
// Make newform 2736.2.dc.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2736_dc();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2736_dc_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_2736_2_dc_c();".
// 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_2736_2_dc_c();".
// 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, 0, 12150, 0, 15858, 0, 9708, 0, 4243, 0, 1012, 0, 174, 0, 16, 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, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [82286048988, 0, 105715511190, 0, 88135584891, 0, 25255513579, 0, 5119222054, 0, 473907666, 0, 30766372, 0, -481967, 0], [-7319943180, 0, 748162737, 0, 962205174, 0, 904757018, 0, 231730940, 0, 47592867, 0, 4547720, 0, 326606, 0], [34847824212, 0, 131635278990, 0, 72060692802, 0, 35915018668, 0, 8773398223, 0, 1606621224, 0, 150207814, 0, 9939136, 0], [-56770896384, 0, -41971706802, 0, -24565580241, 0, 887860574, 0, 906590177, 0, 381453396, 0, 40373015, 0, 3704024, 0], [0, -2199042758754, 0, -1670986351080, 0, -937524731802, 0, -191131571350, 0, -30249558199, 0, -770314944, 0, 62210210, 0, 26555897], [0, -700940621358, 0, -552063322863, 0, -308344384614, 0, -130718214368, 0, -31353520355, 0, -5463091302, 0, -505462190, 0, -32058956], [71014305663, 0, 123833655000, 0, 100433630874, 0, 43490411282, 0, 10542157166, 0, 1774630644, 0, 163560617, 0, 10005800, 0], [-337029762708, 0, -440628437772, 0, -262263231510, 0, -119231635186, 0, -28836857512, 0, -5141477487, 0, -478042816, 0, -30938182, 0], [0, -151450330770, 0, -254108418858, 0, -231158174538, 0, -96537911477, 0, -23457056468, 0, -3662652030, 0, -328891601, 0, -17666630], [0, 95407816194, 0, 63038114712, 0, 33248580135, 0, -20771370740, 0, -8099390306, 0, -2240929032, 0, -236592485, 0, -19112531], [0, 89111629089, 0, -612664504560, 0, -386283003708, 0, -204003532826, 0, -50123726495, 0, -9318113529, 0, -873827510, 0, -58514042], [0, -201120217002, 0, -263270557980, 0, -144121385604, 0, -71830037336, 0, -17546796446, 0, -3213242448, 0, -300415628, 0, -19878272], [11675287323, 0, -14225798826, 0, -18295668252, 0, -15342294773, 0, -4406204120, 0, -904945566, 0, -89119619, 0, -6210188, 0], [0, 2317137727815, 0, 4206283806420, 0, 2986631978124, 0, 1350470930357, 0, 328205077598, 0, 56522041548, 0, 5217299756, 0, 324198086]];
    Rf_basisdens := [1, 1, 21904094763, 2433788307, 65712284289, 21904094763, 197136852867, 197136852867, 21904094763, 65712284289, 65712284289, 65712284289, 197136852867, 65712284289, 7301364921, 197136852867];
    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_2736_dc();"
function MakeCharacter_2736_dc()
    N := 2736;
    order := 6;
    char_gens := [1711, 2053, 1217, 1009];
    v := [6, 6, 3, 5];
    // 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_2736_dc_Hecke();"
function MakeCharacter_2736_dc_Hecke(Kf)
    N := 2736;
    order := 6;
    char_gens := [1711, 2053, 1217, 1009];
