
// Make newform 6348.2.a.t in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6348_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_6348_2_a_t();".
// 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_6348_2_a_t();".
// 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 := [-23, -323, -74, 714, -46, -354, 57, 65, -14, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [-2070, -3540, 5481, 3380, -2057, -1140, 216, 167, -4, -9], [1035, 2000, 8, -4266, -1766, 1743, 559, -279, -44, 16], [-161, 6203, 3184, -11416, -3530, 3898, 831, -481, -55, 20], [2507, 9815, -2896, -18006, 927, 7123, -165, -1030, 12, 50], [506, -7706, -9146, 17423, 7250, -7917, -1672, 1331, 117, -76], [-161, -8885, -8086, 19427, 7993, -9051, -2044, 1589, 152, -95], [-1081, -13617, -10733, 26577, 9757, -11206, -2316, 1791, 163, -99], [-2277, -33058, -20943, 59953, 22023, -25289, -5563, 4119, 405, -233]];
    Rf_basisdens := [1, 1, 529, 529, 529, 529, 529, 529, 529, 529];
    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_6348_a();"
function MakeCharacter_6348_a()
    N := 6348;
    order := 1;
    char_gens := [3175, 2117, 4237];
    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_6348_a_Hecke(Kf)
    return MakeCharacter_6348_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, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [2, 0, 1, 1, 0, 0, -1, 1, 1, 0], [1, 0, 0, 0, 0, 1, -1, 0, 0, 0], [0, -1, 1, 0, -1, 1, 0, 0, 0, 0], [-1, 1, -1, -3, 1, 0, 2, -2, -3, 0], [4, -2, 0, 1, 1, -1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, -1, 1, -2, 0, 0, 0], [3, -1, 1, 2, -1, -1, -1, 2, 1, -1], [-1, -1, 1, 0, -3, 1, 3, 1, 1, -3], [5, 0, -1, 3, 1, -1, -3, 2, 0, 2], [5, 1, -1, 0, 2, 0, -1, 0, 3, 1], [-4, 1, 1, -2, -2, 1, 3, -1, -3, -1], [1, -2, -1, 2, 0, 0, 0, -2, 0, -1], [-2, 3, -2, -6, 1, 1, 3, -5, -1, -1], [1, 1, 0, -5, 2, 0, 2, -5, -4, 1], [2, 2, -1, 0, 0, 0, 0, -1, -2, 1], [-3, 1, -1, -1, -2, 0, -2, 5, 3, -2], [-4, -2, 0, 3, 0, 1, 0, 0, -2, 1], [5, 0, 1, 2, -2, 0, -3, 6, 2, -2], [2, 1, 0, -2, 0, 0, 3, 1, 0, 1], [6, -2, -1, 7, 0, -2, -2, 2, 1, 2], [4, 0, -2, -3, 0, -1, 0, 1, -3, 1], [3, 0, 0, 3, 0, -1, 4, 1, 4, 1], [4, 0, -1, 4, -1, 0, 0, 1, -3, 2], [5, 2, 0, -3, 1, -2, -1, -2, 3, -1], [3, -1, 1, 0, 0, 0, 0, 0, -3, 0], [6, 0, 1, 3, 1, -2, -6, 5, 3, -2], [-9, 4, 1, -4, 0, -1, 4, -1, -5, -3], [-4, -1, -3, 