
// Make newform 676.2.f.i in Magma, downloaded from the LMFDB on 08 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_676_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_676_f_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_676_2_f_i();".
// 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_676_2_f_i();".
// 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 := [256, -512, 320, -64, 80, -64, -48, 64, -36, 32, -12, -8, 5, -2, 5, -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, 0, 0], [-832, -1056, -240, 992, 448, -344, -324, -184, 128, 134, 29, -56, -39, 14, 9, 4], [1152, 2112, -2592, -1360, 544, 1072, 408, -412, -184, -172, 146, 163, -12, -49, -26, 19], [-1216, -2976, 1744, 800, 480, -872, -612, 264, 80, 266, -87, -106, -7, 12, 37, -10], [1024, -2976, 2160, 336, 384, -792, -420, 492, -48, 198, -105, -105, 21, 9, 39, -25], [1600, -2240, 640, 96, 672, -40, -520, 152, -124, 166, 24, -57, -6, -21, 28, -7], [-1600, 2944, -640, -96, -672, 40, 520, -152, 124, -166, -24, 57, 6, 21, -28, 7], [-4224, -8960, 7808, 3344, 1152, -3104, -1472, 1132, 360, 808, -368, -355, 6, 77, 100, -59], [3200, -6592, 3744, 368, 992, -1136, -952, 876, -160, 420, -150, -191, 32, 13, 78, -47], [16384, -20160, 4640, -80, 4384, -176, -3288, 1492, -1120, 1100, 146, -393, 28, -109, 206, -81], [-6336, 10400, -3984, -176, -1840, 680, 1564, -892, 492, -622, 39, 239, -27, 33, -109, 51], [-19200, 25088, -4288, -624, -6496, 352, 4432, -1316, 1296, -1496, -300, 481, 82, 153, -272, 81], [-24064, 33792, -10624, 112, -7296, 1504, 5616, -2780, 1712, -2000, -92, 747, -50, 179, -376, 147], [13568, -18432, 4160, -80, 4544, -480, -3072, 1252, -960, 1068, 112, -365, -6, -109, 204, -73], [19712, -26784, 7792, -176, 6144, -920, -4404, 2092, -1424, 1574, 87, -577, 21, -143, 295, -113], [55808, -65344, 11168, -1840, 17344, 368, -10872, 4140, -4088, 3788, 714, -1183, -64, -483, 726, -247]];
    Rf_basisdens := [1, 704, 1408, 704, 704, 352, 352, 1408, 704, 1408, 704, 1408, 1408, 704, 704, 1408];
    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_676_f();"
function MakeCharacter_676_f()
    N := 676;
    order := 4;
    char_gens := [339, 509];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_676_f_Hecke();"
function MakeCharacter_676_f_Hecke(Kf)
    N := 676;
    order := 4;
    char_gens := [339, 509];
