
// Make newform 2352.2.k.i in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2352_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2352_k_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_2352_2_k_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_2352_2_k_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 := [6561, -13122, 13851, -10206, 5265, -1296, -846, 1332, -962, 444, -94, -48, 65, -42, 19, -6, 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], [-19683, -34992, 38880, -41067, 28890, -846, -7392, 8138, -3546, 1288, 258, -662, 363, -142, 30, 11], [150903, -195372, 244944, -199989, 114858, -19998, -30276, 38170, -24594, 7988, -462, -2374, 1965, -962, 246, 1], [-54675, 173502, -184680, 144099, -72198, 16074, 14538, -21322, 13134, -5162, 306, 1042, -963, 512, -162, 35], [113724, -147987, 126846, -70227, 23193, 5787, -14727, 11651, -5787, 1483, 429, -737, 489, -238, 75, -22], [-113724, 129762, -136323, 98010, -58185, 19359, 6351, -13421, 10815, -5437, 1251, 563, -729, 565, -234, 61], [754515, -1106622, 862164, -496287, 167832, 29736, -96318, 81940, -42648, 12350, 2040, -5176, 3567, -1742, 660, -131], [75087, -105948, 95580, -60237, 24354, 42, -10040, 9482, -5482, 1912, 74, -542, 433, -226, 82, -19], [-255879, 374220, -331290, 197181, -85716, 1116, 33490, -32084, 18728, -6358, -100, 1772, -1463, 784, -302, 61], [244215, -310554, 266814, -157923, 55692, 8184, -30032, 25460, -13408, 4048, 572, -1544, 1105, -562, 178, -37], [-4617, 7452, -7803, 5247, -2499, 349, 661, -779, 497, -195, 17, 41, -38, 21, -8, 2], [-365229, 581742, -554526, 356805, -151794, 5526, 57756, -54982, 32070, -11048, -90, 3166, -2511, 1268, -468, 107], [125388, -181521, 162810, -104409, 41427, 681, -17341, 16161, -9161, 2997, 175, -939, 731, -378, 137, -30], [465831, -650754, 579150, -357399, 141174, -786, -59656, 56962, -31994, 11108, 142, -3082, 2537, -1364, 512, -125], [140697, -202662, 190512, -124497, 55620, -5868, -18226, 19032, -11588, 4386, -236, -1020, 917, -498, 188, -45], [-993627, 1397736, -1239624, 796365, -332046, 3594, 129248, -125070, 72382, -24984, -326, 7074, -5725, 3042, -1090, 243]];
    Rf_basisdens := [1, 69984, 69984, 34992, 17496, 34992, 69984, 7776, 23328, 11664, 432, 23328, 5832, 23328, 5832, 23328];
    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_2352_k();"
function MakeCharacter_2352_k()
    N := 2352;
    order := 2;
    char_gens := [1471, 1765, 785, 2257];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2352_k_Hecke();"
function MakeCharacter_2352_k_Hecke(Kf)