0, 2, -1, 6, -2, 1, 1], [6, 1, -1, 1, 3, -4, -2, 2, 9, 1], [1, -1, -4, -8, 3, -2, 3, -5, -4, 3], [-7, 4, 0, -3, -2, 3, 4, 5, -6, -4], [3, -1, 1, -2, 3, -4, 3, -5, -2, 2], [-2, 4, -3, -1, 2, 3, 5, -6, -1, 1], [0, -2, -2, 1, 2, -2, 0, 2, 5, 3], [0, -5, -2, -4, 1, -4, -1, -3, -5, 3], [1, -3, -2, 1, 2, -1, -5, -5, 1, 1], [1, -4, -1, 6, -4, 1, 1, 0, -4, -4], [0, 2, 3, 3, -1, 2, 0, -5, -2, -2], [2, 1, 1, 2, 0, 1, 2, -3, 7, 0], [-2, 0, -3, -2, 2, 4, 2, -3, 0, -2], [9, -5, 1, -2, 0, -4, 3, 5, 2, 1], [3, -7, 2, 0, -3, 0, 5, -5, -6, 0], [-2, -3, -1, 3, -3, -1, 8, 3, -2, -4], [1, 3, -1, -6, 3, -1, 5, -4, 0, 4], [7, 0, 2, 13, -4, 2, -8, 9, 2, -4], [1, -4, 0, -8, 1, 0, 5, -7, -4, 2], [-5, 3, -2, -1, 4, -2, -4, -5, 2, 2], [0, 3, -1, 6, -2, 4, -1, -1, 6, -4], [3, 0, -1, -6, 2, -2, 12, 2, -1, -3], [-5, 0, 0, 3, 0, 6, 2, -1, -6, 2], [-1, 2, 0, -4, 5, 0, -2, 7, 1, -3], [8, 0, 3, 0, -2, 2, -11, 8, 6, 0], [-3, 2, 1, 1, 2, -2, 2, -3, 0, 4], [-8, 1, 0, -5, -4, 4, 4, 1, -10, -5], [0, 3, 2, 3, 2, 5, 4, -3, -7, 0], [-4, 3, -1, -3, -3, 4, 2, 0, 3, -4], [7, 2, -3, 0, -1, -1, -1, -5, 5, -3], [-6, -3, 5, 1, -5, -2, 4, -6, -4, 0], [11, -4, 0, 12, -3, 0, -12, 8, 9, 1], [-6, 0, 0, -6, 6, -2, 2, -9, -14, 4], [7, 2, 2, 2, 0, 5, -4, 8, 5, -1], [8, 3, 1, 8, 1, 3, -5, 7, -3, 1], [7, -6, 1, 9, -1, -2, -9, -1, 7, 1], [5, 1, 1, -3, -3, 1, 3, -2, -4, -4], [-1, 2, 0, 6, 2, 3, -4, 7, -1, -2], [4, 5, 2, 8, 1, 3, -7, 5, 14, 0], [-2, 0, 1, -1, 4, -1, -4, -11, 4, 0], [5, 4, 0, -8, 2, -1, 3, 7, 6, -2], [-4, 5, 2, -10, 3, -4, 9, -6, -2, -1], [11, -4, 1, 4, 1, -3, -4, 7, -2, 5], [4, 0, 3, 9, -5, 3, 1, -1, 0, -2], [-3, 3, 0, 7, -2, 4, -4, -6, 1, -2], [3, -1, 1, 1, 4, -3, -10, 6, 5, 2], [7, -4, 2, 2, 7, -6, -7, 8, 12, 4], [-4, -2, -3, -4, 3, -1, -9, -5, -1, 3], [7, -1, -1, 1, 3, -4, -1, 11, 1, 0], [5, -1, -3, 4, -1, 0, -1, 6, 14, 1], [10, -6, 0, 6, 0, -1, -10, 11, 0, 4], [-13, 2, -2, -12, 3, -1, 9, 0, -11, -2], [12, -11, 2, 8, -2, -4, 1, 6, 1, -1], [-7, 4, 1, -4, 0, 4, 2, -2, -8, 1], [-10, 6, -3, -14, 6, 3, 6, -15, -8, 0], [-2, 8, 7, 4, -6, 3, -3, 4, 2, -7], [2, -2, -5, 0, 0, -4, -6, -10, -3, 3], [7, -2, 2, 4, -3, -2, 8, 5, 1, -4], [13, -6, 3, 10, 0, -5, 0, 9, -4, 5], [-4, -4, 3, 7, 2, -3, 15, -10, -4, 4], [-11, 1, -4, -8, 3, -2, 0, -19, -9, 0], [6, 1, 4, 1, 1, 1, 7, 4, 6, 0], [-5, 4, -1, -16, 9, -2, 8, -10, 3, -3], [-1, -5, -1, -1, -3, -2, 9, -5, -5, 