
// Make newform 7168.2.a.bi in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7168_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_7168_2_a_bi();".
// 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_7168_2_a_bi();".
// 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 := [196, 0, -1640, 0, 2008, 0, -968, 0, 221, 0, -24, 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, 0, 0, 0], [0, -68, 0, -6, 0, 177, 0, -95, 0, 17, 0, -1], [0, -278, 0, 36, 0, 177, 0, -95, 0, 17, 0, -1], [16, 0, -116, 0, 79, 0, -16, 0, 1, 0, 0, 0], [0, 464, 0, -1154, 0, 772, 0, -207, 0, 24, 0, -1], [0, -830, 0, 1720, 0, -1117, 0, 316, 0, -41, 0, 2], [0, 264, 0, -426, 0, 322, 0, -115, 0, 18, 0, -1], [-26, 0, -232, 0, 286, 0, -113, 0, 18, 0, -1, 0], [22, 0, -244, 0, 286, 0, -113, 0, 18, 0, -1, 0], [-54, 0, 318, 0, -310, 0, 115, 0, -18, 0, 1, 0], [72, 0, -370, 0, 353, 0, -127, 0, 19, 0, -1, 0], [0, 152, 0, 4402, 0, -4776, 0, 1789, 0, -276, 0, 15]];
    Rf_basisdens := [1, 42, 42, 6, 84, 42, 12, 6, 6, 6, 6, 84];
    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_7168_a();"
function MakeCharacter_7168_a()
    N := 7168;
    order := 1;
    char_gens := [1023, 5125, 1025];
    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_7168_a_Hecke(Kf)
    return MakeCharacter_7168_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, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 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, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0], [0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0], [1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, 1, 0, 0, 0, 0, 0, 0, 0], [2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, -1, -1, 0, -2, 1, 1, 0, 0, 0, 0, -1], [2, 0, 0, -1, 0, 0, 0, 1, 1, 1, 0, 0], [0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1], [0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1], [0, -1, 0, 0, -2, -1, 0, 0, 0, 0, 0, 1], [0, 1, 2, 0, -1, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0], [-1, 0, 0, -1, 0, 0, 0, 0, -2, 1, -1, 0], [2, 0, 0, -1, 0, 0, 0, -1, 1, 0, -1, 0], [0, -1, -3, 0, -3, 1, 2, 0, 0, 0, 0, -1], [0, 0, 0, 3, 0, 0, 0, -1, -1, 0, -1, 0], [4, 0, 0, 2, 0, 0, 0, -1, 1, -1, 1, 0], [0, 2, -1, 0, 5, -1, -1, 0, 0, 0, 0, -1], [0, 0, 0, 3, 0, 0, 0, -1, 1, 3, -2, 0], [0, 0, -3, 0, -5, 0, 0, 0, 0, 0, 0, 1], [0, 2, -1, 0, -2, -1, 0, 0, 0, 0, 0, 1], [3, 0, 0, 1, 0, 0, 0, 1, 0, -2, -1, 0], [4, 0, 0, 0, 0, 0, 0, -2, 3, 2, 1, 0], [0, 3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [9, 