -516, -292, 411, -8, -101, 18, 8, -2], [186, 164, -992, -788, 1350, 1126, -764, -540, 221, 104, -33, -7, 2], [-21, -427, 129, 1740, -411, -2043, 591, 960, -312, -194, 67, 14, -5], [267, 122, -1696, 66, 2700, -833, -1384, 586, 263, -140, -11, 11, -1], [-12, -305, -463, 1675, 1362, -2912, -489, 1604, -91, -348, 58, 26, -6], [-141, 152, 905, -1556, -1532, 2895, 506, -1604, 91, 348, -58, -26, 6], [-233, 149, 1747, -1551, -3112, 3278, 1321, -1924, -51, 432, -56, -33, 7]];
    Rf_basisdens := [1, 1, 1, 1, 17, 17, 17, 17, 17, 17, 17, 17, 17];
    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_2013_a();"
function MakeCharacter_2013_a()
    N := 2013;
    order := 1;
    char_gens := [1343, 1465, 1222];
    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_2013_a_Hecke(Kf)
    return MakeCharacter_2013_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], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, -1], [1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, 1, 0], [0, 0, 0, -1, 0, 1, 1, 1, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1], [0, -1, 0, -1, -1, 1, 1, 1, -1, -2, 1, -1, 0], [1, 0, -1, 0, -1, -1, -2, 0, 0, 1, 0, 0, 0], [2, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1], [1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0], [1, -1, -1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0], [2, -1, 1, 0, -2, 0, -1, -1, -1, -2, 0, -1, -2], [3, -2, 0, -1, 0, 0, -1, -2, 0, -2, -2, 0, -2], [2, 0, 1, -2, 2, 2, 3, 1, 0, 0, 1, -2, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 2, 2, -1, -1, -1, 0, 2, 2, 1, 1, 0], [5, -1, -1, -1, 1, -2, -1, -2, 0, 0, -2, 0, -4], [-4, 1, 0, -1, 2, 1, 4, 0, 2, 0, 1, -2, 1], [1, -1, 1, 2, 0, 0, -2, -1, -1, 0, 0, 1, -2], [3, -1, 1, 2, -2, 0, 1, 3, -1, 0, 1, 2, -1], [2, 1, 0, 1, 3, -1, 1, -2, 2, 2, -2, -1, 2], [1, -1, -1, 0, 1, -2, 0, 0, 0, 3, 1, 3, 2], [0, 0, 2, 1, -2, 0, 0, -2, 2, -3, -2, -2, -3], [1, 1, 1, 1, -3, 0, -2, -1, -1, 0, 0, -2, -1], [0, 1, 0, -1, 2, -1, 2, -2, 2, 1, 1, -2, 1], [-3, -2, -2, 1, 1, -1, 0, -3, 1, 1, 0, -1, 1], [2, -3, 2, 0, 0, 2, 1, 4, -2, -1, 2, 0, 1], [-3, 1, -3, -2, 2, -1, -2, -3, 1, 1, -2, -2, 1], [6, -1, 1, 3, 1, -1, 2, 1, -1, 0, -1, 3, 1], [-1, -4, -4, 0, -3, 2, 0, 0, 0, -1, -1, 0, 1], [-4, -4, 0, 1, -3, 3, 1, 0, 0, -1, 3, -3, 0], [-2, 1, -1, 3, -2, 0, -2, -1, 0, 2, 1, 0, 3], [1, -1, 0, -4, 4, 1, 1, 1, -1, 1, 2, -1, 0], [1, -2, 