
// Make newform 357.2.k.b in Magma, downloaded from the LMFDB on 27 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_357_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_357_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_357_2_k_b();".
// 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_357_2_k_b();".
// 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 := [4, 0, 3920, 0, 22224, 0, 48276, 0, 53608, 0, 34148, 0, 13121, 0, 3072, 0, 426, 0, 32, 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, 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], [-7424, -125460, -3417592, -2688720, -14177288, -7693996, -21418240, -8127156, -15808248, -4186454, -6392876, -1155388, -1483756, -169405, -197026, -11305, -13920, -85, -406, 17], [7424, -125460, 3417592, -2688720, 14177288, -7693996, 21418240, -8127156, 15808248, -4186454, 6392876, -1155388, 1483756, -169405, 197026, -11305, 13920, -85, 406, 17], [183892, -5767660, 213940, -33904536, -6339716, -67236060, -11785792, -66745560, -7984012, -37768438, -2547520, -12881216, -387635, -2688045, -21013, -335075, 811, -22873, 93, -657], [183892, 5767660, 213940, 33904536, -6339716, 67236060, -11785792, 66745560, -7984012, 37768438, -2547520, 12881216, -387635, 2688045, -21013, 335075, 811, 22873, 93, 657], [0, 2034172, 0, 7096832, 0, 9112434, 0, 7011006, 0, 3979738, 0, 1631879, 0, 434332, 0, 68424, 0, 5724, 0, 195], [-368, 0, -112992, 0, -645676, 0, -1213762, 0, -1090814, 0, -537850, 0, -151965, 0, -24295, 0, -2033, 0, -69, 0], [23296, 0, -105104, 0, -645676, 0, -1213762, 0, -1090814, 0, -537850, 0, -151965, 0, -24295, 0, -2033, 0, -69, 0], [125460, 3857928, 2688720, 27070456, 7693996, 68182016, 8127156, 83688200, 4186454, 56985812, 1155388, 22868820, 169405, 5504606, 11305, 776736, 85, 58986, -17, 1856], [-125460, 3857928, -2688720, 27070456, -7693996, 68182016, -8127156, 83688200, -4186454, 56985812, -1155388, 22868820, -169405, 5504606, -11305, 776736, -85, 58986, 17, 1856], [0, -247648, 0, -1398932, 0, -3227630, 0, -3841122, 0, -2603766, 0, -1055167, 0, -258329, 0, -37159, 0, -2875, 0, -92], [-183588, -5757608, -225372, -33679164, 5126080, -72362140, 14100488, -80846048, 15716856, -53485294, 8940326, -21821542, 2776963, -5465008, 472715, -807790, 41273, -64146, 1443, -2100], [0, 4062736, 0, 15490296, 0, 16648736, 0, 67858, 