0, 0, 3, 0, 0, 0, 1, -1, -1, 1, 0], [0, 1, 1, 0, 4, -1, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, 5, 0, -1, 0, 0, 0, 0, 1], [-4, 0, 0, 4, 0, 0, 0, -1, -2, 0, -1, 0], [0, 0, 1, 0, -5, -1, 1, 0, 0, 0, 0, 1], [0, 0, -3, 0, 3, 0, -2, 0, 0, 0, 0, -1], [0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0], [0, -2, 2, 0, -2, 0, -1, 0, 0, 0, 0, 0], [0, 3, 2, 0, 1, -2, -3, 0, 0, 0, 0, 0], [0, -3, -2, 0, 4, 3, 0, 0, 0, 0, 0, -1], [-2, 0, 0, 0, 0, 0, 0, 1, -3, 2, -2, 0], [8, 0, 0, 2, 0, 0, 0, 0, 1, 1, 2, 0], [0, 4, 5, 0, 4, -1, 0, 0, 0, 0, 0, 1], [5, 0, 0, -1, 0, 0, 0, 2, -2, -2, -2, 0], [0, 1, -3, 0, 0, 1, 1, 0, 0, 0, 0, 1], [-1, 0, 0, 2, 0, 0, 0, -1, -1, 1, 0, 0], [0, 1, -2, 0, 11, 0, 0, 0, 0, 0, 0, -1], [0, -4, -1, 0, 3, 1, 1, 0, 0, 0, 0, 1], [3, 0, 0, 1, 0, 0, 0, -3, -1, 1, -1, 0], [0, 0, 0, 2, 0, 0, 0, 0, 1, 4, 1, 0], [-3, 0, 0, 4, 0, 0, 0, -2, 0, 2, 1, 0], [0, -5, -1, 0, -3, -1, 2, 0, 0, 0, 0, -1], [13, 0, 0, -2, 0, 0, 0, -1, -1, 1, 0, 0], [-6, 0, 0, -1, 0, 0, 0, -2, -2, -2, 1, 0], [0, 2, 4, 0, -2, 2, -1, 0, 0, 0, 0, 0], [5, 0, 0, -4, 0, 0, 0, 1, 1, -3, 0, 0], [0, -5, -3, 0, 1, 2, 1, 0, 0, 0, 0, -3], [6, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [0, -3, 2, 0, -4, 4, 2, 0, 0, 0, 0, -2], [0, 2, 6, 0, 8, 0, -3, 0, 0, 0, 0, 0], [0, -3, -3, 0, -5, 1, -2, 0, 0, 0, 0, 1], [3, 0, 0, 2, 0, 0, 0, -3, 3, 3, 0, 0], [8, 0, 0, -3, 0, 0, 0, 0, 4, -1, 0, 0], [0, 4, 6, 0, 7, -4, -2, 0, 0, 0, 0, 2], [0, -9, -7, 0, -4, 1, 1, 0, 0, 0, 0, 1], [3, 0, 0, 5, 0, 0, 0, 1, -2, 4, -1, 0], [0, -3, 0, 0, 3, 2, 1, 0, 0, 0, 0, 2], [0, 1, 7, 0, 1, -2, -4, 0, 0, 0, 0, 2], [7, 0, 0, 2, 0, 0, 0, 1, 3, 1, -4, 0], [4, 0, 0, -2, 0, 0, 0, -1, 0, -4, -1, 0], [12, 0, 0, -2, 0, 0, 0, 0, -2, -2, -2, 0], [0, -3, 2, 0, 2, -3, 1, 0, 0, 0, 0, 0], [0, -1, 1, 0, 2, -1, 1, 0, 0, 0, 0, 3], [11, 0, 0, 1, 0, 0, 0, 4, 2, 0, 2, 0], [0, 0, -5, 0, -6, 5, 2, 0, 0, 0, 0, -3], [0, -9, -3, 0, -5, 2, 2, 0, 0, 0, 0, -2], [10, 0, 0, -4, 0, 0, 0, 0, 1, 1, 2, 0], [1, 0, 0, 3, 0, 0, 0, -3, -1, -4, 0, 0], [0, -1, 1, 0, 3, 1, 0, 0, 0, 0, 0, 3], [0, 5, -5, 0, -3, 2, 3, 0, 0, 0, 0, -1], [-18, 0, 0, 4, 0, 0, 0, -2, 1, 2, -3, 0], [4, 0, 0, 0, 0, 0, 0, -1, 3, 5, -1, 0], [6, 0, 0, -2, 0, 0, 0, 0, 2, 4, -2, 0], [0, -3, -5, 0, -4, -1, -3, 0, 0, 0, 0, 1], [3, 0, 0, -1, 0, 0, 0, 1, 1, -3, 1, 0], [-1, 0, 0, -3, 0, 0, 0, 3, 0, -2, -1, 0], [0, -5, 7, 0, 1, -4, 0, 0, 0, 0, 0, 0], [-6, 0, 0, 2, 0, 0, 0, 1, 5, 4, -2, 0], [0, -1, 4, 0, 13, -2, 0, 0, 0, 0, 0, 1], [4, 0, 0, -1, 0, 0, 0, -5, 1, -3, -4, 0], [-10, 