0, -1, 3, -1, 2, 3, -1, -2, 0, -1, 1], [2, 3, -3, -1, 1, -3, 0, -1, 2, 3, -1, 1, 3], [2, 0, -3, -1, 5, -3, -2, 0, -1, 1, -3, 1, 1], [2, 5, 0, 0, -2, 0, 2, 1, 1, 4, 0, 2, 2], [5, -3, -1, 3, -2, -3, -5, -1, 0, 1, -1, 3, -2], [-2, 1, 3, -1, 0, 0, 1, 2, 2, -1, 3, -2, 1], [4, 1, 1, 1, -1, 2, 1, 4, 0, 1, 3, 3, 3], [-2, -3, 5, -1, -4, 5, 4, 4, -1, -4, 2, -1, -1], [1, -1, 1, 1, 3, 0, 3, -2, 1, -1, -2, -3, -2], [4, 0, 2, 0, -3, 3, 0, -1, 0, -3, -3, -1, -4], [-1, 3, 3, 2, -1, -4, -4, -1, 0, 4, 0, 2, 0], [5, -2, 3, 4, 1, 0, 2, 2, 1, 4, 2, 2, 3], [5, -1, -4, -1, 2, 0, 1, -2, 3, 2, -1, 0, 2], [3, -1, 1, 0, 0, 1, -2, -3, -1, 0, 0, -1, 3], [0, -3, 4, 0, 1, 2, 3, 2, 0, -1, 0, 1, -1], [5, 5, 3, -1, 1, 2, 3, 5, -3, -1, 1, 1, 0], [-4, -2, 1, 1, 1, -3, -1, -4, 1, 0, 0, 0, -2], [-1, -3, 1, 0, -1, 3, -1, -3, 2, 2, 1, -2, 3], [-4, 4, 2, -3, -1, 1, 3, 1, 0, -3, 0, -3, -1], [2, -4, 1, 1, 4, -1, -3, -5, 2, 4, 1, -3, -1], [5, -4, -5, 1, 0, 0, -3, 4, -3, 3, 1, 5, 1], [0, 3, -1, 0, 3, -2, 1, -3, 1, 4, -4, 4, -1], [-4, 4, 3, -2, 0, 5, 4, 0, 1, -1, 1, -2, 4], [3, 4, -2, -1, 6, -7, -4, -4, -1, 6, -3, 1, -1], [1, 5, -2, -1, 1, -2, -2, -3, 0, 1, -5, -2, -1], [1, -2, -1, 3, -1, -4, -4, 1, -3, 4, 3, 2, 1], [4, -2, 2, 1, 0, 2, 2, 5, -3, -1, -1, 1, 3], [-2, -2, -3, 1, -4, 1, -4, -1, -1, -2, 0, -1, -1], [-1, 4, 2, 3, -2, -1, 0, -2, 1, 3, -1, 1, -5], [1, 4, 0, 0, 4, -4, 3, 3, 1, 6, 1, 4, 1], [5, 1, 4, 4, 0, -1, -3, 0, -1, 4, 5, 2, 3], [-11, 4, -5, -4, 5, -3, 0, -5, 1, 2, -3, -4, 0], [2, 3, 4, 1, 0, 0, 0, 2, 0, 1, 1, 2, -4], [0, -1, 0, 3, 4, -1, 1, -5, 2, -3, -3, -5, -1], [-2, 0, -3, -1, -1, -1, 3, 1, -2, -2, 1, 1, 2], [6, 0, -4, -2, 8, -3, 1, 0, -2, -1, -3, -1, 0], [3, 8, -2, 1, 3, -2, -2, -6, 1, 4, -3, -3, -3], [-1, -1, 0, 0, 4, -1, 3, 1, 2, 1, -2, -3, 0], [-7, 3, -1, 0, 3, 1, 8, 1, 1, 0, 2, 0, 1], [0, 0, 0, -1, -1, 4, 4, 7, -4, 0, 1, 4, 7], [8, 5, -6, -1, 0, -4, -3, 1, -2, 2, -3, 7, 1], [6, 3, -3, -1, 0, -2, -3, 4, -2, 4, -3, 7, 4], [2, -1, 0, 5, -2, 1, 2, -2, 1, 1, 1, -2, 2], [-4, 1, 1, 0, -1, -1, -1, -6, 2, 0, 1, -2, -4], [7, 0, -2, -2, -5, -2, -10, -3, -2, -4, -2, 1, -1], [-8, -3, -3, -3, -2, 3, -1, -1, 0, -6, -2, -4, 1], [-4, 6, 1, -3, 1, -3, -2, 2, 1, 3, 1, 0, 2], [-3, 3, -3, 5, -4, 0, -1, 4, -1, 4, 0, 4, 3], [-1, -4, 0, 2, 4, -4, -3, -3, 2, -2, -3, 0, -2], [4, -5, 1, -2, 1, -1, 1, -3, 2, -4, 2, -2, 3], [-1, -3, 1, 0, -4, 5, 6, 3, -3, -6, -2, -1, -5], [-8, -1, 4, -5, -8, 