    N := 2352;
    order := 2;
    char_gens := [1471, 1765, 785, 2257];
    char_values := [[1, 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, 0, 0, 0], [-1, 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, 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+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, 0, 0, 0], [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, 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, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, -1], [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, 0, 0, 1, 0, -1, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, -1, 0, 0], [0, -1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1], [-1, 0, 0, -1, 0, -2, 0, 0, 0, 0, 2, -1, 0, -1, 0, 0], [0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, -1, 0, 1, 0, 0, 0, 0, -1, -1, 0, -1, 0, 0], [0, -1, 1, 0, 1, 0, 0, 0, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, -1, 1, 0], [0, -1, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1], [1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, 0, -1, 0], [0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1], [3, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 0, -1, 0, 0], [0, -2, 0, 0, -1, 0, 0, -3, 1, 2, 0, 0, 1, 0, 0, 1], [0, 2, -2, 0, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, -1], [0, -1, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, -2, 0, 0, 1], [0, -2, 0, 0, -1, 0, 0, -3, 2, 2, 0, 0, 1, 0, 0, 1], [0, -2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 2, -1, 0, 0, 0, 2, 2, 0, -2, 3, 0], [5, 0, 0, 1, 0, -2, 0, 0, 0, 0, 2, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 3, -2, 0, -2, 0, 0], [0, -3, 1, 0, 0, 0, 0, -3, -1, 1, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 1, -2, 0, 0, 0, 1, -2, 0, 2, 0, 0], [0, -3, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -2], [0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0], [8, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 5, 0, 0, 3, 0, 0, 0, 0, -3, 0, 0, -1], [-2, 0, 0, 0, 0, 2, 0, 0, 0, 0, -2, -1, 0, -1, 0, 0], [0, -2, -1, 0, 1, 0, 0, -1, -2, -3, 0, 0, -2, 0, 0, -2], [0, 0, -1, 0, -2, 0, 0, -2, -2, 3, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, 1, -1, 0], [0, -3, 0, 0, 6, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, -2, 2, 0, 0, 0, -2, -3, 0, 3, -1, 0], [5, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0, -2, -1, 0], [0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 2], [0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 