0, -10924586, 0, -8010836, 0, -2656383, 0, -459813, 0, -40207, 0, -1401], [183588, -5757608, 225372, -33679164, -5126080, -72362140, -14100488, -80846048, -15716856, -53485294, -8940326, -21821542, -2776963, -5465008, -472715, -807790, -41273, -64146, -1443, -2100], [-65534, 0, -696044, 0, -4635064, 0, -11040655, 0, -11535258, 0, -6214991, 0, -1847226, 0, -303983, 0, -25870, 0, -887, 0], [0, -734688, 0, -4083804, 0, -9037214, 0, -10309604, 0, -6720484, 0, -2627651, 0, -623022, 0, -87182, 0, -6592, 0, -207], [-369024, 11440900, -4840668, 75105056, -11971228, 182674764, -11477916, 221519204, -5789606, 150962734, -1840258, 61058196, -402904, 14860657, -58518, 2121877, -4882, 162953, -172, 5179], [369024, 11440900, 4840668, 75105056, 11971228, 182674764, 11477916, 221519204, 5789606, 150962734, 1840258, 61058196, 402904, 14860657, 58518, 2121877, 4882, 162953, 172, 5179], [-118040, 0, 11580160, 0, 51533280, 0, 83620342, 0, 67910398, 0, 30879656, 0, 8160989, 0, 1236549, 0, 99193, 0, 3257, 0]];
    Rf_basisdens := [1, 1, 173536, 173536, 173536, 173536, 43384, 7888, 7888, 173536, 173536, 7888, 173536, 86768, 173536, 43384, 7888, 173536, 173536, 86768];
    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_357_k();"
function MakeCharacter_357_k()
    N := 357;
    order := 4;
    char_gens := [239, 52, 190];
    v := [4, 4, 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_357_k_Hecke();"
function MakeCharacter_357_k_Hecke( : Kf := false)
    N := 357;
    order := 4;
    char_gens := [239, 52, 190];
    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, 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]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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, 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, 1, 0, 0, 0, 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, 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, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [-1, 0, 1, -1, -1, -1, 0, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, -1, 0, 1, -1, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, -1, 0], [0, 0, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 1, 0, 0, 0], [0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 0, -2, 0, -1, 0, 0, 0], [-1, 2, 0, 0, 0, 1, 0, -2, 1, 3, 0, 1, -2, -1, 0, 1, -1, 