    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], [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+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, 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, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, 0, 0], [1, 0, 1, 0, -1, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 1, 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, -1, 0, 0, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0], [1, 0, 2, -1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 2, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 1, 0, 0, 0, -2, -2, 1, -1, 0, 1], [0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 1, 1, 6, -1, 0, 0, -2, -1, 0, 0, 0, 0, -1, 0], [2, 0, -2, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, 1, 0, 0, -1], [-4, 0, -3, 0, -4, 2, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0], [2, 0, 0, 2, -2, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -2, 0, 0, -1, 1, -1, 0, 0, 1], [2, 0, 0, 4, -2, 1, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, -2, -1, 0, 1], [0, 0, 0, 1, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -2, -1, 0, 0, 0, 2], [7, 0, 1, -2, -5, -1, 0, 0, -1, 1, 0, 0, 0, 0, -2, 0], [0, -2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, 0, 2], [-1, 0, -1, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0], [0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 2, 0, -1, -2, 0, -1], [0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 0, 0, -3], [-1, 0, -1, 2, -3, -1, 0, 0, -2, -2, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, -2, -2, 0, 1], [0, 0, 2, 1, -2, 0, 0, 0, -2, -2, 0, 0, 0, 0, -1, 0], [5, 0, -4, 0, 5, -2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-4, 0, 0, -4, 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, 2, 0, 0, 0, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -1, 1, 0, 0, -6, -3, 2, 1, 0, -2], [-6, 0, -2, 2, -4, -3, 0, 0, 0, 2, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 1, -4, 0, 0, 0, 2, 0, 0, 0, 0], [5, 0, 2, 0, -1, -1, 0, 0, -1, 0, 0, 0, 0, 0, -4, 0], [0, -2, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0, 2, 0, 0], [-3, 0, -3, 2, -7, 1, 0, 0, 2, -2, 0, 0, 0, 0, 3, 0], [4, 0, 2, 2, 0, 5, 0, 0, 5, -1, 0, 0, 0, 0, -4, 0], [-3, 0, 1, 0, 5, 5, 0, 0, 5, 0, 0, 0, 0, 0, -2, 0], [0, -1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 4, -1, 0, 2], [-7, 0, 0, -4, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 4, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -4, 0, -1], [0, 2, 0, 0, 0, 0, -1, -3, 0, 0, 0, 3, 0, -2, 0, 0], [-7, 0, -1, 1, -4, 4, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0], [0, -2, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 2, -1, 0, 1], [0, 0, 0, 0, 0, 0, -2, 6, 0, 0, 0, 0, 3, 1, 0, 2], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 2, 0, -3, 0, 0, -3], [0, -3, 0, 0, 0, 0, 2, 2, 0, 0, 2, 4, -2, 0, 0, 1], [-8, 0, -6, 0, -8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-6, 0, 