0, 0, 8, 0, 0, 0, 3, -2, 4, 1, 0], [0, -9, 0, 0, -7, -2, -3, 0, 0, 0, 0, 0], [0, -6, -3, 0, -5, 0, 2, 0, 0, 0, 0, -1], [-6, 0, 0, 0, 0, 0, 0, 0, -6, -4, 0, 0], [0, 1, 3, 0, 7, -2, -2, 0, 0, 0, 0, 0], [8, 0, 0, 5, 0, 0, 0, -4, 2, 5, 0, 0], [0, 3, 0, 0, 13, 4, 0, 0, 0, 0, 0, 1], [0, -9, -3, 0, -7, -2, 1, 0, 0, 0, 0, -1], [0, -10, 2, 0, -2, 3, 2, 0, 0, 0, 0, -4], [0, 0, -2, 0, 5, 4, 2, 0, 0, 0, 0, 0], [0, -5, -3, 0, 0, 5, 0, 0, 0, 0, 0, 0], [7, 0, 0, 1, 0, 0, 0, -1, -4, 0, -3, 0], [0, -3, -3, 0, 0, -1, 3, 0, 0, 0, 0, -1], [1, 0, 0, 2, 0, 0, 0, -5, 1, -3, -2, 0], [0, -13, -9, 0, 4, 3, 2, 0, 0, 0, 0, -2], [15, 0, 0, -8, 0, 0, 0, -1, 3, -5, 2, 0], [-8, 0, 0, 0, 0, 0, 0, 4, 2, 0, 2, 0], [-8, 0, 0, 7, 0, 0, 0, 1, 1, 0, 3, 0], [1, 0, 0, -2, 0, 0, 0, -1, -3, -1, 2, 0], [0, 3, 3, 0, 9, -8, -5, 0, 0, 0, 0, 3], [14, 0, 0, -12, 0, 0, 0, 2, -1, -3, 0, 0], [0, 1, 7, 0, -13, -5, -4, 0, 0, 0, 0, -1], [-6, 0, 0, 2, 0, 0, 0, -4, 2, 0, -2, 0], [-6, 0, 0, -4, 0, 0, 0, -4, -3, -3, -2, 0], [0, 3, -2, 0, -7, -2, -4, 0, 0, 0, 0, 1], [-1, 0, 0, 5, 0, 0, 0, 4, -8, 2, -2, 0], [0, -1, -4, 0, 0, 5, 1, 0, 0, 0, 0, 2], [0, -4, 0, 0, -2, -1, -4, 0, 0, 0, 0, -2], [0, -2, -3, 0, 3, 1, 1, 0, 0, 0, 0, 1], [-16, 0, 0, 6, 0, 0, 0, 0, -2, 4, -2, 0], [0, 1, -1, 0, -9, -2, 0, 0, 0, 0, 0, 0], [0, 8, 11, 0, 11, -6, 0, 0, 0, 0, 0, 1], [0, -3, -1, 0, -5, -7, 0, 0, 0, 0, 0, 3], [0, -1, 1, 0, -1, 2, -1, 0, 0, 0, 0, -1], [0, -14, 2, 0, -7, 2, -2, 0, 0, 0, 0, 0], [5, 0, 0, 1, 0, 0, 0, -2, 0, 0, 4, 0], [-14, 0, 0, -5, 0, 0, 0, -3, 5, 1, 0, 0], [0, -1, -10, 0, -8, 9, 0, 0, 0, 0, 0, -1], [0, 2, -5, 0, 3, -4, 0, 0, 0, 0, 0, 1], [-2, 0, 0, -6, 0, 0, 0, 3, 4, -2, -1, 0], [12, 0, 0, -2, 0, 0, 0, 0, -1, 0, -1, 0], [0, 7, -2, 0, 0, 3, -3, 0, 0, 0, 0, 0], [-3, 0, 0, 1, 0, 0, 0, 3, -1, 0, 0, 0], [5, 0, 0, 2, 0, 0, 0, 6, -4, -4, 5, 0], [0, -1, 4, 0, -14, -1, 2, 0, 0, 0, 0, 1], [0, 1, 2, 0, 23, 0, 0, 0, 0, 0, 0, -1], [0, -12, 4, 0, 8, -4, 3, 0, 0, 0, 0, 0], [9, 0, 0, 1, 0, 0, 0, 5, -6, -4, 7, 0], [0, 3, 6, 0, -4, 0, -2, 0, 0, 0, 0, 2], [0, -11, 2, 0, -8, 1, -3, 0, 0, 0, 0, -2], [-2, 0, 0, 1, 0, 0, 0, 4, 6, 8, 3, 0], [0, -10, -1, 0, 1, 4, -2, 0, 0, 0, 0, -3], [0, -5, -7, 0, -9, 2, 0, 0, 0, 0, 0, -4], [1, 0, 0, 9, 0, 0, 0, -6, 2, 4, -2, 0], [0, 0, 1, 0, 1, -5, -5, 0, 0, 0, 0, -1], [12, 0, 0, 3, 0, 0, 0, 4, 2, 5, 2, 0], [0, 3, -8, 0, -2, 2, 0, 0, 0, 0, 0, -2], [6, 0, 0, -2, 0, 0, 0, -7, 3, 2, -6, 0], [0, 8, 3, 0, 14, -7, -2, 0, 0, 0, 0, 1], [7, 0, 0, 4, 0, 0, 0, 3, -3, -1, 0, 0], [0, 12, 12, 0, -2, -1, -2, 0, 0, 0, 0, 4], [8, 