2], [2, 1, -5, 6, 0, 0, -6, 8, -5, 3], [-2, 0, 6, 5, -1, 1, -6, -2, 4, -2], [-3, -5, 3, 6, -1, -9, 3, 2, -6, 1], [6, -4, 1, 6, -2, 1, -5, 0, 0, 5], [-15, 6, -2, 2, -4, 6, 8, -2, -3, -1], [6, -3, 3, 10, -3, -3, -7, -2, 8, 4], [-11, 2, 0, 2, 8, -4, 6, 5, 3, 4], [4, -2, -1, -3, -1, 2, -6, -2, -7, -5], [-9, 3, -4, -11, 7, 4, 8, -13, -5, 2], [4, 4, 5, -10, -1, -2, 6, -1, 4, -1], [-2, -5, -6, -6, -3, -2, 9, 3, 4, -3], [0, -2, -2, -9, -1, 2, -5, -9, -4, -1], [-14, 2, 2, 10, -10, 6, 5, 8, -6, -9], [-13, 0, 4, -11, -4, 3, 5, 8, -6, -6], [-7, 2, 4, -8, -3, -7, 13, -2, 3, -2], [6, 8, -2, -1, 4, -1, -7, -11, 4, 1], [-8, 1, -3, -10, -7, 1, 3, 3, -1, 0], [0, 5, -2, 8, 7, 1, -12, 2, -1, 5], [15, -7, 8, 13, 0, -2, -4, 5, 8, 0], [-7, -5, 1, 2, 0, 3, 10, -4, -9, 5], [-12, 12, 0, 7, -1, 1, 0, 3, 5, 1], [11, -1, -3, -3, 4, -9, 1, -2, -9, 4], [8, 3, -1, 4, 3, -1, -18, 3, 4, 1], [-1, 5, -9, -15, 8, -5, 10, -5, -2, 7], [5, 0, -3, 8, 5, -2, -6, 2, 6, 8], [-2, 0, 1, -2, -5, -1, 0, 3, 6, -11], [-14, 1, -2, 17, -9, 5, -1, 6, 3, 0], [-2, -4, 3, 2, 1, -5, -9, 8, 16, 1], [-19, 4, -4, 4, -3, 9, 3, 1, -8, 0], [0, 6, 6, 5, -4, -3, -1, 6, 6, 1], [-12, -1, -6, -3, 0, -1, -7, 4, 0, 8], [-4, 0, 5, 11, -3, 6, 3, 9, 0, -2], [-6, 0, 2, -6, -1, 4, 5, -24, -15, -3], [4, 0, 4, 12, -9, 1, -10, 0, 6, 4], [13, -11, 3, 2, 1, -3, -9, 6, 5, 4], [-6, 5, -4, -6, -8, 5, 6, -4, -5, -5], [2, 5, -2, -8, 8, -4, 15, -10, 0, 2], [19, 2, -5, 4, 7, -4, -12, 9, 7, 4], [-22, 7, -3, -29, 5, 2, 15, -18, -14, 5], [22, 3, 2, 10, 9, -3, -23, 6, 11, 7], [-11, -2, -2, -7, -3, 10, 17, -22, -27, -1], [-8, 10, -1, -6, 6, -8, 1, 10, 11, 1], [-3, 5, -1, -7, -8, 10, 7, -4, -3, -7], [-25, 0, -7, -10, 5, 1, 0, -13, -1, 2], [-3, -2, 9, -2, -8, -3, -14, 15, 11, -5], [18, -7, -2, 16, -8, -10, -4, 18, 14, 2], [-9, 2, -7, -7, -1, 2, -8, 3, -7, -1], [-7, -2, -3, -9, 2, 5, 4, -14, 11, 1], [6, -5, -2, 17, 0, 1, -6, 7, 6, -5], [-2, -11, -2, 11, -2, -1, 0, -9, -11, 3], [-2, 5, -3, -8, 6, -1, -9, -3, -2, 0], [-5, -6, 5, -1, -16, 6, 13, -4, -1, -9], [10, -11, -4, 9, 2, 1, -3, -8, -7, 0], [-13, 5, -5, -10, 2, 5, 3, 6, -8, -2], [3, -4, 0, -8, -2, -5, 8, -4, -5, -3], [6, -2, 2, -7, -7, 6, 1, 7, -3, -4], [-22, 5, 6, -6, -2, 8, -3, -8, -6, -7], [-23, 9, -1, -13, -4, 6, 0, -2, -6, -1], [17, -10, -2, 16, 9, -2, -5, 9, 5, 5], [-5, -4, -10, -6, 0, 4, 7, -12, -1, 4], [-16, -4, 3, -9, -3, 2, 