0, -1, 0], [1, -1, -1, 0, 0, 1, 1, 1, -1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, -1], [2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, -2, 1, 1, 0, -1, 0, 0, -1, 0], [0, -3, 0, 0, 3, 0, 1, -3, 1, 0, 0, 0, 0, -1, 1, -1, 1, 1, 0, 1], [0, -1, 0, 0, 1, -1, 1, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, -3, 3, 0, -1, 0, 1, -1, 0, -1, 1, -1], [0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, -3, 0, 1, 0, 0, 1, 0, 0, 0], [0, 1, 0, 0, -1, 1, 1, 0, 0, 0, 0, -2, -1, -1, -1, 0, 0, 0, 0, 0], [0, -1, 0, -3, 1, 0, 3, -1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 0, 3], [3, 0, -1, 1, 1, 1, 0, -2, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 1], [0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -2, 0, 2, 0, 0, 1, 0], [-4, -2, 2, 0, 0, 2, 0, 2, 0, 0, 0, 4, 2, -2, 0, 2, 0, 0, -2, 0], [0, 3, 0, 1, -2, 0, -1, 3, 0, 0, -1, 0, 0, -1, -3, -1, 0, -1, 0, -1], [0, 0, 0, 0, 1, -1, 2, 0, 0, 0, 0, 2, 1, 0, 1, 0, 0, 1, 1, 0], [-4, 0, 0, 0, 0, 0, 0, 2, -2, 1, -1, 0, 1, 0, -1, 0, 0, 1, -1, 2], [4, 0, -2, 0, 0, 0, 1, 0, -1, -4, 0, -4, 0, 1, 0, -1, 1, 0, 2, -1], [2, 0, 2, -2, 0, 0, 0, -1, 0, -1, 1, 0, 1, 0, -1, 3, 0, 2, -2, -1], [3, 0, 3, -3, 1, 1, 0, -2, 0, -1, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0], [0, -1, -3, 0, 0, 3, 2, 1, -1, 5, 0, 0, 2, 0, 0, 0, 1, 0, -2, -2], [4, -2, 0, 3, 0, 0, 0, -2, 0, 0, 7, 4, 0, -3, -1, -3, 0, -3, 0, 0], [1, -4, 0, -1, 4, 0, 2, -4, 4, 0, -2, 1, 0, 1, 2, 1, 4, 1, 0, 2], [0, 3, 0, 0, -2, 2, -4, 0, 0, 2, 2, 6, -2, 2, -2, 0, -5, -2, -2, 0], [-2, 2, 0, 0, 0, -2, 0, -2, 2, 3, 0, 2, 1, 0, 0, 0, -2, 0, 0, 0], [1, 0, 0, 0, -1, -1, 0, -1, -3, 1, -1, 0, 2, 0, -2, 1, 0, -1, 1, 2], [1, 0, 0, 0, 0, -2, -1, 0, 1, 0, 0, -1, -2, 2, 0, -2, -1, 0, 4, 1], [-4, 0, 3, -3, 0, 0, 0, -2, 2, -4, 4, 0, 0, 0, 0, -2, 0, 1, -1, 4], [0, 0, 0, 0, 1, -1, -3, 0, 0, -4, -4, -3, -1, 2, -1, 0, -3, 0, 0, 0], [-8, 0, 0, 0, 1, 1, 0, 0, -3, 0, 0, 0, -1, 0, 1, 1, 0, -2, 2, -1], [6, 2, 0, 0, 0, 0, -4, 2, -2, 0, 2, 6, 0, 0, -2, 0, -2, 0, 0, -4], [2, -1, -6, 0, 0, -3, 1, 1, 1, -8, 0, -2, -1, 5, 0, -5, -1, 0, 3, -1], [2, -1, -6, 0, 0, 0, 1, 1, 2, 1, 0, -2, -1, 1, 0, -1, -2, 0, -2, -1], [0, 0, 1, 1, -1, 1, -2, 0, 0, -3, -3, -6, 0, -2, 0, 0, 0, 0, 0, 0], [0, -2, 0, 2, 4, 0, -1, -2, 1, 0, -10, 0, 0, 3, 2, 3, 1, 2, 0, -1], [-10, 0, 1, -1, -2, -2, 0, 4, 0, 0, 0, 0, -2, 0, 2, -2, 