8, 7, 7, -6, -11, 1, -3, -2], [-5, -1, 4, 1, -6, 4, 2, -3, 2, -1, 3, -3, -1], [2, -5, 5, -2, 5, 0, 5, 1, 4, -2, 2, -1, -3], [4, -2, 2, 1, -2, 2, 3, 1, 3, -4, 0, -3, -3], [-8, 7, -2, -1, 0, 3, -2, -4, 1, -2, -3, -3, 0], [-4, -1, -1, 1, -1, 2, 5, -1, 1, 1, 0, -1, -2], [0, 10, -4, -4, 8, -3, 4, -1, 2, 7, -1, 1, 5], [-3, 0, -7, -5, 8, 1, 0, -6, 0, 2, 2, -4, 2], [-6, 1, 7, -1, -8, 3, 0, 7, -4, -8, 3, -3, -2], [6, 4, -4, 1, 10, -5, 0, -9, 2, 6, -5, -3, -1], [2, 4, 2, -1, -12, 2, -5, 0, -1, -10, 1, 3, -5], [-8, 2, 6, 1, 0, 2, 7, 1, 2, -1, 4, -2, 4], [-12, 0, 2, 0, -9, 6, 0, 5, 1, -3, 5, -1, -1], [-1, -2, -4, -1, -6, 2, -2, 4, -2, 5, 5, 6, 5], [-4, 10, 1, -1, -5, 0, -4, -7, 6, 0, -2, -3, -5], [0, 2, 0, 0, -2, 2, 2, 6, -4, -2, 0, -1, -1], [-1, 5, -1, 0, 2, 3, 2, 4, 4, 6, 5, -1, 7], [0, -5, -5, -1, 0, -2, -1, 2, 0, -1, -5, 3, 1], [-1, 6, 5, 2, 3, -3, 1, -3, 1, 6, 0, 2, 0], [8, 10, -1, 2, 5, 0, 2, 0, -3, 4, -1, 3, 5], [5, -7, 7, -2, -3, 3, 1, 6, -5, 0, 6, 2, -2], [4, 1, -5, 4, -1, -1, -2, -1, 0, 8, -3, 5, 4], [-4, 5, 5, -1, 3, -1, 4, 2, 0, -2, -2, 3, -4], [-8, -4, 8, -3, -6, 9, 5, 8, -4, -9, 7, -2, 3], [5, -2, -1, 0, 1, -1, -2, -1, 2, -8, -1, 1, -3], [-2, 5, -3, 0, -5, 2, -1, -3, 2, 2, -2, -2, -1], [2, 2, -4, -2, 4, -2, 7, -1, 1, 1, -4, 3, 4], [-7, -2, -3, -2, -6, 1, -1, -4, 6, 1, 5, -3, 4], [10, -1, 6, 0, -1, 6, 2, 8, -1, 1, 5, 2, 1], [18, 1, -2, -1, 3, -7, -2, 2, -4, 8, 3, 5, 2], [-2, 3, 3, 3, -7, -3, -9, 2, -1, 0, 1, 2, -1], [4, 6, 5, -1, -1, 1, 5, 4, -1, -2, 2, -2, -3], [8, -2, -1, -3, -1, -3, -3, 0, -1, -1, 0, 6, -4], [-1, 5, 6, -1, -4, 3, 2, 2, 0, -1, 0, -3, -2], [3, -7, 3, 4, -4, 4, -8, 0, 1, 3, 1, 1, 1], [-8, -3, -6, -1, 3, -6, -4, -5, 3, 6, 2, 0, 4], [6, 0, -1, 6, -1, -5, -8, -1, -2, -1, -4, 4, 0], [-1, 1, 3, 1, -6, 3, 1, 7, -5, -2, 1, 1, 2], [0, 8, -4, -2, 14, -7, 5, -3, 3, 5, -2, 3, 3], [-1, -4, 7, 1, -10, 3, 3, 8, -4, -13, 3, 2, -5], [5, 1, -1, -4, 0, -2, -5, 1, -4, -2, 1, 4, -3], [3, 3, 5, -5, -2, 3, -2, -3, 1, -6, 2, -4, -5], [2, 0, 3, 2, -10, 5, 2, 7, -4, -4, 0, 4, -3], [-5, -2, -5, 0, 0, -6, 1, -3, 5, 2, -2, -2, -8], [-5, 2, 4, -4, 5, 2, 1, 1, -6, -7, 1, -4, -2], [0, 1, -1, -2, 4, -2, -3, -7, -3, -4, -8, -7, -4], [3, 5, 2, 1, 2, -4, -3, -4, 4, 1, -1, -4, -4], [1, 4, 3, 2, 6, 2, -1, 1, 4, 3, -4, -3, 5], [1, -11, 8, 0, -4, 6, 4, 5, 2, -4, 3, -5, -3], [8, 2, 0, 1, -2, 3, 2, 7, -1, -3, -1, 6, -5], [10, -9, 0, 5, 0, -6, -6, 