0, -6, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, -1, 0, -2, 0, -2], [12, 0, 0, 0, 12, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, -1, -1, 0, 0, -4, -2, -2, -2, 0, -2], [3, 0, 1, -10, -1, -3, 0, 0, -3, 5, 0, 0, 0, 0, -2, 0], [0, 2, 0, 0, 0, 0, -2, 4, 0, 0, 0, 3, 0, -2, 0, 0], [1, 0, 1, 7, -3, 1, 0, 0, 2, -7, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 4, 0, 0], [4, 0, 0, 1, 8, 0, 0, 0, -7, -2, 0, 0, 0, 0, 0, 0], [-4, 0, -1, -2, 2, -5, 0, 0, -5, 1, 0, 0, 0, 0, 2, 0], [0, -1, 0, 0, 0, 0, 0, -7, 0, 0, 3, 0, -7, -2, 0, 1], [-7, 0, 0, 0, 0, 6, 0, 0, 3, 0, 0, 0, 0, 0, -6, 0], [0, 1, 0, 0, 0, 0, -2, -1, 0, 0, 0, 4, 0, -1, 0, 0], [0, -1, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, -2, -2, 0, -4], [3, 0, 3, 0, -5, 1, 0, 0, 2, 0, 0, 0, 0, 0, -3, 0], [-8, 0, -4, 0, -12, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0], [-4, 0, 2, -2, -10, 0, 0, 0, -3, 4, 0, 0, 0, 0, -1, 0], [0, -5, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -3, -3, -4, 0, 0, -2], [11, 0, 0, 0, 11, 1, 0, 0, -1, -2, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, -6, 1, 0, 0, -1, -2, -1, 0, 0, -3], [-5, 0, 3, -1, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 3, 0], [0, -1, 0, 0, 0, 0, 4, 2, 0, 0, 0, 5, 0, 1, 0, 0], [3, 0, -3, -6, -9, 1, 0, 0, 1, 3, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, -1, -6, 0, 0, -2, 2, -3, -3, 0, 1], [5, 0, 3, 3, 4, -3, 0, 0, 0, 3, 0, 0, 0, 0, 3, 0], [-6, 0, -4, 6, -2, 2, 0, 0, 2, -3, 0, 0, 0, 0, 8, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, -8, -8, 0, 1, 0, 0], [0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 8, 4, 0, 4], [3, 0, 0, 7, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0], [0, 1, 0, 0, 0, 0, 0, -4, 0, 0, 4, 0, -4, 2, 0, -1], [-12, 0, 0, -7, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, -3, 0, 0, 0, 0, -2, 5, 0, 0, 0, -1, 0, 3, 0, 0], [0, 2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, -1, 1, 0, -3], [16, 0, 6, -1, 26, 0, 0, 0, 0, 2, 0, 0, 0, 0, -3, 0], [0, 2, 0, 0, 0, 0, 0, -3, 0, 0, -4, 0, -3, 4, 0, -3], [0, 0, 4, 0, 0, 4, 0, 0, -4, -1, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 2, 6, 1, 0, 2], [0, 0, 0, 0, 0, 0, 3, -2, 0, 0, 2, -2, -1, 5, 0, -3], [0, -2, 0, 0, 0, 0, -1, -2, 0, 0, 0, 0, -4, -2, 0, -2], [17, 0, 1, -4, 9, 0, 0, 0, 0, -4, 0, 0, 0, 0, 1, 0], [-12, 0, -12, -2, -11, 0, 0, 0, 0, 2, 0, 0, 0, 0, 12, 0], [-22, 0, 2, 1, -10, -1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0], [0, -2, 0, 0, 0, 0, -1, 0, 0, 0, -6, -6, -9, -1, 0, -1], [0, 7, 0, 0, 0, 0, -1, 4, 0, 0, 6, 3, 8, 7, 0, -2], [0, 2, 0, 0, 0, 0, 0, 5, 0, 0, 2, 1, 10, 2, 0, 0], [22, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0], [-8, 0, 0, -5, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 2, 0, 0, 0, 0, 0, -4, 0, 0, 3, 0, -4, 4, 0, -2], [0, 2, 0, 0, 0, 0, -3, 0, 0, 0, -5, -5, 5, 1, 0, -3], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 5, -5, 2, 4, 0, 0], [5, 0, 10, -1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, -5, 