0, -1], [0, -3, -3, 0, 0, 0, 0, -3, 1, -3, 0, 0, -3, 0, 0, -3], [0, -1, 0, 0, -1, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 3], [0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -2, 6, 0, 0, 0, -2, -2, 0, 2, 0, 0], [0, 3, 0, 0, 7, 0, 0, -3, 0, 0, 0, 0, -1, 0, 0, 0], [0, 4, -2, 0, -1, 0, 0, 3, 1, 0, 0, 0, -1, 0, 0, -1], [0, 4, 3, 0, 0, 0, 0, 4, 0, 3, 0, 0, 3, 0, 0, 3], [0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 2, 5, 0, -5, 2, 0], [0, -2, 1, 0, 1, 0, 0, -1, -3, -1, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 6, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 3], [3, 0, 0, -3, 0, 2, 0, 0, 0, 0, -2, -3, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, 0, 2, 1, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 2], [0, 2, -1, 0, -2, 0, 0, 0, -3, 3, 0, 0, 1, 0, 0, 1], [0, 3, 0, 0, 10, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 2], [0, -5, 2, 0, 2, 0, 0, -3, -2, -2, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, -1, 0, 0, -4, 0, 0, 0, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, -3, -1, 0, 0, 0, -3, -3, 0, 3, -2, 0], [-9, 0, 0, -1, 0, -3, 0, 0, 0, 0, 3, 2, 0, 2, 0, 0], [-7, 0, 0, 2, 0, -5, 0, 0, 0, 0, 5, 2, 0, 2, 0, 0], [0, 0, 0, 0, 0, -2, 6, 0, 0, 0, -2, -3, 0, 3, 1, 0], [0, 4, 0, 0, -2, 0, 0, -4, 0, 0, 0, 0, -2, 0, 0, 0], [0, 4, 3, 0, 2, 0, 0, 6, 10, -1, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, -4, 0, 0, 0, 0, -4, -3, 0, 3, -4, 0], [0, 3, 0, 0, 8, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 1], [5, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 0], [9, 0, 0, -3, 0, 1, 0, 0, 0, 0, -1, 3, 0, 3, 0, 0], [0, 7, 1, 0, -1, 0, 0, 6, 0, 3, 0, 0, 2, 0, 0, 2], [0, 0, 0, 0, 0, 5, -2, 0, 0, 0, 5, 6, 0, -6, 5, 0], [0, -5, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -5], [0, 0, 0, 0, 0, -3, 4, 0, 0, 0, -3, -2, 0, 2, -1, 0], [0, 10, 0, 0, 3, 0, 0, -8, 0, 0, 0, 0, -2, 0, 0, 2], [0, 0, 4, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 2], [1, 0, 0, 1, 0, -1, 0, 0, 0, 0, 1, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, -2, 2, 0, 0, 0, -2, -3, 0, 3, -5, 0], [0, 3, 0, 0, 4, 0, 0, -1, 0, 0, 0, 0, 3, 0, 0, 2], [0, -4, 0, 0, 4, 0, 0, -1, 0, 0, 0, 0, -4, 0, 0, -5], [0, 0, 0, 0, 0, -2, 3, 0, 0, 0, -2, -6, 0, 6, -3, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -6, 0, 6, 1, 0], [-9, 0, 0, -4, 0, -4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0], [0, 2, 2, 0, 2, 0, 0, 4, 4, -2, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 3, 0, 1, 0, 