0, 1, -1, -2], [-7, 0, 0, -1, 0, 0, 1, 0, -1, 0, -5, -7, 0, 1, 3, 1, -1, 3, 0, 1], [4, 3, 0, 1, -2, 0, 0, 3, -1, 0, 0, 4, 0, -2, -2, -2, -1, -3, 0, 0], [1, 1, 5, 0, 0, 1, -4, -1, 2, -4, 0, -1, 1, -1, 0, 1, -2, 0, 0, 4], [1, 2, 0, -1, 2, 0, 0, 2, -2, 0, -3, 1, 0, -1, -1, -1, -2, -1, 0, 0], [0, 2, -5, -5, 3, -3, 0, 0, 0, 1, 1, 3, -3, 3, -3, 0, 0, 6, 6, 0], [-2, -2, 0, -2, -1, 0, -2, -2, 1, 0, -2, -2, 0, 2, 1, 2, 1, 1, 0, -2], [0, 0, -1, -1, 2, -2, 1, 0, 0, 1, 1, 3, 1, -2, 1, 0, 3, 0, 0, 0], [7, -1, 1, 0, 0, 1, 2, 1, -4, -5, 0, -7, 6, -1, 0, 1, 4, 0, 0, -2], [1, 0, -4, 4, -2, -2, 0, -1, 3, 8, -8, 0, -2, 0, 2, -1, 0, 0, 0, 0], [9, 0, 4, 0, 0, 4, 1, 0, 1, -2, 0, -9, -2, 0, 0, 0, -1, 0, 0, -1], [-12, 0, 4, -4, 0, 0, 0, -3, 2, -3, 3, 0, -1, 0, 1, 5, 0, 3, -3, 1], [0, 3, 2, 2, 2, -2, 1, 0, 0, -2, -2, 6, -2, 1, -2, 0, -2, -2, -2, 0], [0, 2, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4, 2, 2, 0], [-2, 2, 2, 0, 0, 1, 0, -2, 1, -6, 0, 2, 3, -2, 0, 2, -1, 0, -1, 0], [8, 0, -2, 2, -1, -1, 0, 4, 3, 2, -2, 0, 2, 0, -2, -1, 0, 2, -2, -1], [-6, -1, 8, 0, 0, 4, -1, 1, -2, 0, 0, 6, 0, -2, 0, 2, 2, 0, -2, 1], [0, 6, 3, 3, -1, 1, 4, 0, 0, 1, 1, -8, 0, 2, 0, 0, 0, 0, 0, 0], [-1, -3, 0, -5, 3, 0, 4, -3, 2, 0, 4, -1, 0, 3, 1, 3, 2, 6, 0, 4], [2, 0, 0, 0, 1, 1, 0, -2, 0, -5, 5, 0, 0, 0, 0, 2, 0, 2, -2, 2], [6, 0, 4, -4, 0, 0, 0, -4, 2, -4, 4, 0, 1, 0, -1, -2, 0, -2, 2, 0], [-8, -5, -2, 0, 0, -8, -1, 5, 0, 0, 0, 8, -4, -1, 0, 1, 0, 0, 4, 1], [1, -1, 0, -1, 3, 0, 2, -1, 0, 0, 0, 1, 0, 3, -1, 3, 0, 0, 0, 2], [5, 1, 0, 2, 2, 0, 2, 1, -3, 0, -10, 5, 0, -1, 0, -1, -3, -2, 0, 2], [0, 1, -1, -1, 2, -2, -2, 0, 0, -7, -7, -6, 1, 4, 1, 0, -1, 1, 1, 0], [-2, 0, 9, 0, 0, -2, 0, 0, 0, 1, 0, 2, 3, -1, 0, 1, 0, 0, 1, 0], [1, -5, 0, -8, 1, 0, 4, -5, 3, 0, 3, 1, 0, 3, 4, 3, 3, 1, 0, 4], [0, -6, -1, -1, 0, 0, -3, 0, 0, 4, 4, -2, 4, -3, 4, 0, 3, 0, 0, 0], [6, 0, -4, 4, 0, 0, 0, 7, -6, 1, -1, 0, 1, 0, -1, -7, 0, -2, 2, -3], [0, -7, 1, 1, -1, 1, 0, 0, 0, 7, 7, 5, 2, -1, 2, 0, 3, 0, 0, 0], [-1, 2, 0, 0, 2, 0, 1, 2, 3, 0, -6, -1, 0, 0, 4, 0, 3, 4, 0, 1], [-1, 0, 1, -1, -3, -3, 0, 4, 3, -5, 5, 0, -2, 0, 2, -4, 0, -1, 1, 3], [1, 1, 4, 0, 0, 0, -1, -1, 2, -4, 0, -1, -4, 4, 0, -4, -2, 0, -2, 1], [0, 5, -8, -8, 2, -2, 1, 0, 0, 0, 0, -6, 0, 3, 0, 0, -2, 1, 1, 0], [0, -2, 0, 0, -3, 3, -2, 0, 0, 3, 3, -10, 0, 2, 0, 0, 2, 3, 3, 0], [-6, -5, 