    char_values := [[-1, 0, 0, 0, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [1, -1, -1, 1, -1, -1, 1, 1, -1, 0, 1, 0, 0, 0, 1, 0], [1, -1, -1, 0, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 3, 0, 0, 0, -1, 1, 0, -1, 1, -1, 0, 0], [0, 0, -2, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, -2, 0, 1], [0, 0, -3, 0, -1, 0, 0, 1, -1, 0, 1, 0, -3, -1, -1, 1], [1, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, -3, -2, -1, -2, 0, -1, 0, 2, 0, 0, -1, 3, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, -1, 4, 0, -1, 2, 0, 1, 0, 0, 2, -1, -1], [-1, 1, -2, -1, -2, 1, 2, 2, 1, 0, 2, 0, 0, 3, -1, 0], [-1, -1, -2, 0, 0, 0, 1, 0, -2, 0, 0, 0, 2, 2, 0, 0], [-1, 1, 3, -1, 3, 1, -3, -3, 5, 0, -3, 0, 4, -2, -1, 0], [1, -2, -3, 0, 0, 0, 2, 0, -3, 0, 0, 0, 3, 3, 0, 0], [0, 0, 0, 1, -2, 0, -2, 0, 3, 0, 2, 0, -3, 0, -1, 3], [2, -2, -2, 2, 1, 2, 1, 0, 1, 0, -1, 0, 1, 0, -2, -1], [-1, -2, 0, 0, -1, 2, 2, 0, 1, 0, -2, 0, 1, 2, 0, 0], [0, 0, 0, 0, 0, 0, 4, 2, -4, 0, 0, 0, 0, 4, 0, -2], [2, -2, 0, 5, 3, 2, 3, 0, 2, 0, -3, 0, 0, 2, -5, 0], [-3, 3, 0, 0, -4, -3, -3, 0, 0, -1, 3, 0, -3, -3, 0, 0], [0, 1, 3, 0, 0, 1, -1, 0, 1, 0, -1, -5, 0, -2, 0, 0], [-1, 0, 3, 0, 5, 4, 0, 0, 1, 1, -4, -1, 3, 1, 0, 0], [0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 3, 0, 0], [2, -2, 2, 7, 0, 0, 1, -2, 2, 0, 0, 0, -2, -2, 0, 2], [-5, -2, 0, 0, 5, -2, 2, 0, -2, 0, 2, -6, 0, -2, 0, 0], [3, 2, -1, 0, 0, 0, -2, 0, -1, 0, 0, 0, 1, 1, 0, 0], [0, 0, 3, 0, 5, 0, 0, -1, 1, 0, -5, 0, 3, 1, -3, -1], [0, 0, 3, 6, 0, 0, -2, -4, 5, 0, 0, 0, -3, -5, 0, 4], [-2, 3, 0, 0, 1, -3, -3, 0, -1, 3, 3, 0, -2, -3, 0, 0], [-2, 2, -1, -3, 0, 0, -5, -2, 5, 0, 0, 0, 1, -5, 0, 2], [1, -2, -2, 0, -1, -2, 2, 0, -2, 0, 2, 5, 0, 0, 0, 0], [-1, 1, 0, 1, -2, 1, 2, 0, 1, 0, 2, 0, 0, 1, 1, 0], [3, 3, 3, 0, 0, 0, -3, 0, 3, 4, 0, 4, -3, -3, 0, 0], [1, -1, -3, -5, 3, -1, -3, 3, -5, 0, -3, 0, -4, 2, -5, 0], [-1, 1, -3, -2, -2, 1, 2, 3, 1, 0, 2, 0, 0, 4, -2, 0], [-5, 0, -4, 0, 0, -3, 0, 0, 1, 5, 3, -5, -4, 1, 0, 0], [0, 0, -4, 0, -9, -6, 0, 3, -4, 0, 9, 0, -4, -4, 9, 3], [2, 0, -8, 0, -13, -5, 0, 0, 3, -2, 5, 2, -8, 3, 0, 0], [-6, 6, 4, 1, 0, 0, -5, -6, 8, 0, 0, 0, -4, -8, 0, 6], [4, -4, 0, 0, -5, 4, 4, 0, -4, -9, -4, 0, 8, 4, 0, 0], [0, -2, -5, 0, 0, -2, 2, 0, -2, 0, 2, 1, 0, 3, 0, 0], [0, 0, 1, 0, 5, -2, 0, -1, -3, 0, -5, 0, 1, -3, -1, -1], [-6, 6, 2, 3, 0, 0, 5, -2, 2, 0, 0, 0, -2, -2, 0, 2], [0, 0, 6, -1, 4, 0, 4, 0, -5, 0, -4, 0, 5, 6, 1, -5], [0, 0, 4, 2, 1, 0, 1, 0, 1, 0, -1, 0, -1, 4, -2, 1], [9, -2, 0, 0, 7, 2, 2, 0, -2, -2, -2, 0, 4, 2, 0, 0], [-10, 0, 0, 0, 