0, 0, 0, -1, -1, 0, -1, 0, 0], [0, -5, -1, 0, 1, 0, 0, -4, 4, -3, 0, 0, -2, 0, 0, -2], [0, 11, 1, 0, -1, 0, 0, 10, 6, 3, 0, 0, 2, 0, 0, 2], [9, 0, 0, 4, 0, 5, 0, 0, 0, 0, -5, -3, 0, -3, 0, 0], [0, 0, 0, 0, 0, -4, -3, 0, 0, 0, -4, -1, 0, 1, -2, 0], [-11, 0, 0, -1, 0, -1, 0, 0, 0, 0, 1, -2, 0, -2, 0, 0], [0, 4, 0, 0, -2, 0, 0, 2, -6, 4, 0, 0, 2, 0, 0, 2], [0, -4, 0, 0, 2, 0, 0, -2, -1, -4, 0, 0, -2, 0, 0, -2], [0, -2, 2, 0, 1, 0, 0, -1, -6, 0, 0, 0, 1, 0, 0, 1], [0, 2, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, -7, 0, 0, -4], [-1, 0, 0, -3, 0, -4, 0, 0, 0, 0, 4, 3, 0, 3, 0, 0], [-20, 0, 0, -2, 0, -6, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, -1, 0, 0, 0, -3, 3, 0, -3, 2, 0], [0, -1, 4, 0, 1, 0, 0, 0, -5, 2, 0, 0, 3, 0, 0, 3], [0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 4, 0], [-13, 0, 0, -1, 0, -5, 0, 0, 0, 0, 5, -2, 0, -2, 0, 0], [0, 8, 0, 0, -3, 0, 0, -6, 0, 0, 0, 0, 2, 0, 0, 2], [0, 2, -3, 0, 2, 0, 0, 4, 1, -7, 0, 0, -5, 0, 0, -5], [0, -4, 0, 0, -1, 0, 0, -5, 0, 2, 0, 0, 1, 0, 0, 1], [0, 0, 0, 0, 0, -2, -4, 0, 0, 0, -2, -1, 0, 1, 2, 0], [0, 6, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 5, 0, 0, 5], [0, 9, 0, 0, -2, 0, 0, -6, 0, 0, 0, 0, 5, 0, 0, 3], [20, 0, 0, 4, 0, 7, 0, 0, 0, 0, -7, -2, 0, -2, 0, 0], [0, 0, 0, 0, 0, -2, 8, 0, 0, 0, -2, -6, 0, 6, -4, 0], [0, -2, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, 7, 0, 0, -4], [14, 0, 0, -3, 0, -2, 0, 0, 0, 0, 2, -1, 0, -1, 0, 0], [0, 0, 0, 0, 0, 5, -4, 0, 0, 0, 5, 8, 0, -8, 11, 0], [0, -1, 0, 0, -2, 0, 0, -5, 0, 0, 0, 0, -1, 0, 0, -6], [0, 7, -4, 0, -4, 0, 0, 3, -2, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, -5, 0, 0, 0, 5, 1, 0, -1, 4, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, -4, 0, 4, 1, 0], [0, -1, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, -1], [-9, 0, 0, 2, 0, 9, 0, 0, 0, 0, -9, 2, 0, 2, 0, 0], [0, 0, 3, 0, 1, 0, 0, 1, 9, 1, 0, 0, 2, 0, 0, 2], [0, 0, 0, 0, 0, 2, -11, 0, 0, 0, 2, 4, 0, -4, -3, 0], [0, -6, 0, 0, -10, 0, 0, 10, 0, 0, 0, 0, -1, 0, 0, 4], [0, 0, 0, 0, 0, -4, 3, 0, 0, 0, -4, -1, 0, 1, 1, 0], [4, 0, 0, 8, 0, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0], [0, 9, 4, 0, -2, 0, 0, 7, -4, 8, 0, 0, 6, 0, 0, 6], [0, -8, 0, 0, -10, 0, 0, 7, 0, 0, 0, 0, 1, 0, 0, -1], [0, 3, 0, 0, -7, 0, 0, -2, 0, 0, 0, 0, 2, 0, 0, 1], [3, 0, 0, -7, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 4, -4, 0, 4, -2, 0], [19, 0, 0, -2, 0, 5, 0, 0, 0, 0, -5, 3, 0, 3, 0, 0], [18, 0, 0, -2, 0, -2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, -8, 