14, -15, -16, 2], [4, 4, 4, -4, -1, 0, -1, 5, -16, -2], [-13, 4, 2, -9, 0, 9, 6, 7, 1, -2], [18, -6, 3, 17, -2, -6, -1, 5, 17, -1], [-20, 0, -2, -9, 2, -1, -2, 8, -9, 5], [2, -1, 0, -5, 3, 7, -5, -5, -12, 0], [-15, -1, 7, -12, -9, 2, 9, -19, -8, -12], [-6, -1, -1, 5, 7, -3, -22, 1, 6, 3], [-1, -6, 2, 6, 1, 5, -8, -5, 6, -1], [-15, 4, 1, -8, 0, -2, -9, 11, 1, -8], [14, -7, 0, 1, -4, -3, 2, 10, -1, 3], [3, -13, 4, 2, 5, -1, -9, -11, -2, 0], [-2, 14, -8, -13, 4, -3, 17, -11, -10, 1], [-3, 5, 0, -16, -4, 4, 0, 5, 12, -4], [-18, 3, -7, 1, 10, -1, 12, -9, -8, 4], [-9, -8, -9, -6, -4, 1, 13, -14, -13, 4], [-1, 2, 10, 5, -7, -2, -8, 7, -11, -3], [2, 12, -1, -13, 15, -3, 2, -13, -7, 4], [1, -9, -1, -3, 1, 1, -3, -15, 2, -2], [-5, 0, 1, -25, 7, -3, 26, -20, -18, 1], [-9, -1, -1, 15, -3, 3, -13, -1, 2, 4], [13, 0, 4, -4, 4, -8, -7, 16, 4, -1], [-2, -10, -4, 24, -8, -2, -6, 7, 12, 6], [10, 1, -1, -2, 15, -4, 1, -16, -2, 10], [-1, 3, -5, -19, 7, -8, -5, 4, 10, 2], [9, 4, 4, -3, -1, 6, -10, 15, 1, -5], [-13, 5, -3, -5, 6, 0, 11, -23, -2, 5], [9, -1, 0, -7, 2, -3, 5, 1, 5, 2], [30, -13, 0, 19, 8, -7, -28, 16, 18, 13], [5, 10, -6, 14, 2, -2, -4, 19, 2, -1], [-2, -7, 2, 1, 4, -1, 4, -8, 4, 0], [-22, 14, 1, -8, 1, 3, 12, 10, -13, -6], [-8, 10, -4, -8, -4, 6, 5, 2, -13, -6], [-18, 8, -2, -10, 8, -2, 13, -20, -15, -1], [-2, 7, 1, -7, 6, 3, -5, 5, 5, 6], [-3, -1, -7, -14, 7, -2, 10, -7, 5, 5], [-9, -11, -1, 7, -5, 1, 6, 6, 10, -6], [-1, 5, -2, 0, -3, 2, 3, 7, -3, -4], [-16, 8, -7, -30, 1, 7, 7, -18, -11, -6], [3, 2, -5, -17, 8, -4, -2, -18, -12, 11], [4, -8, 2, 18, -5, -2, 3, 15, 13, -8], [8, -8, -4, 8, 10, -11, -4, -15, 1, 9], [-26, 14, 0, 4, -4, 7, 15, -10, -10, -8], [-18, 6, 8, -14, 7, 1, 7, -4, 2, -5], [-13, 7, -2, -21, -4, 0, -1, -4, 0, -14], [-18, 10, 0, -8, 14, -5, 5, -8, 5, 10], [14, -8, 5, 22, -3, -1, 4, 13, 12, 2], [-13, 0, 6, -9, -6, -1, 10, -21, -7, -8], [8, -2, -4, -14, 8, -9, -9, -6, 2, 5], [-7, 0, 7, 3, -13, 5, 9, 14, 11, -11], [6, -10, 7, -6, 1, -5, -1, -11, 7, 6], [-3, 5, -5, 1, 0, 5, -11, -11, 8, -1], [-27, 10, -7, -23, 6, -3, 16, -17, -10, -2], [-2, 2, 10, 10, 0, -2, -3, -5, -8, 7], [11, 8, 0, 21, -9, 6, -13, 23, 14, -12], [-17, -8, -2, -2, 5, -4, 9, -3, -16, 4], [-17, -9, 4, 1, -16, 1, 10, 12, 6, -16], [-20, 2, 7, -8, -10, 13, 3, 3, 1, -9], [16, -5, 5, 8, 3, -4, -15, -2, 5, 8], [-27, -10, -2, -3, -1, 