0, 0, 7, 0, 0, 0, 1, -3, -3, 2, 0], [0, -7, 0, 0, 15, -2, 3, 0, 0, 0, 0, 0], [-16, 0, 0, 4, 0, 0, 0, -2, -7, -2, -1, 0], [14, 0, 0, 3, 0, 0, 0, 0, -4, -4, 3, 0], [8, 0, 0, -8, 0, 0, 0, -3, 5, -3, -1, 0], [-8, 0, 0, 1, 0, 0, 0, -9, 5, 4, -9, 0], [0, 2, 2, 0, 12, 0, 1, 0, 0, 0, 0, 4], [0, -8, -10, 0, -8, 5, 4, 0, 0, 0, 0, 0], [14, 0, 0, -4, 0, 0, 0, -2, 2, 6, 0, 0], [-20, 0, 0, 4, 0, 0, 0, -5, 0, -2, -5, 0], [0, -1, -5, 0, 0, -3, 2, 0, 0, 0, 0, -2], [3, 0, 0, -1, 0, 0, 0, -1, 3, 7, -1, 0], [11, 0, 0, -5, 0, 0, 0, -4, 0, 0, -6, 0], [0, 0, 0, -7, 0, 0, 0, 3, 3, 4, -3, 0], [0, 5, 11, 0, 7, -6, -6, 0, 0, 0, 0, 2], [-16, 0, 0, 0, 0, 0, 0, 0, -1, -9, 4, 0], [0, 4, 2, 0, 4, 2, 5, 0, 0, 0, 0, 0], [0, 11, 6, 0, 19, 4, -3, 0, 0, 0, 0, 2], [0, -5, 0, 0, -8, 3, 6, 0, 0, 0, 0, 1], [-4, 0, 0, 5, 0, 0, 0, 0, 0, 4, -1, 0], [-12, 0, 0, 2, 0, 0, 0, 2, -4, 4, 0, 0], [12, 0, 0, 9, 0, 0, 0, -3, 1, 9, -4, 0], [9, 0, 0, 1, 0, 0, 0, -4, 8, 1, -3, 0], [0, 4, 3, 0, 17, -2, 0, 0, 0, 0, 0, -1], [0, 15, 5, 0, 7, 2, -1, 0, 0, 0, 0, 1], [5, 0, 0, 2, 0, 0, 0, 3, 1, -5, 2, 0], [0, 2, 12, 0, 6, -2, -7, 0, 0, 0, 0, 2], [2, 0, 0, -8, 0, 0, 0, 2, -5, -6, 7, 0], [0, -3, 5, 0, -20, -1, -2, 0, 0, 0, 0, -2], [0, -6, 2, 0, -1, 2, 2, 0, 0, 0, 0, -6], [4, 0, 0, 1, 0, 0, 0, 3, -3, -6, 1, 0], [0, 0, 0, -10, 0, 0, 0, -2, -1, 6, -1, 0], [0, 6, -4, 0, -12, 2, 1, 0, 0, 0, 0, -2], [0, 10, 8, 0, 11, -8, 0, 0, 0, 0, 0, 0], [0, 7, -1, 0, 18, 1, 4, 0, 0, 0, 0, 6], [-15, 0, 0, 3, 0, 0, 0, -3, -5, 3, 5, 0], [44, 0, 0, -1, 0, 0, 0, -1, 1, 1, 0, 0], [-10, 0, 0, 2, 0, 0, 0, -2, 4, 10, -2, 0], [0, 0, -6, 0, 8, 3, 2, 0, 0, 0, 0, -2], [0, -4, -14, 0, 6, 7, 6, 0, 0, 0, 0, 0], [0, -4, 1, 0, 7, -1, 3, 0, 0, 0, 0, 1], [0, -4, -6, 0, 2, 7, 2, 0, 0, 0, 0, -4], [13, 0, 0, -5, 0, 0, 0, -2, 10, -3, 3, 0], [-9, 0, 0, 1, 0, 0, 0, 1, 10, 4, 3, 0], [0, -5, 1, 0, 11, 2, -1, 0, 0, 0, 0, -1], [13, 0, 0, 2, 0, 0, 0, 3, 5, 3, -4, 0], [5, 0, 0, 2, 0, 0, 0, 1, -11, -1, 0, 0], [0, 8, 13, 0, 16, -1, 0, 0, 0, 0, 0, -3], [-18, 0, 0, 11, 0, 0, 0, 3, -3, 7, 8, 0], [0, 9, 4, 0, -14, -3, 0, 0, 0, 0, 0, -1], [-15, 0, 0, -4, 0, 0, 0, -1, 1, 9, 0, 0], [0, 3, -7, 0, -6, -3, 4, 0, 0, 0, 0, 2], [0, 0, 6, 0, 12, -2, 3, 0, 0, 0, 0, -6], [8, 0, 0, -5, 0, 0, 0, -7, -1, -7, -8, 0], [0, 15, 8, 0, 3, -6, -1, 0, 0, 0, 0, 0], [-6, 0, 0, 4, 0, 0, 0, 0, -7, 9, -4, 0], [0, -1, 9, 0, -1, 1, 2, 0, 0, 0, 0, -3], [2, 0, 0, 6, 0, 0, 0, 0, 1, -3, 4, 0], [0, -3, 12, 0, 0, -7, 0, 0, 0, 0, 0, 1], [-4, 0, 0, 0, 0, 0, 0, -1, -2, -4, 7, 0], [0, -1, -1, 0, -11, -1, 4, 0, 0, 0, 0, 