3, 0, 4, 0, 0, -4, 0, -5, 0, 0, -1, 0, 0, -1], [0, 0, 0, 0, -8, 0, 0, -3, 0, 0, 0, 0, -5, 0, 0, -3], [0, -10, 4, 0, 3, 0, 0, -7, 2, -2, 0, 0, 1, 0, 0, 1], [0, -1, 0, 0, -16, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 8, -6, 0, -3, 0, 0, 5, -3, 0, 0, 0, -3, 0, 0, -3], [0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 1, -4, 0, 4, 3, 0], [0, -2, 0, 0, 16, 0, 0, 4, 0, 0, 0, 0, -2, 0, 0, 2], [0, 0, 0, 0, 0, 1, 9, 0, 0, 0, 1, -5, 0, 5, 1, 0], [2, 0, 0, -2, 0, 4, 0, 0, 0, 0, -4, -3, 0, -3, 0, 0], [0, 0, 0, 0, 0, -8, 7, 0, 0, 0, -8, -3, 0, 3, -5, 0], [0, 5, 0, 0, -3, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 7], [0, 4, 2, 0, 0, 0, 0, 4, 6, 2, 0, 0, 2, 0, 0, 2], [0, -5, 0, 0, -8, 0, 0, -3, 0, 0, 0, 0, -3, 0, 0, -8], [0, -2, 2, 0, 1, 0, 0, -1, -6, 0, 0, 0, 1, 0, 0, 1], [0, 5, 0, 0, -2, 0, 0, -7, 0, 0, 0, 0, -2, 0, 0, -2], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -2, -5, 0, 5, -6, 0], [-16, 0, 0, 2, 0, -5, 0, 0, 0, 0, 5, -4, 0, -4, 0, 0], [0, -8, 7, 0, 5, 0, 0, -3, 6, -3, 0, 0, 2, 0, 0, 2], [21, 0, 0, 1, 0, 3, 0, 0, 0, 0, -3, 1, 0, 1, 0, 0], [0, -9, -6, 0, -2, 0, 0, -11, 4, -2, 0, 0, -4, 0, 0, -4], [3, 0, 0, 5, 0, 5, 0, 0, 0, 0, -5, 6, 0, 6, 0, 0], [0, 0, 0, 0, 0, 2, 4, 0, 0, 0, 2, 8, 0, -8, -1, 0], [11, 0, 0, -1, 0, -3, 0, 0, 0, 0, 3, -10, 0, -10, 0, 0], [0, 2, -5, 0, 0, 0, 0, 2, 2, -5, 0, 0, -5, 0, 0, -5], [0, -2, 0, 0, 14, 0, 0, -1, 0, 0, 0, 0, -5, 0, 0, -3], [0, 6, 2, 0, -4, 0, 0, 2, -3, 10, 0, 0, 6, 0, 0, 6], [0, 0, 0, 0, 0, -4, 6, 0, 0, 0, -4, 3, 0, -3, -4, 0], [0, 0, 0, 0, 0, -4, 4, 0, 0, 0, -4, 0, 0, 0, -5, 0], [-3, 0, 0, -4, 0, -1, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0], [0, 9, 4, 0, 1, 0, 0, 10, 11, 2, 0, 0, 3, 0, 0, 3], [0, 0, 0, 0, 0, -3, -8, 0, 0, 0, -3, -4, 0, 4, -10, 0], [0, -7, -5, 0, -1, 0, 0, -8, 10, -3, 0, 0, -4, 0, 0, -4], [-9, 0, 0, 2, 0, -13, 0, 0, 0, 0, 13, -1, 0, -1, 0, 0], [0, -1, 0, 0, -1, 0, 0, 10, 0, 0, 0, 0, -2, 0, 0, 9], [3, 0, 0, -4, 0, -7, 0, 0, 0, 0, 7, -4, 0, -4, 0, 0], [0, -14, -1, 0, 1, 0, 0, -13, -5, -3, 0, 0, -2, 0, 0, -2], [0, 0, 0, 0, 0, 2, 7, 0, 0, 0, 2, -10, 0, 10, -2, 0], [0, -6, 0, 0, -14, 0, 0, 8, 0, 0, 0, 0, 2, 0, 0, 2], [0, 0, 0, 0, 0, 2, -4, 0, 0, 0, 2, 3, 0, -3, 4, 0], [-15, 0, 0, 0, 0, -2, 0, 0, 0, 0, 2, -4, 0, -4, 0, 0], [0, 11, 0, 0, 10, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0, 9], [0, 10, 0, 0, -4, 0, 0, 6, 0, 8, 0, 0, 4, 0, 0, 4], [-21, 0, 0, 7, 0, -5, 0, 0, 0, 0, 5, -7, 0, -7, 0, 0], [0, 0, 0, 0, 0, 1, -3, 0, 0, 0, 1, -3, 0, 3, 3, 0], [0, -2, 0, 0, -4, 0, 