14, 0, 0, 0, 0, 10, 0, -10, 0, 0, 0, 0], [0, 0, -4, -3, -7, 0, -7, 0, 4, 0, 7, 0, -4, -4, 3, 4], [-2, -3, 0, 0, -9, 3, 3, 0, -5, -7, -3, 0, 8, 3, 0, 0], [0, 0, 11, 0, 7, 0, 0, -5, 1, 0, -7, 0, 11, 1, 3, -5], [11, 0, -6, 0, -9, 2, 0, 0, 8, -11, -2, 11, -6, 8, 0, 0], [-10, 10, 5, -11, 0, 0, -9, -4, 3, 0, 0, 0, -5, -3, 0, 4], [0, 1, -3, 0, 0, 0, -1, 0, -3, 10, 0, 10, 3, 3, 0, 0], [-1, 1, -3, 4, -8, 1, 8, 3, -5, 0, 8, 0, -6, 4, 4, 0], [-1, 0, 3, 0, -5, 0, 0, 0, -3, 1, 0, -1, 3, -3, 0, 0], [-8, -1, 0, 0, -8, 1, 1, 0, -6, 0, -1, 0, 7, 1, 0, 0], [0, 0, 12, 0, 9, 4, 0, -5, 6, 0, -9, 0, 12, 6, -3, -5], [-4, -1, 0, 0, 7, 1, 1, 0, -7, 11, -1, 0, 8, 1, 0, 0], [1, -1, -8, -5, -2, -1, 2, 8, -7, 0, 2, 0, -6, 7, -5, 0], [0, 0, -9, 0, -12, -4, 0, 4, -5, 0, 12, 0, -9, -5, 8, 4], [8, -10, -2, 0, 0, 0, 10, 0, -2, -10, 0, -10, 2, 2, 0, 0], [0, 7, 3, 0, 0, 7, -7, 0, 7, 0, -7, -2, 0, 4, 0, 0], [-1, 1, 4, 0, 6, -1, 6, 0, -6, 0, -6, 0, 5, 3, 0, -5], [6, 0, -3, 0, 1, 5, 0, 0, 8, -6, -5, 6, -3, 8, 0, 0], [6, -6, -4, 3, 0, 0, 9, 6, -8, 0, 0, 0, 4, 8, 0, -6], [-13, -1, 0, 0, 4, 1, 1, 0, 2, 17, -1, 0, -1, 1, 0, 0], [1, 5, 18, 0, -1, 5, -5, 0, 5, 0, -5, -1, 0, -13, 0, 0], [8, -8, -4, 7, -3, -8, 3, 4, -10, 0, 3, 0, -2, -4, 7, 0], [0, 0, -4, 1, 0, 0, 13, 6, -8, 0, 0, 0, 4, 8, 0, -6], [-15, -2, 4, 0, 0, 0, 2, 0, 4, 2, 0, 2, -4, -4, 0, 0], [-4, 4, 10, 1, 8, -4, 8, 0, -7, 0, -8, 0, 3, 6, -1, -3], [1, -1, -4, 4, -6, 1, -6, 0, 10, 0, 6, 0, -9, -3, -4, 9], [12, 0, 8, 0, -21, 5, 0, 0, -3, -12, -5, 12, 8, -3, 0, 0], [-3, 2, 0, 0, 4, -2, -2, 0, 3, 7, 2, 0, -5, -2, 0, 0], [1, 3, 0, 0, -8, -3, -3, 0, 9, -9, 3, 0, -12, -3, 0, 0], [-7, -2, 0, 0, 7, -2, 2, 0, -2, 0, 2, -5, 0, -2, 0, 0], [4, -4, -8, -7, 0, 0, 5, 8, -8, 0, 0, 0, 8, 8, 0, -8], [12, -5, -9, 0, -12, -5, 5, 0, -5, 0, 5, 0, 0, 4, 0, 0], [3, -3, -2, -5, -5, 3, -5, 0, 0, 0, 5, 0, 3, 1, 5, -3], [-3, 0, -4, 0, 7, 0, 0, 0, 4, 3, 0, -3, -4, 4, 0, 0], [0, 0, 2, 0, 5, -12, 0, 1, -8, 0, -5, 0, 2, -8, 5, 1], [16, -16, -8, 6, 0, 0, 10, 10, -12, 0, 0, 0, 8, 12, 0, -10], [9, -2, 0, 0, 8, 2, 2, 0, 3, -1, -2, 0, -1, 2, 0, 0], [10, 4, -6, 0, -10, 4, -4, 0, 4, 0, -4, 3, 0, 10, 0, 0], [-13, -2, 0, 0, 13, -2, 2, 0, -2, 0, 2, 3, 0, -2, 0, 0], [-6, 6, 4, 1, -3, 6, 3, -4, 6, 0, 3, 0, 0, 2, 1, 0], [0, 0, -10, 0, -6, -8, 0, 8, -2, 0, 6, 0, -10, -2, -2, 8], [0, 0, 11, 2, 9, 0, -9, -11, 8, 0, -9, 0, 8, -11, 2, 0], [-1, 1, -2, -8, 2, -1, 2, 0, -10, 0, -2, 0, 9, -3, 8, -9], [0, 0, -6, 0, -5, 0, 0, 3, 0, 0, 5, 0, -6, 0, -1, 