0], [-18, 0, -6, -4, -30, 0, 0, 0, 5, 8, 0, 0, 0, 0, 3, 0], [4, 0, -4, 0, 4, 1, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0], [0, 4, 0, 0, 0, 0, 1, 0, 0, 0, -4, -4, -2, 2, 0, 1], [-8, 0, 0, -2, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, -1, 0], [-1, 0, -1, 4, 13, 2, 0, 0, 4, -4, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 5, 10, 0, 0, 0, 1], [2, 0, 0, 0, 2, -6, 0, 0, 6, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 10, 0, 0, -2, 2, 5, -1, 0, 2], [18, 0, 2, -2, 10, -4, 0, 0, 0, -2, 0, 0, 0, 0, 2, 0], [-1, 0, 6, 3, -8, 0, 0, 0, -1, -6, 0, 0, 0, 0, -3, 0], [0, 2, 0, 0, 0, 0, 3, -3, 0, 0, 4, 2, -6, 2, 0, 6], [0, -3, 0, 0, 0, 0, 3, -2, 0, 0, 8, 4, -4, -3, 0, 6], [4, 0, 0, 0, 0, 8, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 0, -1, 0, -4, 0, 0, -2], [3, 0, 3, 11, 6, 3, 0, 0, 6, -11, 0, 0, 0, 0, -3, 0], [0, -1, 0, 0, 0, 0, 0, -5, 0, 0, 1, 0, -5, -2, 0, -1], [-4, 0, 0, 0, -4, -3, 0, 0, 3, -9, 0, 0, 0, 0, 0, 0], [-12, 0, 0, -2, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, -4, 0, 0, 4, 5, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -8, 2, 0, 0, -1, -2, -2, 0, 0, -4], [2, 0, 2, 3, 2, -2, 0, 0, 0, 3, 0, 0, 0, 0, 2, 0], [4, 0, -7, 0, 4, 3, 0, 0, -3, 5, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 8, 0, 0, 0, 0, 0, 1, 0, 0], [4, 0, 4, 0, 13, -2, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0], [0, 0, 0, 0, 0, 0, 5, -4, 0, 0, -1, 1, -2, -2, 0, -5], [10, 0, -2, 0, 22, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0], [0, -7, 0, 0, 0, 0, -1, -4, 0, 0, -2, -1, -8, -7, 0, -2], [0, 1, 0, 0, 0, 0, -4, 8, 0, 0, 0, 3, 0, -1, 0, 0], [-9, 0, -1, 7, -5, 6, 0, 0, 0, 7, 0, 0, 0, 0, -1, 0], [0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, -6, 2, 0, 2], [-1, 0, -1, -9, -1, -2, 0, 0, -4, 9, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 2, -4, 0, 0, -4, 4, -2, -3, 0, -2], [3, 0, -2, -9, 8, 0, 0, 0, -3, 18, 0, 0, 0, 0, 1, 0], [0, 8, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 4, 0, 0, 1], [0, 0, 9, 0, 0, -4, 0, 0, 4, -1, 0, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, 0, -6, 5, 0, 0, 2, 4, -5, 0, 0, -3], [-4, 0, -4, -3, 8, -4, 0, 0, -8, 3, 0, 0, 0, 0, 4, 0], [0, 3, 0, 0, 0, 0, 3, -1, 0, 0, -8, -4, -2, 3, 0, 6], [12, 0, 1, -6, -10, -8, 0, 0, -8, 3, 0, 0, 0, 0, -2, 0], [0, 2, 0, 0, 0, 0, -4, 6, 0, 0, 0, 5, 0, -2, 0, 0], [-4, 0, -4, 13, 4, 1, 0, 0, 2, -13, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 3, 4, 0, 0], [6, 0, 0, 0, -6, -4, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, -7, 0, 0, 0, 0, 4, 3, 0, 0, -10, -5, 6, -7, 0, 8], [-10, 0, 0, -1, 0, 12, 0, 0, 6, 0, 0, 0, 0, 0, -10, 0], [0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 4, 2, 0, -2], [-6, 0, -6, -8, -5, -1, 0, 0, -2, 8, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 0, -2, 4, 0, 0, 0, 0, 2, 5, 0, 2], [0, 1, 0, 0, 0, 0, 0, -11, 0, 0, 2, 0, -11, 2, 0, -5], [0, 3, 0, 0, 0, 0, 2, -7, 0, 0, -1, -2, 7, 0, 0, 1], [4, 0, -2, 