5], [28, 0, 0, -10, 0, 0, 0, 4, 2, -4, 8, 0], [-21, 0, 0, 5, 0, 0, 0, -3, 5, 6, 4, 0], [-26, 0, 0, 2, 0, 0, 0, -1, 5, -4, 4, 0], [-13, 0, 0, 0, 0, 0, 0, 0, -10, 0, 1, 0], [-6, 0, 0, 10, 0, 0, 0, 2, -2, -2, -2, 0], [0, 11, 4, 0, 16, -3, 5, 0, 0, 0, 0, 2], [0, -15, -4, 0, -2, 7, 7, 0, 0, 0, 0, -4], [-28, 0, 0, -2, 0, 0, 0, 0, -6, 8, -4, 0], [-6, 0, 0, 10, 0, 0, 0, 4, -6, 4, 0, 0], [24, 0, 0, -2, 0, 0, 0, 0, -3, 0, 5, 0], [0, 1, 6, 0, 3, -4, -4, 0, 0, 0, 0, 7], [0, 9, 10, 0, -4, 1, -1, 0, 0, 0, 0, 4], [9, 0, 0, 7, 0, 0, 0, 6, -2, -1, 1, 0], [-18, 0, 0, 0, 0, 0, 0, 2, -3, 6, 5, 0], [-17, 0, 0, 12, 0, 0, 0, -5, -1, 3, -4, 0], [0, -2, 11, 0, -13, 4, 0, 0, 0, 0, 0, -3], [0, 1, 2, 0, 22, -3, 3, 0, 0, 0, 0, -2], [0, -11, -13, 0, 1, 3, 4, 0, 0, 0, 0, -3], [-22, 0, 0, -2, 0, 0, 0, -1, 1, 0, 4, 0], [-12, 0, 0, 0, 0, 0, 0, 0, 6, 4, -6, 0], [0, 9, 11, 0, 21, -7, -2, 0, 0, 0, 0, 3], [5, 0, 0, -4, 0, 0, 0, -7, -3, 5, -4, 0], [-12, 0, 0, -6, 0, 0, 0, 7, -5, -9, 3, 0], [0, -3, -5, 0, -15, 10, 3, 0, 0, 0, 0, -3], [0, -7, -12, 0, -7, 10, -1, 0, 0, 0, 0, -2], [-2, 0, 0, -12, 0, 0, 0, 0, 0, 4, -2, 0], [0, 13, 4, 0, -7, -8, -7, 0, 0, 0, 0, 0], [0, -7, -5, 0, -12, -3, -6, 0, 0, 0, 0, 2], [-7, 0, 0, 4, 0, 0, 0, -5, 3, -7, 0, 0], [0, -2, 3, 0, -10, -1, 2, 0, 0, 0, 0, 3], [2, 0, 0, 10, 0, 0, 0, -4, 6, -4, 0, 0], [0, 0, 0, -7, 0, 0, 0, -3, -1, -5, -4, 0], [-13, 0, 0, 0, 0, 0, 0, -7, 3, 5, -12, 0], [0, -11, 2, 0, 20, 4, 8, 0, 0, 0, 0, -4], [0, 2, -1, 0, -13, -2, -4, 0, 0, 0, 0, 5], [-20, 0, 0, -1, 0, 0, 0, -7, 5, 0, -1, 0], [0, 9, -6, 0, 14, 1, -3, 0, 0, 0, 0, 4], [-12, 0, 0, 10, 0, 0, 0, -4, 2, 4, 2, 0], [20, 0, 0, -8, 0, 0, 0, -2, 2, 2, 2, 0], [-13, 0, 0, -5, 0, 0, 0, 8, 0, -7, 1, 0], [0, -17, -6, 0, -10, 1, 2, 0, 0, 0, 0, 1], [0, 7, 2, 0, 11, 4, 9, 0, 0, 0, 0, -2], [0, 11, 13, 0, 23, -2, -1, 0, 0, 0, 0, 1], [0, -9, -5, 0, -18, 1, 8, 0, 0, 0, 0, -2], [0, 9, 15, 0, 1, -6, -4, 0, 0, 0, 0, 4], [-11, 0, 0, 5, 0, 0, 0, -5, 5, 4, -6, 0], [-6, 0, 0, -12, 0, 0, 0, 6, -1, -6, -5, 0], [0, -9, 1, 0, -22, 3, -3, 0, 0, 0, 0, 3], [0, 19, 6, 0, -2, 1, -4, 0, 0, 0, 0, -3], [0, -7, 6, 0, 2, 3, 2, 0, 0, 0, 0, -1], [13, 0, 0, -2, 0, 0, 0, -4, -2, 4, 1, 0], [0, -1, -9, 0, 2, -9, -4, 0, 0, 0, 0, 2], [0, -5, -1, 0, 15, -8, 1, 0, 0, 0, 0, -1], [-5, 0, 0, 3, 0, 0, 0, -2, 6, 3, -5, 0], [0, -10, 2, 0, 0, 9, 8, 0, 0, 0, 0, -4], [0, 2, 6, 0, -13, -2, -2, 0, 0, 0, 0, 0], [0, 6, 1, 0, 7, 9, 9, 0, 0, 0, 0, -1], [0, -17, -7, 0, -14, 1, -7, 0, 0, 0, 0, 3], [-20, 0, 0, -3, 0, 0, 0, -1, -1, 6, -1, 0], [20, 0, 0, 2, 0, 0, 0, 