3], [5, -5, 8, -1, 2, 5, 2, 0, -3, 0, -2, 0, 8, 13, 1, -8], [-8, 8, 5, -3, 0, 0, 1, -2, -1, 0, 0, 0, -5, 1, 0, 2], [-9, -1, 6, 0, 9, -1, 1, 0, -1, 0, 1, 1, 0, -7, 0, 0], [-15, 11, 10, 0, 0, 0, -11, 0, 10, 2, 0, 2, -10, -10, 0, 0], [-2, 2, 5, -3, 0, 0, -13, -6, 7, 0, 0, 0, -5, -7, 0, 6], [-7, 0, 0, 0, 3, 0, 0, 0, 3, 10, 0, 0, -3, 0, 0, 0], [-10, 10, 7, 6, 0, 0, 4, -6, 5, 0, 0, 0, -7, -5, 0, 6], [-9, 5, 0, 0, 0, -5, -5, 0, 3, 9, 5, 0, -8, -5, 0, 0], [0, 0, 2, 0, 1, 4, 0, -3, 0, 0, -1, 0, 2, 0, 1, -3], [1, 0, 2, 0, -30, 1, 0, 0, -1, -1, -1, 1, 2, -1, 0, 0], [0, 0, 4, 0, -8, -8, 0, 2, 0, 0, 8, 0, 4, 0, 12, 2], [18, 1, 1, 0, -18, 1, -1, 0, 1, 0, -1, 24, 0, 0, 0, 0], [7, -7, -4, 3, 5, 7, 5, 0, -2, 0, -5, 0, 9, 3, -3, -9], [-1, 0, 0, 0, 15, 0, 0, 0, -5, 16, 0, 0, 5, 0, 0, 0], [-2, 2, 5, -8, 0, 0, -14, -4, 3, 0, 0, 0, -5, -3, 0, 4], [-1, 0, 3, 0, 0, 0, 0, 0, 3, -14, 0, -14, -3, -3, 0, 0], [10, -10, 2, 15, 0, 0, 9, 2, -6, 0, 0, 0, -2, 6, 0, -2], [0, -2, -18, 0, 0, -2, 2, 0, -2, 0, 2, 5, 0, 16, 0, 0], [-10, 7, 16, 0, 10, 7, -7, 0, 7, 0, -7, -11, 0, -9, 0, 0], [-4, 4, 8, 9, -5, 4, 5, -8, 8, 0, 5, 0, 4, -4, 9, 0], [5, -5, -11, -1, -5, -5, 5, 11, -11, 0, 5, 0, -6, 6, -1, 0], [15, 0, 3, 0, -21, 8, 0, 0, 5, -15, -8, 15, 3, 5, 0, 0], [1, -1, -12, 4, -8, 1, -8, 0, 12, 0, 8, 0, -11, -11, -4, 11], [0, 0, -5, 0, -1, 10, 0, -3, -1, 0, 1, 0, -5, -1, -3, -3], [16, 1, -11, 0, 0, 0, -1, 0, -11, 6, 0, 6, 11, 11, 0, 0], [4, -4, 5, 3, 0, 0, -13, -6, 7, 0, 0, 0, -5, -7, 0, 6], [0, 6, 0, 0, 11, -6, -6, 0, -4, 11, 6, 0, -2, -6, 0, 0], [7, 0, -7, 0, -1, 0, 0, 0, 7, -7, 0, 7, -7, 7, 0, 0], [18, -4, -14, 0, 0, 0, 4, 0, -14, 10, 0, 10, 14, 14, 0, 0], [0, 0, -1, 9, -10, 0, 10, 1, -6, 0, 10, 0, -6, 1, 9, 0], [-7, 7, -6, -13, -11, -7, -11, 0, -2, 0, 11, 0, -5, -13, 13, 5], [4, 0, -8, 0, -10, -8, 0, 0, 0, -4, 8, 4, -8, 0, 0, 0], [-7, 10, 0, 0, -14, -10, -10, 0, 12, -7, 10, 0, -22, -10, 0, 0], [0, 0, -12, 0, -15, 0, 0, 11, 10, 0, 15, 0, -12, 10, -7, 11], [0, 0, 16, 0, 8, 4, 0, -8, 4, 0, -8, 0, 16, 4, 4, -8], [-13, -2, 13, 0, 13, -2, 2, 0, -2, 0, 2, -18, 0, -15, 0, 0], [7, -7, 1, 8, 0, -7, 0, -1, -7, 0, 0, 0, 0, -8, 8, 0], [3, -3, -8, 7, -5, 3, -5, 0, 12, 0, 5, 0, -9, -5, -7, 9], [0, 0, -10, 0, -13, -8, 0, 5, -8, 0, 13, 0, -10, -8, 11, 5], [-22, 3, 3, 0, 0, 0, -3, 0, 3, 6, 0, 6, -3, -3, 0, 0], [7, -4, 0, 0, -6, 4, 4, 0, 5, -13, -4, 0, -1, 4, 0, 0], [-6, -10, -3, 0, 6, -10, 10, 0, -10, 0, 10, -13, 0, -7, 0, 0], [14, 3, 15, 0, -14, 3, -3, 0, 3, 