0, 4, 1, 0, 0, -1, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, -1, 0, 0, 0, 0, 1, 0, 0, -1], [0, 8, 0, 0, 0, 0, -1, 0, 0, 0, 6, 3, 0, 8, 0, -2], [0, -1, 0, 0, 0, 0, -4, 3, 0, 0, 2, 4, -3, 0, 0, -2], [15, 0, -1, -4, -17, -3, 0, 0, -3, 2, 0, 0, 0, 0, 2, 0], [4, 0, 4, 0, 1, 2, 0, 0, 4, 0, 0, 0, 0, 0, -4, 0], [-14, 0, -4, 12, 6, 12, 0, 0, 12, -6, 0, 0, 0, 0, 8, 0], [0, 6, 0, 0, 0, 0, -4, 0, 0, 0, 5, 5, 5, 3, 0, -4], [0, -10, 0, 0, 0, 0, 2, -1, 0, 0, -8, -4, -2, -10, 0, 4], [-6, 0, -1, 2, 4, -1, 0, 0, -1, -1, 0, 0, 0, 0, 2, 0], [0, -1, 0, 0, 0, 0, 0, -2, 0, 0, -15, 0, -2, -2, 0, 3], [-15, 0, 0, 9, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 7, 0], [12, 0, -4, -3, 4, 2, 0, 0, 0, -3, 0, 0, 0, 0, -4, 0], [0, 2, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1, 1, 0, -3], [-1, 0, -1, 0, -1, -1, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0], [0, 2, 0, 0, 0, 0, 0, 7, 0, 0, -4, 0, 7, 4, 0, -3], [-2, 0, 2, 2, -6, 0, 0, 0, 0, -4, 0, 0, 0, 0, -1, 0], [-21, 0, 6, 0, -21, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0], [3, 0, 3, 3, 7, -3, 0, 0, -6, -3, 0, 0, 0, 0, -3, 0], [0, -2, 0, 0, 0, 0, -2, 1, 0, 0, 14, 7, 2, -2, 0, -4], [23, 0, -3, 5, 10, 8, 0, 0, 0, 5, 0, 0, 0, 0, -3, 0], [0, -6, 0, 0, 0, 0, 0, 4, 0, 0, 1, 2, -4, 0, 0, 0], [0, -1, 0, 0, 0, 0, 1, -10, 0, 0, 0, -7, 0, 1, 0, 0], [0, 4, 0, 0, 0, 0, -5, -3, 0, 0, 0, -12, 0, -4, 0, 0], [11, 0, -1, -1, 5, -1, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0], [4, 0, 6, 6, 8, 4, 0, 0, 4, -3, 0, 0, 0, 0, -12, 0], [0, 0, -2, -1, 2, 0, 0, 0, -13, 2, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, -6, 7, 0, 0, 0, -8, 0, 1, 0, 0], [24, 0, -2, -6, 11, 4, 0, 0, 0, -6, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, 0, -11, -11, -3, 0, 0, 6], [-16, 0, -10, 4, -22, 0, 0, 0, -9, -8, 0, 0, 0, 0, 5, 0], [0, 1, 0, 0, 0, 0, 6, -4, 0, 0, 1, 2, 4, 0, 0, 3], [0, 8, 0, 0, 0, 0, 1, 0, 0, 0, -7, -7, -1, 4, 0, 1], [0, 0, 0, 0, 0, 0, -2, -6, 0, 0, -5, 5, -3, 2, 0, 2], [0, 0, 0, -4, 12, -3, 0, 0, -6, 4, 0, 0, 0, 0, 0, 0], [13, 0, 12, 0, 13, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, -9, 6, 0, 0, 0, 11, 0, -3, 0, 0], [0, 0, 0, 0, 0, 0, -1, 10, 0, 0, -5, 5, 5, -4, 0, 1], [33, 0, -3, 9, 15, 11, 0, 0, 0, 9, 0, 0, 0, 0, -3, 0], [-16, 0, -7, -6, 2, -1, 0, 0, -1, 3, 0, 0, 0, 0, 14, 0], [10, 0, -1, -2, -12, 5, 0, 0, 5, 1, 0, 0, 0, 0, 2, 0], [0, -3, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 1, -6, 0, 1], [0, -6, 0, 0, 0, 0, -4, 0, 0, 0, -1, -1, -7, -3, 0, -4], [15, 0, 15, -1, 11, 0, 0, 0, 0, 1, 0, 0, 0, 0, -15, 0], [0, 0, 0, 0, 0, 0, -4, 6, 0, 0, -9, 9, 3, -10, 0, 4], [0, -5, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, -8, -10, 0, -4], [-24, 0, -6, 4, -42, 0, 0, 0, 4, -8, 0, 0, 0, 0, 3, 0], [10, 0, 11, 0, 10, 3, 0, 0, -3, -7, 0, 0, 0, 0, 0, 0], [0, -14, 0, 0, 0, 0, 3, 0, 0, 0, 3, 3, -2, -7, 0, 3], [-26, 0, 0, 6, 0, 10, 0, 0, 5, 0, 0, 0, 0, 0, 9, 0], [0, 0, 0, 0, 0, 0, -2, -4, 0, 0, 3, -3, -2, 5, 0, 2], [0, -2, 0, 0, 