6, 1, -2, 5, 0], [-5, 0, 0, 0, 0, 0, 0, 1, -3, 1, 0, 0], [36, 0, 0, 2, 0, 0, 0, 2, -2, -2, 8, 0], [0, 14, 16, 0, 12, 1, -2, 0, 0, 0, 0, -2], [0, -4, -9, 0, -4, 1, -4, 0, 0, 0, 0, -3], [-24, 0, 0, 0, 0, 0, 0, -2, -1, -1, 4, 0], [0, 1, -11, 0, 5, 7, 4, 0, 0, 0, 0, 1], [12, 0, 0, 8, 0, 0, 0, 0, -14, -4, 2, 0], [-18, 0, 0, 17, 0, 0, 0, -4, 2, 4, -5, 0], [-13, 0, 0, 11, 0, 0, 0, -8, 14, 10, -6, 0], [0, 8, 2, 0, 16, 2, -5, 0, 0, 0, 0, 6], [0, -13, -9, 0, -3, -9, -4, 0, 0, 0, 0, 1], [2, 0, 0, 2, 0, 0, 0, 4, -5, -16, 5, 0], [-18, 0, 0, 0, 0, 0, 0, -8, -2, 0, -2, 0], [0, 2, 3, 0, -16, -1, -6, 0, 0, 0, 0, 3], [11, 0, 0, -19, 0, 0, 0, 2, 0, -8, -2, 0], [17, 0, 0, -6, 0, 0, 0, 3, 1, -7, 10, 0], [0, 4, -16, 0, 3, 2, -2, 0, 0, 0, 0, -2], [0, -19, -2, 0, 0, 0, 4, 0, 0, 0, 0, 4], [6, 0, 0, -2, 0, 0, 0, 2, 1, 3, 6, 0], [0, -11, -7, 0, 5, 3, 10, 0, 0, 0, 0, -1], [0, 5, 10, 0, 22, -7, 1, 0, 0, 0, 0, 6], [0, -3, 0, 0, 16, 9, -2, 0, 0, 0, 0, -1], [-1, 0, 0, 8, 0, 0, 0, 3, -1, 3, 0, 0], [0, 12, 5, 0, 5, -5, -7, 0, 0, 0, 0, 7], [0, 13, -5, 0, 11, -3, -8, 0, 0, 0, 0, -1], [0, 3, 14, 0, 18, -12, -2, 0, 0, 0, 0, 4], [0, 0, 0, 5, 0, 0, 0, 2, 0, -3, 12, 0], [0, 0, -4, 0, 1, -6, 0, 0, 0, 0, 0, 2], [-12, 0, 0, -14, 0, 0, 0, 3, 3, -2, 2, 0], [0, 5, -3, 0, -6, 3, -1, 0, 0, 0, 0, 1], [0, 12, 14, 0, 18, -3, -12, 0, 0, 0, 0, 6], [23, 0, 0, -11, 0, 0, 0, 11, -1, 3, 5, 0], [0, -21, -14, 0, 1, 6, 3, 0, 0, 0, 0, -2], [0, 9, -3, 0, -3, 4, 4, 0, 0, 0, 0, 0], [16, 0, 0, -2, 0, 0, 0, -11, 0, 6, -5, 0], [0, -3, 6, 0, 0, -7, 1, 0, 0, 0, 0, 0], [12, 0, 0, -12, 0, 0, 0, 6, 4, 6, 0, 0], [0, -20, -9, 0, -10, -3, 2, 0, 0, 0, 0, 3], [-4, 0, 0, -12, 0, 0, 0, 4, 2, -4, -8, 0], [0, -7, -5, 0, -14, 5, 3, 0, 0, 0, 0, 5], [-18, 0, 0, -4, 0, 0, 0, 0, 2, 0, -4, 0], [3, 0, 0, 3, 0, 0, 0, 0, 8, 9, 5, 0], [0, 15, 1, 0, -2, -9, 6, 0, 0, 0, 0, 0], [-36, 0, 0, 12, 0, 0, 0, -1, 5, 6, -6, 0], [-21, 0, 0, 18, 0, 0, 0, -9, 1, -1, 2, 0], [15, 0, 0, 3, 0, 0, 0, -3, 3, 13, -5, 0], [0, 1, 2, 0, 34, 0, -2, 0, 0, 0, 0, 0], [12, 0, 0, -6, 0, 0, 0, -8, 0, 10, -8, 0], [0, -16, -10, 0, -18, 2, -1, 0, 0, 0, 0, -8], [-8, 0, 0, 11, 0, 0, 0, -3, 7, 14, 3, 0], [-7, 0, 0, 5, 0, 0, 0, -7, -6, 8, -9, 0], [7, 0, 0, 10, 0, 0, 0, 12, -4, -2, 11, 0], [0, 8, -3, 0, 5, -2, 6, 0, 0, 0, 0, -7], [0, -19, 3, 0, -11, -3, 6, 0, 0, 0, 0, -1], [-2, 0, 0, -10, 0, 0, 0, 0, 5, -8, -1, 0], [0, -13, -9, 0, -31, -8, 7, 0, 0, 0, 0, -3], [0, -10, 13, 0, 10, -7, 2, 0, 0, 0, 0, 1], [0, 9, 5, 0, 29, 12, -3, 0, 0, 0, 0, -3], [-11, 0, 0, -7, 0, 0, 0, 2, 12, 2, -6, 0], [0, 