0, -3, 13, 0, -12, 0, 0], [0, 0, 13, 3, 10, 0, -10, -13, 8, 0, -10, 0, 8, -13, 3, 0], [-3, 0, 8, 0, 2, 11, 0, 0, 3, 3, -11, -3, 8, 3, 0, 0], [-3, 8, 15, 0, 0, 0, -8, 0, 15, -4, 0, -4, -15, -15, 0, 0], [-3, 3, -4, -1, -4, -3, -4, 0, 3, 0, 4, 0, -6, -7, 1, 6], [-4, 8, 7, 0, 4, 8, -8, 0, 8, 0, -8, 1, 0, 1, 0, 0], [0, 0, 2, 0, -9, 0, 0, -1, 0, 0, 9, 0, 2, 0, 11, -1], [-4, 4, 0, -1, -3, 4, 3, 0, -2, 0, 3, 0, -6, 4, -1, 0], [-9, 0, 5, 0, -1, -4, 0, 0, -9, 9, 4, -9, 5, -9, 0, 0], [6, -6, -7, 5, -6, -6, 6, 7, 4, 0, 6, 0, 10, 1, 5, 0], [14, -5, 0, 0, 14, 5, 5, 0, 6, 0, -5, 0, -1, 5, 0, 0], [-4, 0, 8, 0, 13, 9, 0, 0, 1, 4, -9, -4, 8, 1, 0, 0], [-2, 2, -1, -2, 0, 0, 24, 12, -23, 0, 0, 0, 1, 23, 0, -12], [-4, 4, 12, -5, 11, -4, 11, 0, -16, 0, -11, 0, 12, 8, 5, -12], [-10, 3, -4, 0, 10, 3, -3, 0, 3, 0, -3, 13, 0, 7, 0, 0], [-5, 0, 8, 0, 25, 0, 0, 0, -8, 5, 0, -5, 8, -8, 0, 0], [0, 0, 3, 0, 0, 0, 0, 4, 11, 0, 0, 0, 3, 11, -8, 4], [8, -8, 3, 11, 0, 0, -1, -2, 1, 0, 0, 0, -3, -1, 0, 2], [0, 0, -24, 0, -11, -12, 0, 17, -2, 0, 11, 0, -24, -2, -11, 17], [6, -6, -2, -2, 0, 0, 8, 8, -14, 0, 0, 0, 2, 14, 0, -8], [12, -12, -14, 9, 0, 0, 9, 6, 2, 0, 0, 0, 14, -2, 0, -6], [1, 2, -14, 0, -1, 2, -2, 0, 2, 0, -2, -9, 0, 16, 0, 0], [-9, 5, 1, 0, 0, 0, -5, 0, 1, -6, 0, -6, -1, -1, 0, 0], [3, 0, 8, 0, -3, 0, 0, 0, 0, 0, 0, -8, 0, -8, 0, 0], [0, 0, -9, -4, -5, 0, 5, 9, -8, 0, 5, 0, -8, 9, -4, 0], [1, 0, 0, 0, 12, 1, 0, 0, 1, -1, -1, 1, 0, 1, 0, 0], [-9, 9, -2, -7, -2, -9, -2, 0, -5, 0, 2, 0, -4, -11, 7, 4], [-6, 6, 6, 3, 0, 0, 17, 4, -14, 0, 0, 0, -6, 14, 0, -4], [24, 0, -4, 0, -24, 0, 0, 0, 0, 0, 0, 15, 0, 4, 0, 0], [7, -7, -4, 9, -6, -7, 6, 4, -11, 0, 6, 0, -4, -3, 9, 0], [-8, 8, 6, -1, 0, 0, 11, 2, -10, 0, 0, 0, -6, 10, 0, -2], [9, -14, -5, 0, 0, 0, 14, 0, -5, -20, 0, -20, 5, 5, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_676_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_676_2_f_i();".
// 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_676_2_f_i(:prec:=16)
    chi := MakeCharacter_676_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_676_2_f_i();".
// 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_676_2_f_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_676_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![52, 0, 112, 0, 65, 0, 14, 0, 1]>,<5,R![4, -24, 72, 0, 5, -18, 18, -6, 1]>,<7,R![43264, 0, 0, 0, 361440, 0, 0, 0, 28745, 0, 0, 0, 486, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;