4, 17, -10, -2, -1], [-21, -1, 2, -6, -3, 0, 4, -26, -19, -5], [1, 8, -10, 5, 8, 2, -16, 10, 1, 9], [-17, 15, -3, -27, 10, 2, 5, -5, -13, -6], [3, -5, 1, 6, -3, -7, 25, 8, -7, 1], [0, 13, 1, 7, 1, -3, -18, 13, 3, -5], [3, 3, 3, 6, -3, 11, -17, 9, 19, 2], [11, -11, 2, -3, 2, -5, -6, -2, 8, 6], [24, -5, 7, 37, -2, 1, -16, 11, 26, 1], [4, 6, 0, -20, 11, -3, 10, -17, 11, 2], [0, -7, 10, 22, -1, -8, -15, 21, 21, 0], [-7, -3, 0, -9, 8, -5, 16, -10, 6, 5], [-2, -11, 5, -9, 6, -4, 7, -15, -4, 7], [17, 9, 0, 25, 3, -3, -20, 18, 26, 4], [-11, 5, 5, -15, 1, 1, -5, -5, 6, 1], [-22, 9, 1, 7, 4, 0, 5, -10, 1, 2], [21, 2, 7, 4, 0, 9, -17, 3, -2, 2], [6, 6, 3, -20, 6, -2, 15, -8, -12, 6], [-15, -5, 4, 2, -12, 4, -4, -1, -21, -10], [4, -1, 4, -4, 1, -1, -11, 15, 11, 5], [-3, -2, 0, -7, 2, -1, 4, -7, 3, 0], [9, 4, -9, 1, 10, -1, -5, -11, 12, 6], [2, -10, -3, 11, -4, 3, 3, 17, 1, 5], [-5, -3, -6, -19, 12, -4, 2, -15, 1, 11], [23, -8, -2, -2, 22, -8, -12, -9, 3, 7], [-2, -17, -2, 12, -7, 3, 9, -8, -8, 3], [7, 7, 3, 23, -4, 2, -19, 8, 10, -1], [20, -5, 1, 2, 18, -5, -1, 1, 17, 14], [-12, -1, -6, -16, 14, -2, 4, -17, -23, 4], [17, -2, 6, 10, -1, -2, 7, 9, 11, -11], [3, -1, -8, 11, -4, 10, -20, 15, 1, 3], [1, 0, 1, 7, -3, 3, 6, -4, 15, 4], [13, -15, 8, 15, -15, 4, -5, 8, -4, -4], [-18, 1, 4, 14, -12, 14, 5, 5, 20, -9], [9, 3, 0, 27, -7, -1, -10, 20, 3, -8], [19, -2, -1, 1, -1, -4, -14, 0, -4, -1], [1, -3, 5, 18, -6, -1, 4, 20, 14, -4], [-19, 0, 5, 12, -6, -2, -10, 7, -1, 1], [10, -2, 2, -11, 1, -2, -15, 20, -8, 4], [11, 7, -9, 4, 2, -4, -5, 21, 13, -6], [1, 1, -8, -6, 8, -14, 7, -3, -2, -2], [13, 16, 4, -8, -4, 3, -15, 13, 3, -5], [-15, 5, -5, -4, 8, -6, -6, -7, 26, 3], [1, -8, -3, 2, 7, -1, -2, -22, -4, 5], [13, 6, -5, 4, 5, -7, 10, 2, 9, 4], [-13, -5, -6, 4, 1, -4, 11, -3, -6, 6], [-13, -8, -2, -3, 1, -3, 4, 3, -21, -2], [-24, 16, -5, -31, 7, 7, 5, -24, -17, 0], [12, -10, -4, 6, 4, -5, 3, -1, -4, 6], [-2, -4, -1, 4, -1, -6, -3, 3, 1, 0], [14, -10, -10, 20, 11, -5, 2, 3, -3, 17], [-7, 9, -3, 0, 12, 1, -11, -8, -6, 6], [-3, -7, 8, 15, 7, -11, -8, 5, 0, 1], [-31, 10, 6, -16, 0, 4, 13, 7, -13, -8], [7, 4, -6, -7, -7, -4, 19, 7, -8, 3], [-29, 7, -4, -5, -4, 11, 22, -16, -16, -2], [5, -1, 10, 24, -1, -2, -3, 21, 16, -6], [-9, -13, 9, 16, -16, 6, 18, 1, -18, -2], [-11, 4, 0, 6, -1, 0, 4, 11, 3, 7], [28, -8, -7, 24, -3, -1, -9, 