0, 4, 0, 0, 0, 0, -2, 0, 0, 2], [0, 5, 0, 0, -9, 0, 0, -2, 0, 0, 0, 0, 12, 0, 0, 3], [5, 0, 0, 8, 0, 5, 0, 0, 0, 0, -5, 3, 0, 3, 0, 0], [0, 2, -5, 0, -2, 0, 0, 0, -13, -1, 0, 0, -3, 0, 0, -3], [1, 0, 0, -9, 0, -3, 0, 0, 0, 0, 3, -5, 0, -5, 0, 0], [0, 14, 5, 0, 0, 0, 0, 14, 0, 5, 0, 0, 5, 0, 0, 5], [0, 0, 0, 0, 0, -2, 4, 0, 0, 0, -2, -1, 0, 1, 8, 0], [0, 6, 7, 0, 1, 0, 0, 7, 3, 5, 0, 0, 6, 0, 0, 6], [-24, 0, 0, -6, 0, 3, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, -10, 0, 0, -4, 0, 0, 14, 0, 0, 0, 0, -3, 0, 0, 4], [21, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 5, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 11, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 4], [0, 0, 0, 0, 0, 4, 5, 0, 0, 0, 4, 7, 0, -7, -1, 0], [-10, 0, 0, -8, 0, -8, 0, 0, 0, 0, 8, -7, 0, -7, 0, 0], [0, 2, 3, 0, 4, 0, 0, 6, 6, -5, 0, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 2, 8, 0, -8, 1, 0], [0, -8, -6, 0, -3, 0, 0, -11, -4, 0, 0, 0, -3, 0, 0, -3], [-17, 0, 0, -1, 0, -6, 0, 0, 0, 0, 6, -5, 0, -5, 0, 0], [4, 0, 0, 6, 0, -5, 0, 0, 0, 0, 5, -4, 0, -4, 0, 0], [0, -14, -4, 0, 0, 0, 0, -14, 0, -4, 0, 0, -4, 0, 0, -4], [0, -7, 5, 0, 3, 0, 0, -4, -10, -1, 0, 0, 2, 0, 0, 2], [0, 0, 0, 0, 0, -3, 9, 0, 0, 0, -3, -11, 0, 11, -8, 0], [0, -5, 0, 0, -18, 0, 0, 3, 0, 0, 0, 0, 3, 0, 0, -2], [0, -6, 0, 0, 11, 0, 0, -3, 0, 0, 0, 0, -13, 0, 0, -9], [0, 0, 0, 0, 1, 0, 0, -5, 0, 0, 0, 0, 5, 0, 0, -5], [0, 0, -6, 0, -4, 0, 0, -4, -14, 2, 0, 0, -2, 0, 0, -2], [0, 2, 7, 0, 5, 0, 0, 7, 3, -3, 0, 0, 2, 0, 0, 2], [0, -1, 0, 0, 30, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0], [0, 0, 0, 0, 0, -8, 1, 0, 0, 0, -8, -2, 0, 2, -4, 0], [0, 0, 0, 0, 0, -4, -6, 0, 0, 0, -4, -2, 0, 2, -4, 0], [0, -11, 0, 0, -16, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, -10], [-23, 0, 0, -3, 0, 8, 0, 0, 0, 0, -8, 1, 0, 1, 0, 0], [0, -4, -5, 0, 0, 0, 0, -4, 1, -5, 0, 0, -5, 0, 0, -5], [-14, 0, 0, -1, 0, -2, 0, 0, 0, 0, 2, -7, 0, -7, 0, 0], [0, -7, 2, 0, 3, 0, 0, -4, -13, -4, 0, 0, -1, 0, 0, -1], [0, -3, 7, 0, 5, 0, 0, 2, 0, -3, 0, 0, 2, 0, 0, 2], [0, -5, 0, 0, 1, 0, 0, 10, 0, 0, 0, 0, -2, 0, 0, 5], [16, 0, 0, 5, 0, 4, 0, 0, 0, 0, -4, 8, 0, 8, 0, 0], [0, -4, -4, 0, -1, 0, 0, -5, -4, -2, 0, 0, -3, 0, 0, -3], [0, 0, 0, 0, 0, -6, 12, 0, 0, 0, -6, 3, 0, -3, -5, 0], [0, 0, 0, 0, 0, -2, -6, 0, 0, 0, -2, 0, 0, 0, -7, 0], [-3, 0, 0, -9, 0, -6, 0, 0, 0, 0, 6, 1, 0, 1, 0, 0], [0, -13, 0, 0, -4, 0, 0, 17, 0, 0, 0, 0, -5, 0, 0, 4], [0, 0, 0, 0, 0, 3, 4, 0, 0, 0, 3, 0, 0, 0, -7, 0], [20, 0, 0, -3, 