11, -2, 0, 7, -6, -4, 0, 0, 0, 0, -1], [0, -18, -16, 0, -16, 9, 4, 0, 0, 0, 0, -4], [0, -17, -9, 0, -8, -3, 6, 0, 0, 0, 0, -2], [0, -12, -4, 0, -15, 2, 12, 0, 0, 0, 0, -6], [31, 0, 0, -8, 0, 0, 0, 5, 3, 5, -2, 0], [8, 0, 0, -1, 0, 0, 0, 1, 3, 14, -5, 0], [14, 0, 0, 5, 0, 0, 0, -5, 1, 5, -4, 0], [0, 10, 2, 0, -9, 2, 0, 0, 0, 0, 0, -6], [-3, 0, 0, 11, 0, 0, 0, -7, 2, 6, -1, 0], [0, -1, 3, 0, -9, 4, 4, 0, 0, 0, 0, -10], [-40, 0, 0, -8, 0, 0, 0, 6, 2, -2, 2, 0], [0, -1, 7, 0, 19, -12, -5, 0, 0, 0, 0, 5], [0, 11, 3, 0, 14, -5, 1, 0, 0, 0, 0, -1], [0, -1, -10, 0, 18, 1, 6, 0, 0, 0, 0, 3], [0, -17, 5, 0, -6, -3, -3, 0, 0, 0, 0, 3], [-7, 0, 0, -10, 0, 0, 0, -3, -3, 7, -4, 0], [0, -13, -12, 0, -6, -1, -4, 0, 0, 0, 0, -3], [0, -15, -19, 0, -23, 5, 8, 0, 0, 0, 0, 1], [8, 0, 0, 14, 0, 0, 0, 0, 0, 0, 2, 0], [0, -14, -6, 0, 15, -2, -4, 0, 0, 0, 0, -2], [-4, 0, 0, 17, 0, 0, 0, 3, -3, -3, -2, 0], [0, 4, 11, 0, 22, -7, 4, 0, 0, 0, 0, 7], [8, 0, 0, -9, 0, 0, 0, 1, -7, -4, -5, 0], [-5, 0, 0, -1, 0, 0, 0, 0, -4, 0, -10, 0], [0, -7, 8, 0, -7, -10, 2, 0, 0, 0, 0, 3], [7, 0, 0, 11, 0, 0, 0, -3, 1, 9, -3, 0], [2, 0, 0, -4, 0, 0, 0, 1, 6, 4, -13, 0], [0, 1, -1, 0, 9, 10, -5, 0, 0, 0, 0, -7], [8, 0, 0, -1, 0, 0, 0, 1, 15, 6, -1, 0], [40, 0, 0, -14, 0, 0, 0, 2, 1, -2, 5, 0], [0, 1, 4, 0, -27, 2, 7, 0, 0, 0, 0, -2], [0, -21, -8, 0, 5, 2, 4, 0, 0, 0, 0, 1], [14, 0, 0, 0, 0, 0, 0, 10, 5, 7, 0, 0], [0, -16, 6, 0, 0, 2, -9, 0, 0, 0, 0, 4], [8, 0, 0, -6, 0, 0, 0, -5, -10, -14, -1, 0], [-19, 0, 0, -1, 0, 0, 0, -3, -6, -4, -7, 0], [0, 16, 26, 0, 18, -13, -6, 0, 0, 0, 0, 8], [0, -3, -4, 0, -11, -4, -14, 0, 0, 0, 0, -1], [6, 0, 0, 5, 0, 0, 0, -3, -3, 6, 5, 0], [0, -15, -10, 0, -10, 5, -7, 0, 0, 0, 0, 0], [15, 0, 0, -10, 0, 0, 0, 5, -9, -3, 2, 0], [0, 11, -3, 0, 7, -10, -7, 0, 0, 0, 0, 7], [0, 17, 3, 0, 10, -5, -10, 0, 0, 0, 0, 0], [-8, 0, 0, 2, 0, 0, 0, -8, 9, -4, 1, 0], [30, 0, 0, -4, 0, 0, 0, 4, -14, -14, 4, 0], [2, 0, 0, 2, 0, 0, 0, -11, 3, 9, -11, 0], [0, 0, 0, -5, 0, 0, 0, 8, 2, 3, 4, 0], [0, 15, -3, 0, 1, -14, -8, 0, 0, 0, 0, 8], [9, 0, 0, 7, 0, 0, 0, -3, -8, 6, 1, 0], [-26, 0, 0, -7, 0, 0, 0, 0, 4, 0, -7, 0], [19, 0, 0, -15, 0, 0, 0, 1, -5, 1, -7, 0], [0, 19, 15, 0, 8, -1, 0, 0, 0, 0, 0, 8], [54, 0, 0, -10, 0, 0, 0, 4, -10, -6, -4, 0], [-36, 0, 0, 11, 0, 0, 0, 1, 3, 2, -5, 0], [0, 3, 12, 0, -8, -5, -2, 0, 0, 0, 0, -1], [0, -3, -21, 0, -42, 13, 5, 0, 0, 0, 0, -5], [37, 0, 0, -9, 0, 0, 0, 0, -4, 0, 4, 0], [-3, 0, 0, 1, 0, 0, 0, -1, -14, -4, 3, 0], [0, 13, -15, 0, 5, 0, 6, 0, 0, 0, 0, -4], [0, 4, -19, 