1, -3, 1, -2, 8, -5], [1, 3, 5, -6, -6, 5, 6, 4, -4, -8, 1, -3, -4], [1, -1, -1, 0, -1, -3, 5, 5, 4, 4, 3, 3, 3], [5, -2, -7, -5, 2, 1, 1, 6, -6, -2, -1, 4, 3], [6, 2, 12, 4, -6, 8, 3, 6, -6, 2, 7, -1, -1], [6, -3, -8, 5, 1, -8, -8, -3, -1, -1, -10, 4, -7], [2, -1, 5, 0, 2, 3, 6, 1, 3, 2, -1, -2, 4], [5, 5, -8, 2, 0, -7, -1, 2, -1, -1, -3, 3, 3], [9, -1, -5, 4, -4, -2, -3, -8, 1, 4, -2, 3, -3], [1, -10, -4, 0, 9, 0, 5, 1, -3, 1, -1, -4, 10], [5, 1, -8, -6, -3, 3, -2, -2, 0, 2, 0, -4, -2], [-5, -6, 1, 3, 3, 4, 10, 5, 3, -5, 3, -4, 4], [5, -12, -2, -5, -7, 3, 1, 7, -3, -5, 5, 6, 2], [-7, -5, -6, 0, -5, 1, -8, 1, -4, -4, 3, 5, 3], [-8, 9, 0, 4, 2, 1, 3, 1, 0, 4, -1, -1, 2], [0, -4, 3, 0, -3, 3, 13, 4, 2, -6, 4, -3, -1], [10, 10, -9, -3, 1, -4, -10, -8, 5, 8, -3, 3, 4], [10, 3, 6, 4, 4, 0, 2, -1, -1, 1, 0, -2, -5], [3, 3, 2, 3, 0, -8, -9, -3, 2, -2, -10, 3, -8], [-7, 7, -3, -3, 12, -2, 10, 7, -3, 8, 4, 1, 6], [2, -5, -5, -3, 5, -4, 1, -8, 7, 7, 3, -4, -2], [2, 1, -4, -1, 8, -3, -2, -6, -3, 4, -1, -7, -4], [-4, 4, 6, 11, -5, 1, 1, 1, 1, 3, 6, 5, 3], [6, 1, -6, 2, -4, 3, 0, 6, -3, 3, 3, 5, 1], [2, 11, 6, -1, 6, -2, 3, -13, 7, 2, -1, -8, 1], [0, -9, 5, 3, -2, 4, 2, 5, 4, -5, -3, 0, 3], [-10, 9, 4, -2, -9, 10, 4, -2, 5, -5, 5, -7, 3], [1, -4, 1, 2, -8, 8, -2, 7, -3, -4, 3, -4, 2], [4, -1, -2, 7, 0, -7, -5, -4, 6, 2, -6, 4, -5], [-1, 16, 1, 2, 6, -4, -4, -6, -1, 9, 2, 0, -3], [9, -3, -7, -2, 0, -5, -5, 3, -9, -2, -1, 7, -3], [13, -2, -6, -2, -7, 2, -7, 8, -6, 0, 2, 1, -2], [-4, -2, -1, -10, 2, 6, 2, 2, -7, -8, -3, -1, -3], [7, 6, -7, 4, -3, -3, -1, -6, 4, 3, -5, 4, -1], [-16, -1, -5, -3, -3, 7, 4, 3, -2, -8, -3, -3, 1], [3, 4, -4, 8, 1, -10, -10, -3, 6, 10, -4, 8, 4], [-5, -8, -2, 5, 1, -1, 1, 4, 4, 7, 6, 5, 8], [6, -12, -8, -5, -8, 0, 0, 6, -3, -6, -2, 6, 2], [20, -1, 3, 1, -2, 4, -3, 6, 3, 5, -1, 1, 6], [-16, -10, -5, 3, -8, -2, -4, -1, 3, 0, 4, 3, 1], [1, -4, -8, -6, 7, -7, -2, -1, -3, -2, -6, -3, 1], [9, 0, 0, -2, -6, 0, -13, -6, -7, -9, -6, -4, -4], [-10, -9, -6, -4, 7, 5, 0, -2, -2, -2, 1, -1, 2], [7, 2, -3, -3, -3, 2, -2, -1, 0, 4, 1, 2, 5], [1, 10, 1, -1, 3, 2, 10, 0, 6, 8, 2, -4, 3], [-4, 6, 8, 1, -4, 10, 7, 5, 5, 2, 4, -4, 6], [-4, -3, 4, 2, -5, 0, -3, 1, -5, 5, 3, -5, 1], [5, -4, -8, 1, 4, -5, -5, -5, 1, 3, -1, -2, -5], [-5, 8, -5, 9, -5, -2, 1, 2, 2, 3, -5, 8, 2], [-12, 6, 5, 8, -1, 0, 6, 0, 5, 3, -3, -1, -1], [20, 1, -9, -1, 