0, -4, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0], [0, -10, 3, 0, 4, 0, 0, -6, -11, -5, 0, 0, -1, 0, 0, -1], [-10, 0, 0, 2, 0, 6, 0, 0, 0, 0, -6, 8, 0, 8, 0, 0], [0, -4, 2, 0, -3, 0, 0, -7, -8, 8, 0, 0, 5, 0, 0, 5], [0, 0, 0, 0, 0, -10, -6, 0, 0, 0, -10, -11, 0, 11, -10, 0], [0, 1, 0, 0, 2, 0, 0, 8, 0, 0, 0, 0, -5, 0, 0, 9], [0, 0, 0, 0, 0, -6, 2, 0, 0, 0, -6, -7, 0, 7, -5, 0], [-3, 0, 0, 11, 0, 8, 0, 0, 0, 0, -8, 3, 0, 3, 0, 0], [0, 11, 0, 0, -8, 0, 0, -19, 0, 0, 0, 0, -12, 0, 0, -8], [11, 0, 0, -6, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -3, 2, 0, 0, 0, -3, 6, 0, -6, -4, 0], [0, -1, 0, 0, -10, 0, 0, -1, 0, 0, 0, 0, 11, 0, 0, -2], [0, -1, 5, 0, -3, 0, 0, -4, -4, 11, 0, 0, 8, 0, 0, 8], [0, 12, 0, 0, -7, 0, 0, -16, 0, 0, 0, 0, -6, 0, 0, -4], [0, 0, 0, 0, 0, 3, -1, 0, 0, 0, 3, -1, 0, 1, -4, 0], [0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 4, -6, 0, 6, 0, 0], [0, 10, 1, 0, -1, 0, 0, 9, 10, 3, 0, 0, 2, 0, 0, 2], [0, 0, 0, 0, 0, -2, 2, 0, 0, 0, -2, 1, 0, -1, -9, 0], [0, 4, 0, 0, -4, 0, 0, -6, 0, 0, 0, 0, 9, 0, 0, -2], [0, -5, 0, 0, 16, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, -7], [16, 0, 0, -4, 0, 13, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0], [0, 0, 2, 0, -2, 0, 0, -2, -10, 6, 0, 0, 4, 0, 0, 4], [0, -3, 13, 0, 3, 0, 0, 0, 10, 7, 0, 0, 10, 0, 0, 10], [0, -2, 0, 0, 2, 0, 0, -1, 0, 0, 0, 0, 9, 0, 0, -3], [0, -17, 5, 0, 5, 0, 0, -12, 0, -5, 0, 0, 0, 0, 0, 0], [-42, 0, 0, 2, 0, 8, 0, 0, 0, 0, -8, 5, 0, 5, 0, 0], [0, 0, 0, 0, 0, -2, -8, 0, 0, 0, -2, 8, 0, -8, -2, 0], [-17, 0, 0, 1, 0, 13, 0, 0, 0, 0, -13, 7, 0, 7, 0, 0], [0, -10, -3, 0, -2, 0, 0, -12, 6, 1, 0, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 4, 9, 0, -9, 10, 0], [0, -1, 0, 0, -8, 0, 0, 15, 0, 0, 0, 0, 7, 0, 0, 14], [0, -6, -9, 0, -3, 0, 0, -9, -13, -3, 0, 0, -6, 0, 0, -6], [0, 0, 0, 0, 0, 8, -3, 0, 0, 0, 8, 10, 0, -10, 10, 0], [-1, 0, 0, 9, 0, -6, 0, 0, 0, 0, 6, 5, 0, 5, 0, 0], [0, 6, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 6], [0, 4, 8, 0, 3, 0, 0, 7, 5, 2, 0, 0, 5, 0, 0, 5], [0, 13, 0, 0, 19, 0, 0, -9, 0, 0, 0, 0, -1, 0, 0, 4], [-27, 0, 0, -7, 0, 5, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, -5, 0, 0, 0, -2, -5, 0, 5, 2, 0], [0, 19, 0, 0, 5, 0, 0, -20, 0, 0, 0, 0, 2, 0, 0, -1], [0, 7, 0, 0, -8, 0, 0, -1, 0, 0, 0, 0, 11, 0, 0, 6], [0, -8, -2, 0, -5, 0, 0, -13, -2, 8, 0, 0, 3, 0, 0, 3], [0, -18, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 1, 0, 0, -8], [0, 0, 0, 0, 0, -4, 1, 0, 0, 0, -4, -11, 0, 11, -8, 0], [0, -10, 3, 0, 7, 0, 