2, 13, 3], [-22, 3, 2, -2, -13, 6, 1, -10, -5, -7], [-10, 15, -5, -6, 8, -2, 5, 5, -5, -7], [-11, 0, 2, 6, -1, -6, 3, -6, 2, -2], [18, -11, 9, 11, -10, 7, -12, 1, 1, 1], [14, 2, -6, -15, 14, -7, -8, 11, 10, 4], [-2, -13, 6, 5, -7, -1, 11, 11, 8, -9], [-10, -4, -4, -9, -3, -8, 4, -15, -19, -1], [-31, 11, 1, -31, 0, -1, 10, -21, -30, -3], [-5, -3, 3, -16, -5, 6, 10, 2, -8, -14], [4, 12, 2, -5, 10, -3, -3, -1, 14, -3], [-13, -7, -3, -12, -13, 8, 16, 10, -25, -1], [6, 1, 5, 12, -6, -5, -22, 18, 10, -3], [2, -6, -5, -8, 14, -8, 5, -16, -3, 2], [-40, 15, -2, -14, -8, 11, 18, -22, -16, -13], [-14, 0, -7, -8, -4, 6, 4, -17, -14, -7], [-6, 1, -1, -13, -15, -1, 25, -5, -11, 0], [-14, 6, 2, 13, 3, -6, -24, 23, 25, -1], [10, 1, -9, -5, 17, -8, 8, 2, 12, 8], [-18, 4, -1, -9, -2, 9, 11, -6, 10, 5], [21, 1, 9, 9, -1, -11, -30, 23, 28, 4], [-4, 13, -10, 5, 3, 3, -8, -7, 3, 4], [-23, 18, 6, -13, -7, 3, 4, -10, -14, -4], [-48, 0, -4, 1, 4, 10, 6, -7, -21, 3], [-10, 2, -11, -20, 11, -7, 8, -17, -10, 11], [2, 8, -7, -14, -2, 5, 4, 9, 3, -2], [-29, 2, 2, -28, -11, 2, 37, -26, -18, -9], [12, 1, 7, 17, -6, 7, 0, 12, 4, -7], [-2, -5, -6, 14, -6, 2, -10, 23, 29, 1], [12, 15, -11, -12, 14, -10, -4, -2, 6, 6], [2, 4, -7, 2, 14, -5, -1, -7, -3, 16], [-3, 7, 4, 5, -7, -3, 9, 26, 10, -10], [9, -9, -7, 8, -7, -14, -7, 1, 10, 3], [-1, 3, 10, 7, -12, -1, -9, 23, 13, -11], [12, 11, -4, -12, 4, 2, -9, 6, 4, 0], [2, -2, -5, -3, 2, -4, 24, -10, -24, -4], [-23, -1, -3, -18, -1, 7, -2, -5, -13, 4], [-18, -1, -4, -3, -12, 7, 9, -19, -16, -2], [2, -1, -6, 3, 8, -15, 14, 1, 2, 2], [-13, 0, -7, 4, -1, -4, 32, -4, 3, -9], [19, 6, -13, 16, 4, -6, 7, -3, 14, -2], [9, -5, 0, 40, -12, 9, -20, -1, 1, -6], [13, 13, 2, 1, 4, 3, -11, -3, -6, 0], [3, -1, -1, 4, 0, -6, -3, 8, 11, 14], [-7, 19, -6, -25, -3, 4, 5, -2, -4, -3], [8, -13, 4, 14, -3, -2, -22, -9, 23, 3], [23, -5, 16, 41, -6, 1, -21, 21, 16, 0], [2, -2, 1, -17, 7, -1, -4, -7, -3, 0], [38, -9, 0, 6, -2, -6, 5, -2, 7, 4], [-17, -12, 5, 24, -20, 3, 5, 7, -1, -4], [-41, 10, -9, -30, 3, 15, 14, -31, -32, -7], [39, -14, 2, 36, 3, -4, -46, 24, 34, 8], [15, -13, 4, 9, 6, -10, -23, 13, 26, 10], [12, -5, -12, 11, -4, 0, -13, 3, 12, -7], [25, -9, 0, 13, 8, -4, 2, 13, -9, -2], [11, -3, 0, -13, -5, 6, 29, -13, -20, -14], [5, 1, -2, 18, 9, -5, -4, -10, -15, 5], [-12, -7, 8, -19, 9, -2, -1, -4, -5, -1], [-14, 