0, 1, 3, 0, 5, -5, 5, 0, -2, -6, 0, -1, 5, -1, 5, 4, 0, 5], [-4, 1, 0, 3, 2, 0, 0, 1, -1, 0, 4, -4, 0, 0, 2, 0, -1, -5, 0, 0], [7, 0, 5, -5, -1, -1, 0, -8, 0, -1, 1, 0, -2, 0, 2, -3, 0, 2, -2, 4], [4, -1, 0, 0, -5, 0, -1, -1, -1, 0, 7, 4, 0, -1, 2, -1, -1, -9, 0, -1], [-8, 0, -1, 1, 0, 0, 0, -3, 3, 7, -7, 0, 1, 0, -1, 6, 0, 5, -5, 2], [-5, 2, 0, -2, -4, 0, -4, 2, -1, 0, 6, -5, 0, 3, -6, 3, -1, 0, 0, -4], [0, 10, 0, 0, -3, 3, 1, 0, 0, 4, 4, 6, -5, 1, -5, 0, 1, 0, 0, 0], [-9, 1, -3, 0, 0, -1, -4, -1, 2, 2, 0, 9, -5, 1, 0, -1, -2, 0, 2, 4], [7, 0, -1, 1, -1, -1, 0, 1, -1, 11, -11, 0, 0, 0, 0, 7, 0, 0, 0, -2], [4, -1, 0, -4, -3, 0, -4, -1, -2, 0, 9, 4, 0, 2, -2, 2, -2, 1, 0, -4], [0, -5, -2, -2, -2, 2, -6, 0, 0, 4, 4, 4, 0, -2, 0, 0, 1, -2, -2, 0], [0, 3, 4, 4, -5, 5, -1, 0, 0, 4, 4, -4, -1, 1, -1, 0, -6, -1, -1, 0], [7, 0, 3, -3, 1, 1, 0, -10, 1, -1, 1, 0, 0, 0, 0, 2, 0, 1, -1, -5], [0, 4, 0, 0, -1, 1, 4, 0, 0, 6, 6, -8, -1, 4, -1, 0, 2, 1, 1, 0], [8, 0, -8, 0, 0, -3, 0, 0, -1, -12, 0, -8, 3, 6, 0, -6, 1, 0, 7, 0], [0, 0, 0, -2, -2, 0, -2, 0, 2, 0, -2, 0, 0, -2, 0, -2, 2, 4, 0, -2], [0, -10, -4, -4, 4, -4, 2, 0, 0, 2, 2, 6, 6, -6, 6, 0, 0, 2, 2, 0], [-1, 1, 0, 9, 2, 0, 0, 1, 3, 0, -7, -1, 0, -3, 3, -3, 3, 1, 0, 0], [-2, 3, 0, -6, 2, 0, 3, 3, 2, 0, -17, -2, 0, 3, 1, 3, 2, 4, 0, 3], [4, 0, 6, -6, 2, 2, 0, -8, 8, 0, 0, 0, -6, 0, 6, 0, 0, 1, -1, 6], [2, 0, 0, 4, -1, 0, -2, 0, -3, 0, 6, 2, 0, -2, -1, -2, -3, -1, 0, -2], [-6, 0, -2, 2, 4, 4, 0, -6, 2, 2, -2, 0, -3, 0, 3, 0, 0, -4, 4, 2], [0, -1, -12, 0, 0, -4, 3, 1, -2, 2, 0, 0, -6, 4, 0, -4, 2, 0, 6, -3], [4, -3, 9, 0, 0, 2, 3, 3, -4, 5, 0, -4, -5, -3, 0, 3, 4, 0, 3, -3], [-3, 0, 2, -2, 4, 4, 0, 7, 1, 2, -2, 0, 2, 0, -2, -1, 0, 0, 0, 0], [0, -6, 2, 2, 3, -3, -4, 0, 0, -10, -10, 6, 1, 2, 1, 0, -4, -5, -5, 0], [0, 2, 7, 7, -4, 4, -4, 0, 0, 0, 0, 14, -4, -4, -4, 0, -2, -5, -5, 0], [-2, 2, 0, 4, -2, 0, 0, 2, 0, 0, 6, -2, 0, 0, -8, 0, 0, 2, 0, 0], [-3, 0, -5, 5, -1, -1, 0, 4, 0, 9, -9, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, -5, 4, 4, 5, -5, -3, 0, 0, 4, 4, 14, 3, -5, 3, 0, -2, -2, -2, 0], [0, -1, -4, -4, 0, 0, -1, 0, 0, 8, 8, 2, 0, -1, 0, 0, 0, 4, 4, 0], [-4, 0, -3, 3, 1, 1, 0, -2, 2, 3, -3, 0, 2, 0, -2, 0, 0, 0, 0, 2], [-6, 3, 0, 1, -9, 0, -1, 3, 3, 0, 0, -6, 0, -1, 3, -1, 3, 4, 0, -1], [-2, -1, -1, 0, 0, -5, 1, 1, -1, -6, 0, 2, 5, -1, 0, 1, 1, 