0, -3, -2, -11, 0, 0, -4, 0, 0, -4], [-32, 0, 0, -8, 0, -4, 0, 0, 0, 0, 4, -8, 0, -8, 0, 0], [0, 0, 0, 0, 0, 5, -3, 0, 0, 0, 5, 9, 0, -9, 0, 0], [0, -5, 0, 0, -17, 0, 0, -4, 0, 0, 0, 0, 2, 0, 0, -9], [2, 0, 0, 0, 0, -12, 0, 0, 0, 0, 12, -9, 0, -9, 0, 0], [0, 2, 0, 0, -5, 0, 0, 2, 0, 0, 0, 0, -4, 0, 0, 4], [17, 0, 0, -4, 0, -5, 0, 0, 0, 0, 5, 5, 0, 5, 0, 0], [0, 9, 0, 0, -18, 0, 0, -10, 0, 0, 0, 0, -6, 0, 0, -1], [0, -5, 0, 0, 14, 0, 0, 5, 0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 4, -6, 0, 6, 7, 0], [-5, 0, 0, -5, 0, -2, 0, 0, 0, 0, 2, 11, 0, 11, 0, 0], [11, 0, 0, 3, 0, 4, 0, 0, 0, 0, -4, 1, 0, 1, 0, 0], [0, 5, 3, 0, -1, 0, 0, 4, -12, 5, 0, 0, 4, 0, 0, 4], [0, -11, 6, 0, 4, 0, 0, -7, 10, -2, 0, 0, 2, 0, 0, 2], [7, 0, 0, 0, 0, -3, 0, 0, 0, 0, 3, -9, 0, -9, 0, 0], [0, -24, 0, 0, 2, 0, 0, -22, -6, -4, 0, 0, -2, 0, 0, -2], [0, 7, 0, 0, 4, 0, 0, -13, 0, 0, 0, 0, -5, 0, 0, -6], [0, 11, 0, 0, -14, 0, 0, -7, 0, 0, 0, 0, -2, 0, 0, 4], [0, 0, 0, 0, 0, -4, 8, 0, 0, 0, -4, -11, 0, 11, -4, 0], [-5, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -10, 0, -10, 0, 0], [0, 0, 0, 0, 0, 2, 7, 0, 0, 0, 2, -3, 0, 3, 1, 0], [0, -3, 0, 0, 12, 0, 0, -4, 0, 0, 0, 0, -15, 0, 0, -7], [0, 2, -3, 0, -1, 0, 0, 1, 12, -1, 0, 0, -2, 0, 0, -2], [0, 0, 0, 0, 0, 7, 6, 0, 0, 0, 7, -4, 0, 4, 2, 0], [0, 5, 9, 0, 2, 0, 0, 7, 15, 5, 0, 0, 7, 0, 0, 7], [0, 0, 0, 0, 0, -1, 8, 0, 0, 0, -1, -8, 0, 8, -9, 0], [0, 14, -4, 0, -4, 0, 0, 10, 6, 4, 0, 0, 0, 0, 0, 0], [-33, 0, 0, 5, 0, -3, 0, 0, 0, 0, 3, 3, 0, 3, 0, 0], [0, 24, -7, 0, -3, 0, 0, 21, -1, -1, 0, 0, -4, 0, 0, -4], [0, 10, 0, 0, 2, 0, 0, -16, 0, 0, 0, 0, -3, 0, 0, -6], [0, 14, 1, 0, 3, 0, 0, 17, 4, -5, 0, 0, -2, 0, 0, -2], [0, 11, 1, 0, 3, 0, 0, 14, 16, -5, 0, 0, -2, 0, 0, -2], [0, 1, 0, 0, 4, 0, 0, -11, 0, 0, 0, 0, -17, 0, 0, -10], [0, -6, 0, 0, 5, 0, 0, 5, 0, 0, 0, 0, 13, 0, 0, -1], [0, 0, 0, 0, 0, 7, -3, 0, 0, 0, 7, 1, 0, -1, -6, 0], [-28, 0, 0, -5, 0, -4, 0, 0, 0, 0, 4, 6, 0, 6, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2352_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2352_2_k_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_2352_2_k_i(:prec:=16)
    chi := MakeCharacter_2352_k();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2352_2_k_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_2352_2_k_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2352_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, 0, -763, 0, 263, 0, -29, 0, 1]>,<13,R![4096, 0, 3584, 0, 836, 0, 55, 0, 1]>],Snew);
    return Vf;
end function;