12, -4, -8, -5, -2, -25, 19, 16, -9], [1, 3, -14, -8, 11, -12, -2, -15, -7, 9], [-2, -16, 13, 17, -18, 5, 1, 12, -16, -1], [-8, -22, -3, 18, -15, 4, 11, -7, -11, -1], [-2, 14, 5, -6, -5, -3, 5, 2, -3, -13], [-10, 9, -6, -22, 17, 4, 9, -30, -1, -5], [11, 10, -5, -8, 20, 0, -6, -9, 14, 3], [-1, 5, 9, -20, -2, 1, 23, -16, -7, -10], [-14, 17, 6, -16, -8, 8, -4, 4, -14, -10], [22, -4, -4, 10, 12, -15, -10, -1, 14, 13], [23, 5, 8, 3, 4, -4, -5, 7, 19, 3], [-3, -3, 2, -3, -3, 1, -4, -8, 14, -4], [-15, 15, 6, -3, -6, -1, -2, 23, 21, 1], [24, -4, 12, 14, 6, 3, -8, 28, 1, 2], [15, 5, -5, -6, 10, 4, 1, -25, -15, 14], [5, -2, 5, 1, -8, -5, -12, 9, -24, -3], [-12, -6, -4, 10, 0, 7, 11, -21, -17, 3], [-20, 0, 11, -11, -5, 10, 1, -1, -17, 2], [8, -4, -4, 16, 5, 1, -12, -1, -8, -2], [-21, -11, -5, -3, -4, -11, 1, -3, -3, 1], [-20, -6, -9, -4, 2, 1, 13, -16, -9, -6], [1, -11, 2, 28, -2, -2, 0, -2, 8, 6], [-16, -7, -9, -6, -11, 6, 16, 12, 7, 2], [11, 8, -7, -22, 4, 10, 15, -13, -19, -6], [4, -6, 11, -4, -8, 0, 5, -20, -14, -6], [2, -2, -4, -21, 1, 1, 21, -21, -18, 1], [-24, -5, 0, -28, -6, -1, 36, -20, -34, 0], [23, -11, 12, -9, 11, -3, 19, -2, 0, 3], [6, -5, -9, 7, 7, -4, 13, 3, -3, 5], [4, 0, 2, 13, 8, -2, 15, -6, 0, 12], [-1, -12, 1, 19, 8, -4, 14, -6, -4, 3], [5, -1, 11, 15, -13, 2, 0, 6, -3, -10], [44, 0, 5, 30, -8, 7, -19, 7, 30, 3], [-21, 4, 5, -17, 0, -4, 10, -15, -32, -4], [15, -2, 9, 27, 1, -2, -10, 34, -1, 6], [8, 5, 8, -2, 8, 4, -15, -13, 14, 10], [-26, -4, 4, 17, -11, -1, 2, -8, -24, -8], [-7, 5, -2, 0, -14, 10, -16, 28, 5, -19], [43, 5, 3, -11, 13, -1, -15, 18, 21, 6], [-8, -8, 4, -5, -2, -2, 25, 0, 8, -13], [-4, -3, 5, 2, 4, -8, -19, 2, -4, 10], [30, -10, -10, 16, 6, -7, 4, 15, 19, 8], [1, 2, 0, 28, 9, 9, -7, 2, -26, 6], [20, -9, -1, 36, 5, -4, -2, 18, 21, -5], [11, 2, 7, 2, 0, 7, -24, 2, 13, 7], [10, -8, 7, -5, -1, 1, 5, -18, -11, 6], [-15, 6, -7, -6, -5, -5, 16, 2, -25, -10], [23, -4, 3, -8, 3, -15, 9, 7, -9, 11], [-11, 21, -12, 0, 9, 6, -5, -4, -7, 7], [-24, 6, -7, -44, 5, -3, 14, -34, -9, 6], [27, -5, -3, 17, -8, -9, 9, 28, 5, -5], [4, 12, 12, -20, 9, 0, 11, 4, 18, -8], [-30, 9, -9, -16, 13, -2, 0, -20, -26, 8], [16, -7, -9, -10, 5, 5, -10, -25, -1, 13], [-7, 9, 5, 11, -8, 20, -7, -11, -16, -8], [-28, 1, -10, 11, -2, 4, -11, -2, 11, 0], [8, -7, -3, 24, -6, -2, 7, -28, 3, 6], [24, -8, 10, -3, 1, 