0, 0, 1, 5, 0, 0, 10, 5, 10, -2, 0, 2], [-21, 0, -3, 1, -12, 14, 0, 0, 0, 1, 0, 0, 0, 0, -3, 0], [21, 0, 3, 0, 21, 2, 0, 0, -2, 13, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, -4, -1, 0, 0, -4, -2, -2, 1, 0, -8], [0, -1, 0, 0, 0, 0, 2, 0, 0, 0, 12, 6, 0, -1, 0, 4], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, -2, 0, 3], [-7, 0, -1, 7, -4, 7, 0, 0, 0, 7, 0, 0, 0, 0, -1, 0], [8, 0, 10, 2, 6, 0, 0, 0, 16, -4, 0, 0, 0, 0, -5, 0], [0, -2, 0, 0, 0, 0, 0, -5, 0, 0, -4, -8, 5, 0, 0, 0], [11, 0, -5, 0, 11, 10, 0, 0, -10, 10, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -8, 7, 0, 0, -3, -6, -7, 0, 0, -4], [4, 0, 4, 16, -9, 9, 0, 0, 18, -16, 0, 0, 0, 0, -4, 0], [0, -2, 0, 0, 0, 0, 1, -7, 0, 0, 4, 2, -14, -2, 0, 2], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 5, 10, -2, 0, 0, 0], [-9, 0, -3, 18, 3, 2, 0, 0, 2, -9, 0, 0, 0, 0, 6, 0], [0, -4, 0, 0, 0, 0, -4, -9, 0, 0, 0, -3, 0, 4, 0, 0], [7, 0, 7, 9, 4, 8, 0, 0, 16, -9, 0, 0, 0, 0, -7, 0], [-12, 0, 4, -5, -4, -6, 0, 0, 0, -5, 0, 0, 0, 0, 4, 0], [18, 0, -2, -1, 38, 0, 0, 0, -7, 2, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 6, 2, 0, 0, 8, 4, 4, -1, 0, 12], [4, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, -6, 0, 0, 0, 0, 0, -11, 0, 0, -6, 0, -11, -12, 0, 0], [-9, 0, 0, 7, 0, -10, 0, 0, -5, 0, 0, 0, 0, 0, 3, 0], [0, 4, 0, 0, 0, 0, -6, -8, 0, 0, 0, 6, 0, -4, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 5, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 16, 0, 0, -6, 6, 8, -6, 0, 1], [0, 1, 0, 0, 0, 0, 10, -4, 0, 0, -2, -4, 4, 0, 0, 5], [-6, 0, 0, 11, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, -15, 0], [-6, 0, -6, -3, -8, 7, 0, 0, 14, 3, 0, 0, 0, 0, 6, 0], [1, 0, -5, 3, -2, 8, 0, 0, 0, 3, 0, 0, 0, 0, -5, 0], [0, -4, 0, 0, 0, 0, 6, -3, 0, 0, 1, 2, 3, 0, 0, 3], [0, 3, 0, 0, 0, 0, -2, -1, 0, 0, 0, 11, 0, -3, 0, 0], [-4, 0, -4, -14, 16, -10, 0, 0, -20, 14, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 8, -8, 3, -1, 0, -4], [-29, 0, -3, -12, 23, -8, 0, 0, -8, 6, 0, 0, 0, 0, 6, 0], [0, -4, 0, 0, 0, 0, 6, 0, 0, 0, -1, -1, 4, -2, 0, 6], [9, 0, 0, 3, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, -11, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 11, 0, 2, 0, 0, -8], [0, -5, 0, 0, 0, 0, -4, 7, 0, 0, 0, -1, 0, 5, 0, 0], [2, 0, -6, 10, -2, 1, 0, 0, 0, 10, 0, 0, 0, 0, -6, 0], [0, -6, 0, 0, 0, 0, 0, -8, 0, 0, -2, 0, -8, -12, 0, -5], [0, 4, 0, 0, 0, 0, -4, 0, 0, 0, -10, -10, 12, 2, 0, -4], [-3, 0, -3, -7, -36, 1, 0, 0, 2, 7, 0, 0, 0, 0, 3, 0], [-28, 0, -6, 2, -50, 0, 0, 0, -2, -4, 0, 0, 0, 0, 3, 0], [0, -6, 0, 0, 0, 0, 4, -2, 0, 0, -4, -8, 2, 0, 0, 2], [4, 0, 4, 4, 27, 1, 0, 0, 2, -4, 0, 0, 0, 0, -4, 0], [-34, 0, -2, 7, -18, -4, 0, 0, 0, 7, 0, 0, 0, 0, -2, 0], [0, -1, 0, 0, 0, 0, 8, -2, 0, 0, 7, 14, 2, 0, 0, 4], [22, 0, 15, 0, 22, -5, 0, 0, 5, -3, 0, 0, 0, 0, 0, 0], [2, 0, 1, -10, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, -2, 0], [0, 4, 0, 0, 0, 0, 5, 0, 0, 0, 6, 6, -12, 2, 0, 5], [22, 