3, -11, -7, -11, 2, 7, -2, 4, -2], [-11, -1, 2, -5, 1, 5, 8, 0, 9, 2, 8, -1, -1], [15, -5, 7, 3, -5, 0, -1, 6, -2, 2, 2, 2, -6], [-6, 2, 1, -4, 8, 3, 5, 5, 5, -3, 3, -3, 4], [7, -8, 1, -10, -10, 2, -4, 0, -5, -11, -2, -1, -6], [0, 2, 6, -3, -10, 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-2, 0, 4, 3, -7], [2, -7, 5, 1, -1, -2, 5, 9, 0, 8, -4, 3, 8], [-10, -8, 5, -2, -15, 10, 1, -15, 8, -5, 1, -9, -11], [-2, 7, -11, -11, 2, 0, -1, 6, -6, 5, 3, 0, 11], [-8, 14, 8, -3, 14, -1, 4, 2, 8, 7, 0, -7, 9], [-5, 15, -2, 5, 8, -4, -2, -8, 6, 8, -12, -12, 0], [-4, 14, 0, 5, 0, -17, -6, 2, -3, 10, -6, 8, -2], [-17, -5, 7, -2, -1, 2, 7, -8, 7, -9, -6, -9, -4], [2, -3, -9, 6, -8, -5, 7, 6, 5, -5, -1, 2, 2], [10, -2, 1, 2, -3, 1, -1, 4, 4, -4, 0, 3, 6], [7, -2, -6, -2, 5, -11, -13, -8, 1, 12, 2, 6, 10], [10, -3, -7, 1, -1, -1, 0, 5, -3, 9, 1, 9, 5], [-2, 14, 1, -11, -3, 10, 8, 15, -4, -14, 6, -4, 0], [-19, 10, -4, 7, 17, 5, 10, 5, 7, 14, 0, -1, 13], [-2, -18, -2, 8, -2, 3, 4, 2, 7, 15, -4, 4, 7], [-13, 7, 7, 11, -7, 9, 1, -5, 7, -7, -5, 4, -2], [30, 11, -8, -7, 10, -11, 0, 7, -11, 14, 6, 15, 7], [-10, 10, -11, 0, 2, -8, -13, -12, 9, 7, -6, -2, 12], [-3, -9, 16, 6, -10, 3, 2, 7, 4, -7, 1, -3, -10], [20, 9, 14, 2, 5, 0, -9, -3, -3, 3, 1, 0, 0], [0, 0, -2, 10, 2, -4, -13, -5, -1, 5, -4, -7, 4], [-29, 23, -3, -8, 13, -6, -2, -1, 1, 15, 0, 1, 13], [5, 5, 13, -2, 9, 5, 8, -10, 6, -10, 3, -11, -3], [18, -14, 2, -5, -15, 9, -2, 11, -16, -20, 7, 4, -6], [-5, -2, -6, 2, 5, -5, 0, 1, -2, -10, -4, -2, -5], [-4, 4, 12, -11, -17, 8, -8, 15, -12, -23, 7, 2, -8], [7, -12, -7, -5, 4, -2, -4, 0, 2, -2, -4, 0, -6], [-3, 3, -8, -2, -2, -4, -7, -8, -8, 2, -7, 6, -2], [9, -1, 5, -1, 16, -7, 4, 5, 3, 9, 8, -1, 11], [8, 0, 10, -6, 2, 3, 3, 3, -7, -15, -2, -1, -9], [-9, -18, 7, -13, -4, 4, 2, 9, -5, -19, 7, 4, -6], [34, 0, -2, 5, 4, 1, -12, -6, -2, 11, -8, 4, -5], [10, 3, 0, -16, -9, 5, -11, -1, -1, -17, -1, 1, -3], [2, 12, 0, 6, 14, -4, 4, 3, 5, 14, -1, 4, 7], [2, 11, 10, 2, 9, 6, 11, -8, 10, -2, 8, -9, -1], [12, -16, -4, -2, 0, 7, -5, 0, -2, -8, 3, -10, -8], [11, 0, -28, 2, 8, -17, -8, -23, 7, 13, -16, 8, -2], [-11, -6, -6, -7, 8, -5, 7, 2, -9, -2, -8, 7, -9], [-7, 3, 17, -1, -12, 23, 23, 13, -5, -16, 11, -16, -3], [-8, 9, -7, -5, -17, -1, -3, -2, -9, 0, -7, 10, -13], [24, -3, 1, -5, -7, 10, 16, 7, 1, -7, 11, -1, 3], [33, -4, 7, 11, -3, 1, -7, 2, -6, 15, -1, 9, -1], [-12, 