0, 8, -12, -12, 0], [21, -9, 5, 16, -30, 5, -8, 23, 24, -15], [26, 6, -3, 16, 13, -3, -25, 2, 18, 8], [-2, 18, -4, -14, 14, 8, -13, -10, 21, 2], [-2, 10, -4, -9, -2, 4, -11, -12, -18, 12], [38, -2, 1, 30, 14, -13, -31, -1, 30, 9], [-7, 4, -8, -39, 17, 3, 6, -42, -14, -2], [13, 5, 7, -3, -1, -3, -10, 15, -6, -5], [27, -13, -2, 17, 1, -4, 1, -2, 10, 16], [16, -8, -14, -20, -10, 5, 14, 0, -14, -12], [8, 8, 4, -18, -1, 2, -8, 18, -1, 3], [-23, 6, -2, 6, 1, 2, 28, -32, -28, -3], [-33, 1, 1, -23, 5, 1, 0, -43, -39, 7], [20, -9, 7, -21, 0, -6, 13, -6, -7, -8], [37, -8, -8, 6, 10, -10, -10, -14, 6, 16], [9, 9, 5, 6, -10, 0, -8, 13, 0, -11], [6, 13, 10, -4, 15, -3, -17, 29, 16, -2], [11, 8, -4, -11, 12, -9, -7, -18, 16, 2], [-12, 16, -7, -14, 12, 1, 13, -3, 9, 10], [-19, 5, 5, -6, -7, 4, 9, 29, 0, -14], [33, -1, -2, 23, -1, 3, -21, 29, 12, -4], [24, 1, 8, -15, 18, -6, -3, -13, -19, 3], [-2, -5, 4, 5, -3, -3, -28, 34, 10, 3], [28, 2, -9, 19, 10, 7, -19, 24, 4, -2], [2, 20, -1, 2, -11, 12, 0, 17, 2, -10], [-16, 15, 1, -17, 10, 1, 6, -39, 12, -8], [72, -21, 10, 24, -3, -8, 1, 14, 25, 8], [-22, -1, 5, 18, -9, 15, -3, -17, -12, -1], [17, -7, 0, 32, -20, 4, -15, 24, 29, -9], [-20, 11, -4, -23, 2, 4, -4, -25, -4, 1], [-24, 22, 0, -44, 2, 9, 16, -27, -49, 1], [8, 3, 7, 1, -17, 13, 9, 1, -24, -10], [22, -2, 22, 33, -9, 1, -7, 21, 10, -3], [6, 4, 0, 16, -3, 0, 1, -11, 20, -2], [-14, 6, 15, 10, 1, -8, -13, 17, 1, -5], [10, -5, 19, 1, 4, 1, -9, 14, -9, 5], [13, -6, 9, 19, -27, 12, -10, 19, 14, -4], [-23, -5, 0, -15, -14, 13, 20, 5, -33, -11], [19, 4, -6, -10, 11, 4, 28, -1, 7, 2], [-10, 16, -9, -37, 17, -6, 1, -34, -6, 13], [-6, 2, 16, 10, -5, -3, -9, 17, 7, 3], [12, -8, -5, 26, -11, 0, -4, -6, 22, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6348_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6348_2_a_t();".
// 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_6348_2_a_t(:prec:=10)
    chi := MakeCharacter_6348_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_6348_2_a_t();".
// 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_6348_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6348_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![373, -1198, 172, 1528, -709, -457, 245, 52, -28, -2, 1]>,<7,R![23, 44, -1119, 33, 1493, -154, -455, 121, 23, -11, 1]>],Snew);
    return Vf;
end function;