0, 8, 1, 36, 0, 0, 0, 8, -2, 0, 0, 0, 0, -4, 0], [0, -7, 0, 0, 0, 0, 4, 5, 0, 0, -18, -9, 10, -7, 0, 8], [0, 0, 0, 0, 0, 0, 4, 5, 0, 0, 0, -11, 0, 0, 0, 0], [23, 0, 11, 0, 17, -2, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0], [0, 2, 0, 0, 0, 0, -6, 0, 0, 0, 21, 21, 1, 1, 0, -6], [3, 0, 10, 1, -4, 0, 0, 0, 7, -2, 0, 0, 0, 0, -5, 0], [-6, 0, -6, 3, -6, 0, 0, 0, -3, -6, 0, 0, 0, 0, 3, 0], [-10, 0, -4, 0, -10, 4, 0, 0, -4, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 14, 0], [28, 0, -2, -4, 13, -17, 0, 0, 0, -4, 0, 0, 0, 0, -2, 0], [9, 0, 13, 0, 9, -1, 0, 0, 1, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -8, 0, 0, 0, -3, 0, 0, 0, 0], [6, 0, -1, 14, -8, 11, 0, 0, 11, -7, 0, 0, 0, 0, 2, 0], [-5, 0, 5, -5, 0, -6, 0, 0, 0, -5, 0, 0, 0, 0, 5, 0], [0, 12, 0, 0, 0, 0, -3, 0, 0, 0, -6, -6, 5, 6, 0, -3], [0, -4, 0, 0, 0, 0, -3, -5, 0, 0, 6, 3, -10, -4, 0, -6], [-32, 0, 0, 1, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0], [0, 2, 0, 0, 0, 0, -5, 0, 0, 0, -8, -8, -8, 1, 0, -5], [-9, 0, 14, 5, -32, 0, 0, 0, 2, -10, 0, 0, 0, 0, -7, 0], [3, 0, -15, 0, 3, -1, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0], [0, -10, 0, 0, 0, 0, 4, 0, 0, 0, -12, -12, 12, -5, 0, 4], [-33, 0, 0, 0, 0, -4, 0, 0, -2, 0, 0, 0, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, 4, -2, 0, 0, -6, 6, -1, 5, 0, -4], [22, 0, -8, 0, 22, -1, 0, 0, 1, -15, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, 3, 2, 0, 0, -2, -1, 4, 3, 0, 6], [0, -4, 0, 0, 0, 0, -3, -12, 0, 0, 0, -11, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, -5, 12, 0, 0, 0, -8, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, -10, 0, 0, 10, -10, -5, 4, 0, -1], [0, 4, 0, 0, 0, 0, 5, -1, 0, 0, -16, -8, -2, 4, 0, 10], [-13, 0, -6, 12, 1, 14, 0, 0, 14, -6, 0, 0, 0, 0, 12, 0], [0, 8, 0, 0, 0, 0, -1, -2, 0, 0, 0, 4, 0, -8, 0, 0], [-3, 0, 1, -7, -1, -2, 0, 0, 0, -7, 0, 0, 0, 0, 1, 0], [0, -3, 0, 0, 0, 0, 0, 7, 0, 0, -5, 0, 7, -6, 0, -5], [0, -7, 0, 0, 0, 0, 12, -5, 0, 0, 2, 4, 5, 0, 0, 6], [6, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 0, 11, 0], [54, 0, 2, 0, 54, -1, 0, 0, 1, -2, 0, 0, 0, 0, 0, 0], [0, 3, 0, 0, 0, 0, -10, 3, 0, 0, 10, 20, -3, 0, 0, -5], [-3, 0, -3, 2, 29, -8, 0, 0, -16, -2, 0, 0, 0, 0, 3, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2736_dc_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2736_2_dc_c();".
// 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_2736_2_dc_c(:prec:=16)
    chi := MakeCharacter_2736_dc();
    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_2736_2_dc_c();".
// 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_2736_2_dc_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2736_dc();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![104976, 0, -290304, 0, 709504, 0, -238608, 0, 55740, 0, -6848, 0, 612, 0, -30, 0, 1]>,<17,R![26873856, 0, -19408896, 0, 9787392, 0, -2474496, 0, 451008, 0, -38208, 0, 2320, 0, -56, 0, 1]>],Snew);
    return Vf;
end function;