0, 2, -1], [5, 0, -5, 5, -1, -1, 0, 3, 0, -5, 5, 0, 2, 0, -2, 2, 0, 5, -5, -3], [-4, 7, -4, 0, 0, 3, -4, -7, 6, 15, 0, 4, -8, -2, 0, 2, -6, 0, -7, 4], [12, -5, 0, 5, 1, 0, 2, -5, -2, 0, 4, 12, 0, -6, -1, -6, -2, -6, 0, 2], [0, -3, -1, -1, -1, 1, 5, 0, 0, 9, 9, 1, 2, -2, 2, 0, 2, -1, -1, 0], [7, -3, -3, 0, 0, 1, 0, 3, -4, 15, 0, -7, 2, 3, 0, -3, 4, 0, 4, 0], [-2, 4, -2, 0, 0, 0, -2, -4, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, -2, 2], [4, 0, -8, 8, 2, 2, 0, -2, 2, -4, 4, 0, 2, 0, -2, -8, 0, -2, 2, 0], [2, 4, 0, -7, 3, 0, 6, 4, 1, 0, 7, 2, 0, -2, -2, -2, 1, 0, 0, 6], [-15, 0, -9, 9, 3, 3, 0, -1, 1, -1, 1, 0, 0, 0, 0, -1, 0, 4, -4, 0], [0, -12, -2, -2, -1, 1, 1, 0, 0, -6, -6, 8, 1, 7, 1, 0, -1, -2, -2, 0], [0, -7, 0, 4, 4, 0, -1, -7, 0, 0, 16, 0, 0, -4, 8, -4, 0, -6, 0, -1], [-10, -4, 4, 0, 0, -5, -6, 4, 1, -8, 0, 10, -1, -2, 0, 2, -1, 0, -3, 6], [-7, -5, 5, 0, 0, 4, 2, 5, 0, 18, 0, 7, -2, -5, 0, 5, 0, 0, -13, -2], [-6, -9, 0, -3, 1, 0, 1, -9, -3, 0, 6, -6, 0, 3, 1, 3, -3, 0, 0, 1], [8, 0, 6, -6, -1, -1, 0, -2, 0, -15, 15, 0, -2, 0, 2, -6, 0, -1, 1, 0], [-2, 4, 12, 0, 0, -6, -1, -4, 1, -20, 0, 2, -6, -2, 0, 2, -1, 0, -4, 1], [0, -2, 2, 0, 0, 2, 2, 2, -4, -3, 0, 0, -1, -8, 0, 8, 4, 0, 2, -2], [-10, 0, -3, 3, -3, -3, 0, 14, -6, -4, 4, 0, 2, 0, -2, -2, 0, -4, 4, 0], [0, -2, 2, 2, 2, -2, 4, 0, 0, -2, -2, 6, 2, 4, 2, 0, -2, -4, -4, 0], [0, -1, -1, -1, -1, 1, -1, 0, 0, 5, 5, -3, 8, -4, 8, 0, -2, -7, -7, 0], [-10, -2, -8, 0, 0, -1, -2, 2, 0, -3, 0, 10, -2, 8, 0, -8, 0, 0, 9, 2], [4, -7, 3, 0, 0, 6, -1, 7, 4, 13, 0, -4, 3, 3, 0, -3, -4, 0, -3, 1], [0, -1, 3, 3, 0, 0, 10, 0, 0, -5, -5, -8, 5, -2, 5, 0, 9, 3, 3, 0], [2, 0, 10, -10, -2, -2, 0, 2, 0, -2, 2, 0, -2, 0, 2, 8, 0, 3, -3, 4], [-1, 0, 2, -2, 0, 0, 0, 0, 6, 7, -7, 0, 1, 0, -1, 1, 0, 2, -2, 0], [0, 7, 2, 2, -3, 3, -9, 0, 0, 1, 1, 20, 2, -3, 2, 0, -6, -3, -3, 0], [4, 3, 3, 0, 0, 13, -2, -3, 0, 6, 0, -4, -5, -6, 0, 6, 0, 0, -10, 2], [0, 8, 4, 4, -2, 2, 6, 0, 0, 8, 8, 0, -6, -4, -6, 0, -4, -3, -3, 0], [-9, 6, 0, -3, 2, 0, 2, 6, 0, 0, -4, -9, 0, 9, -2, 9, 0, 5, 0, 2], [-14, 2, 8, 0, 0, 2, 0, -2, 8, -8, 0, 14, -6, -2, 0, 2, -8, 0, -10, 0], [-2, 2, 1, 0, 0, -5, 0, -2, 1, 13, 0, 2, 2, 0, 0, 0, -1, 0, 6, 0], [10, 2, 0, -11, 4, 0, 3, 2, -1, 0, -7, 10, 0, 1, 1, 1, -1, -3, 0, 3], [-4, -6, 0, 15, -2, 0, -4, -6, -1, 0, -8, -4, 0, -6, -2, -6, -1, -5, 