10, -6, 5, 12, -11, 0, 0, 1, 16, 12, 5, 11], [-8, 1, 13, 9, -9, -4, 2, -6, 3, -9, 6, -3, -11], [7, 16, 14, 2, 1, 3, -3, -1, 6, 14, 13, 2, 7], [-8, -4, 6, -2, -18, 7, -9, -4, -12, -14, -9, -2, -15], [-10, 3, -4, -2, 6, -1, 3, -6, 7, 13, -10, -13, -2], [1, -2, 2, 3, -5, 1, -17, -13, -5, -4, -7, -6, -12], [3, 6, -9, -7, -4, -4, -19, -10, 1, -10, -3, 5, -2], [4, 7, 6, 2, -3, 5, 7, 6, 1, -1, 14, -7, -7], [5, 6, 11, 5, -4, 11, 7, 9, -9, -11, 6, 0, -3], [-19, 15, -14, -6, 19, -11, 1, -24, 2, 9, -5, -6, 3], [-9, 10, -10, -3, 26, -12, 19, -13, 13, 7, -11, -2, 1], [-23, 7, 5, -4, -8, 9, 6, -11, 2, -2, 2, -5, 3], [14, -2, -3, -1, -9, 5, -9, 13, -8, 7, 9, 8, 2], [17, -10, 16, 7, -5, 16, 12, 14, -1, -12, 7, 0, -8], [-5, 10, 0, 4, -6, -6, -7, 0, -3, -10, -16, 6, -16], [-8, -11, 7, 2, -14, 9, 6, 22, -3, -4, 6, 11, 6], [17, -12, -12, -8, -14, 3, -9, 9, -6, -11, -1, -2, 6], [14, -12, 1, 2, -1, -9, 0, 3, 0, 9, 8, 7, 1], [-24, 2, 23, -9, -6, 28, 22, 22, -11, -26, 13, -7, 4], [-13, -16, 9, -9, -15, 17, 2, 17, -12, -17, 17, 0, 5], [-15, 21, 15, 2, -2, -2, -4, -8, -3, -5, 3, -8, -3], [11, -18, 7, 9, -30, 8, -21, 0, -14, -20, -5, 8, -14], [6, 8, 10, -3, 6, 6, -6, -9, 4, -13, -9, -6, -6], [22, 2, -6, -7, 1, 3, -8, -3, -16, -17, -3, -3, -4], [-9, 5, -1, 10, 15, -3, 12, -3, 9, 12, -4, 1, -4], [9, -17, -15, -7, 2, -4, 1, 5, -2, -7, 1, -1, -3], [-14, -8, -2, 0, -3, -2, 7, 12, -3, -5, 1, 3, -6], [0, -7, 7, -3, -9, 6, 10, 5, -4, -18, 13, 1, -4], [-18, -10, -13, -2, -5, -3, -13, -3, 5, -7, -5, 1, -7], [29, 0, 0, -13, -4, 5, 2, 11, -4, -5, 4, 6, 3], [24, 4, -2, 6, 12, -4, -8, -8, 4, 0, -2, -7, -8], [6, -10, -1, -8, 17, 6, 10, 3, -3, 0, -4, -9, 2], [15, 18, 0, 9, 8, -8, -12, -4, 1, 19, -1, 4, 2], [34, 14, -5, -9, -8, -3, -7, 3, -3, 0, -5, 1, -2], [8, -10, -7, 1, 1, 5, 2, 16, 1, -10, 5, 6, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2013_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2013_2_a_f();".
// 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_2013_2_a_f(:prec:=13)
    chi := MakeCharacter_2013_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_2013_2_a_f();".
// 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_2013_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2013_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-11, -16, 118, 98, -404, -141, 534, 13, -266, 32, 55, -11, -4, 1]>],Snew);
    return Vf;
end function;