0, 5, -4, 4, 0, 0, 0, 0, -1], [0, -7, -12, 0, -10, -2, 6, 0, 0, 0, 0, 0], [12, 0, 0, 0, 0, 0, 0, 1, -10, -4, -3, 0], [-3, 0, 0, 9, 0, 0, 0, 1, 11, 7, 3, 0], [7, 0, 0, 9, 0, 0, 0, 4, 4, 12, 0, 0], [15, 0, 0, -9, 0, 0, 0, 11, -7, 0, -2, 0], [0, 10, 8, 0, 28, -1, -4, 0, 0, 0, 0, 0], [0, 13, -12, 0, -10, -1, -1, 0, 0, 0, 0, 0], [0, 8, 16, 0, 12, 8, 5, 0, 0, 0, 0, -2], [6, 0, 0, 20, 0, 0, 0, -12, 2, 0, -6, 0], [18, 0, 0, -14, 0, 0, 0, 1, 1, -8, -2, 0], [38, 0, 0, -3, 0, 0, 0, 2, -8, -11, 6, 0], [0, -18, -1, 0, 1, 15, -1, 0, 0, 0, 0, -3], [-4, 0, 0, 11, 0, 0, 0, 3, 5, 19, -2, 0], [0, -15, -17, 0, 3, 18, 7, 0, 0, 0, 0, -3], [-2, 0, 0, -8, 0, 0, 0, -2, -1, -9, 4, 0], [0, -3, -1, 0, -7, 3, 12, 0, 0, 0, 0, -3], [0, -3, 8, 0, -9, 6, 6, 0, 0, 0, 0, 5], [-18, 0, 0, 4, 0, 0, 0, -6, 0, -8, -2, 0], [0, -7, 7, 0, -1, -2, 5, 0, 0, 0, 0, 5], [0, 7, 4, 0, 1, -8, -5, 0, 0, 0, 0, 2], [0, -4, 0, 0, -8, 5, -2, 0, 0, 0, 0, -2], [-5, 0, 0, 3, 0, 0, 0, -3, -17, -15, -3, 0], [0, 15, 10, 0, -34, -3, -6, 0, 0, 0, 0, 3], [-54, 0, 0, -5, 0, 0, 0, 1, -3, 0, -3, 0], [34, 0, 0, -5, 0, 0, 0, -5, 3, -17, -6, 0], [40, 0, 0, 10, 0, 0, 0, 5, -9, -13, 9, 0], [-20, 0, 0, 4, 0, 0, 0, -12, 10, 6, -4, 0], [0, -18, -10, 0, 3, -2, 6, 0, 0, 0, 0, 4], [0, 9, -3, 0, -17, 4, 0, 0, 0, 0, 0, -10], [-20, 0, 0, -3, 0, 0, 0, 1, -7, 8, -3, 0], [0, 13, -6, 0, 24, 4, 4, 0, 0, 0, 0, 0], [39, 0, 0, -15, 0, 0, 0, -1, 3, -10, -8, 0], [0, 13, 1, 0, 7, 4, 2, 0, 0, 0, 0, -2], [0, 1, 12, 0, 25, 2, -11, 0, 0, 0, 0, 0], [10, 0, 0, 8, 0, 0, 0, 0, 4, 8, -2, 0], [0, 13, 5, 0, 3, 5, -10, 0, 0, 0, 0, 5], [9, 0, 0, 9, 0, 0, 0, 6, -4, -12, 4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7168_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7168_2_a_bi();".
// 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_7168_2_a_bi(:prec:=12)
    chi := MakeCharacter_7168_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_7168_2_a_bi();".
// 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_7168_2_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7168_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![64, 0, -384, 0, 772, 0, -632, 0, 196, 0, -24, 0, 1]>,<5,R![2304, 0, -10176, 0, 9892, 0, -3688, 0, 580, 0, -40, 0, 1]>,<11,R![541696, 0, -569344, 0, 190080, 0, -28736, 0, 2148, 0, -76, 0, 1]>,<13,R![3211264, 0, -2563072, 0, 694020, 0, -82856, 0, 4548, 0, -112, 0, 1]>,<17,R![-96, -288, 40, 200, -44, -4, 1]>,<23,R![-3184, 2656, 340, -320, -36, 8, 1]>,<31,R![64, -96, -24, 72, -16, -4, 1]>],Snew);
    return Vf;
end function;