0, -4], [0, 0, -4, 4, -2, -2, 0, 4, 6, 6, -6, 0, -2, 0, 2, 0, 0, 2, -2, 2], [2, -7, 0, -2, 9, 0, 5, -7, 5, 0, -6, 2, 0, 5, 3, 5, 5, 1, 0, 5], [5, -3, -7, 0, 0, -5, 1, 3, -3, -6, 0, -5, 3, 2, 0, -2, 3, 0, 10, -1], [-10, 5, 0, 0, -4, 0, 1, 5, 0, 0, 12, -10, 0, -1, -10, -1, 0, 2, 0, 1], [0, -2, -4, -4, 2, -2, -2, 0, 0, -4, -4, -4, -3, 2, -3, 0, 2, -2, -2, 0], [3, 0, 4, -4, -4, -4, 0, 5, -7, -2, 2, 0, -2, 0, 2, -1, 0, 2, -2, -10], [-12, 6, 0, 1, -2, 0, 4, 6, 2, 0, 21, -12, 0, -3, -5, -3, 2, 1, 0, 4], [-6, 4, 10, 0, 0, 8, -6, -4, 6, 2, 0, 6, -8, -2, 0, 2, -6, 0, -12, 6], [10, 0, 9, -9, 2, 2, 0, 3, -5, 5, -5, 0, 3, 0, -3, 14, 0, 3, -3, 8], [10, 3, -4, 0, 0, 1, 3, -3, -1, 4, 0, -10, -1, -1, 0, 1, 1, 0, 1, -3], [3, -8, 0, -6, 8, 0, -1, -8, -1, 0, 10, 3, 0, -4, -2, -4, -1, 0, 0, -1], [-12, 0, 0, 5, 2, 0, -6, 0, 1, 0, -8, -12, 0, 0, 2, 0, 1, 1, 0, -6], [9, 0, 0, 0, 3, 3, 0, -7, -8, 6, -6, 0, -3, 0, 3, 4, 0, -4, 4, -1], [-10, -4, 0, -2, 1, 0, 0, -4, 7, 0, -4, -10, 0, 2, 3, 2, 7, 9, 0, 0], [0, 0, -3, -3, 0, 0, -5, 0, 0, 1, 1, 9, 1, 2, 1, 0, 5, 2, 2, 0], [10, -5, 0, -6, 8, 0, 1, -5, 0, 0, 4, 10, 0, -7, 6, -7, 0, -2, 0, 1], [-8, -2, 20, 0, 0, 5, -8, 2, 0, -9, 0, 8, 2, -4, 0, 4, 0, 0, -5, 8], [-9, 0, -3, 3, -1, -1, 0, -7, 9, 15, -15, 0, -8, 0, 8, 1, 0, 2, -2, 8], [0, 5, 8, 8, -4, 4, -8, 0, 0, -2, -2, -4, 0, -8, 0, 0, -3, -8, -8, 0], [0, -6, 4, 4, 1, -1, 6, 0, 0, -1, -1, 18, 2, 0, 2, 0, 6, -1, -1, 0], [0, 10, -10, -10, 2, -2, 6, 0, 0, 2, 2, 6, 0, 10, 0, 0, 0, 6, 6, 0], [-6, 6, -2, 0, 0, 5, 4, -6, -1, 12, 0, 6, -5, 0, 0, 0, 1, 0, 1, -4], [7, 5, 2, 0, 0, -2, 1, -5, 2, -4, 0, -7, -10, 6, 0, -6, -2, 0, 2, -1], [-14, -2, 0, 0, -4, 0, -6, -2, 4, 0, 8, -14, 0, 2, 8, 2, 4, 2, 0, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_357_k_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_357_2_k_b();".
// 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_357_2_k_b(:prec:=20)
    chi := MakeCharacter_357_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(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_357_2_k_b();".
// 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_357_2_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_357_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, 3920, 0, 22224, 0, 48276, 0, 53608, 0, 34148, 0, 13121, 0, 3072, 0, 426, 0, 32, 0, 1]>],Snew);
    return Vf;
end function;