
// Make newform 864.2.p.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_864_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_864_p_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_864_2_p_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_864_2_p_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 := [256, -384, 448, -384, 256, -96, -32, 72, -68, 36, -8, -12, 16, -12, 7, -3, 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], [384, -1152, 608, -32, -240, 440, -344, 148, 68, -104, 60, -38, 4, 11, -13, 1], [512, -1088, 1184, -416, 128, 8, -272, 244, -68, -8, 24, -46, 38, -25, -1, -1], [-640, 128, -1088, 1024, -720, 480, -24, -172, 316, -144, 40, 40, -44, 33, -25, 3], [-384, 1600, -2400, 2496, -1216, 456, 288, -540, 492, -176, 24, 66, -102, 59, -29, -1], [-704, 544, -256, -352, 440, -424, 192, 4, -92, 116, -54, 2, 16, -19, 11, -3], [64, -1312, 1072, -304, -152, 260, -300, 272, 16, -92, 80, -39, 17, 3, -8, 6], [-1408, 1536, -512, -256, 880, -736, 552, -76, -100, 176, -136, 32, 4, -31, 15, -13], [1536, -3040, 2768, -2144, 888, 52, -648, 592, -316, 60, 100, -117, 100, -47, 18, -3], [-1728, 928, -944, 592, -600, -20, 176, -92, 128, -64, 24, 45, -32, 24, -15, 6], [-768, 1184, -1888, 1520, -920, 184, 212, -408, 284, -100, -22, 76, -71, 48, -16, 5], [-4608, 5760, -4608, 1280, 752, -1696, 1608, -740, -60, 464, -384, 232, -48, -13, 37, -19], [3072, -4288, 3968, -2720, 1216, 104, -680, 652, -324, 120, 88, -150, 116, -59, 29, -13], [-320, -2400, 3152, -3072, 2440, -964, -124, 768, -640, 376, -36, -113, 153, -99, 40, -16], [2816, -3968, 6848, -6432, 4176, -1216, -600, 1300, -1172, 712, -64, -232, 244, -183, 103, -37], [5888, -12288, 14400, -12512, 7856, -1952, -1448, 2932, -2308, 1112, 24, -536, 528, -375, 167, -57]];
    Rf_basisdens := [1, 896, 896, 896, 896, 448, 448, 896, 448, 448, 448, 896, 448, 448, 896, 896];
    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_864_p();"
function MakeCharacter_864_p()
    N := 864;
    order := 6;
    char_gens := [703, 325, 353];
    v := [3, 3, 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_864_p_Hecke();"
function MakeCharacter_864_p_Hecke(Kf)
    N := 864;
    order := 6;
    char_gens := [703, 325, 353];
    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], [0, -1, 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, 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, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [1, -1, 1, 0, 0, -1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, -1, 0], [0, -1, 0, 1, 0, 1, 0, -2, -1, 0, 0, 0, 0, 0, 0, 0], [1, -1, 1, 1, 0, -1, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, -1, 1, 1, 0, -1], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, -1], [0, 0, 0, 0, 1, 0, 1, 0, 0, -2, 0, 1, 1, -1, -1, 0], [0, 0, 0, 0, -1, 0, -1, 0, 0, -1, 0, -2, 0, 1, 1, -2], [4, 3, -1, -1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [2, 1, -1, 1, 0, 2, 0, -2, 1, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, -1, -1, 1], [0, 0, 0, 0, -2, 0, -1, 0, 0, 1, 0, -1, 1, 0, 1, 0], [0, -2, 2, 2, 0, -3, 0, 0, -3, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, -1, 0, -1, 1], [0, -4, 1, -1, 0, -2, 0, 6, 2, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -1, 0, 0, 1, 0, 0, 0, -1, -1, 0], [2, 1, 2, -1, 0, 1, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, 0, -1, 0, -1, 1], [1, 1, -2, -2, 0, 4, 0, -5, -2, 0, -5, 0, 0, 0, 0, 0], [-1, 2, 3, -4, 0, -4, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [4, -1, 4, 5, 0, 1, 0, -3, -4, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 0, -2, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -2, 0, 0, 4, 0, 2, 2, -2, -1, 1], [-5, -7, 1, -3, 0, -3, 0, 5, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, -1, 0, 1, 0], [4, -2, 4, 4, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 2, 0, 0, 2, 0, -1, -1, 1, 2, -2], [-6, -1, -2, -2, 0, 4, 0, 0, 4, 0, -1, 0, 0, 0, 0, 0], [-2, 7, 0, -5, 0, 5, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 6, -4, 4, 0, 5, 0, -4, -5, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 3, -1, -1, 1, 0], [0, 0, 0, 0, 2, 0, 6, 0, 0, -3, 0, 2, -1, 2, -1, 1], [0, 0, 0, 0, 2, 0, -1, 0, 0, 2, 0, 0, 0, 0, 1, 1], [-2, -4, 2, 4, 0, -4, 0, -2, -2, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 4, 0, 0, 0, 0, 4, -2, -2, -1, 3], [0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, -1, -3, 1, 3, 1], [4, 8, 2, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 3, 0, 0, 3, 0, 4, -2, -1, 1, 2], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -5, -2, 2, 2, -2], [0, -1, -3, 3, 0, 9, 0, -6, -9, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, -2, 0, 0, -2, 0, -2, 0], [0, 0, 0, 0, 2, 0, -1, 0, 0, -1, 0, 1, -3, -2, 1, 4], [0, 7, 3, -3, 0, -1, 0, 4, 1, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, -4, 0, 0, 2, 0, -2, 1, 3, 1, -1], [3, -5, 3, 2, 0, -5, 0, 1, 3, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, 6, 0, 0, 0, 0, -1, -1], [1, 5, 5, -3, 0, -3, 0, -5, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, -3, -1, -1, -2, 3], [-11, -11, 6, 0, 0, -6, 0, 4, -6, 0, -4, 0, 0, 0, 0, 0], [-7, -13, 3, -1, 0, -1, 0, -1, -2, 0, 0, 0, 0, 0, 0, 0], [-14, -11, 4, 4, 0, -6, 0, 0, -6, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 5, 0, 5, 4, -3, -4, 2], [0, 0, 0, 0, 2, 0, -1, 0, 0, 1, 0, 3, 3, -4, -1, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, -2, 0, 0, 1, 0, 4, -2, -2, -2, 2], [5, 1, 3, -6, 0, 3, 0, 2, 3, 0, 2, 0, 0, 0, 0, 0], [0, -5, 6, -6, 0, -4, 0, 2, 4, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, -3, 1], [1, 9, -7, -9, 0, 9, 0, -1, 2, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -3, 0, 0, 0, 0, -3, -1, -1, 3, -2], [7, 5, -2, 2, 0, 4, 0, -6, 2, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, -2, 0, 2, 2, -3, -2, 4], [-10, -7, 2, -3, 0, -5, 0, 4, -2, 0, -4, 0, 0, 0, 0, 0], [0, -5, -10, 10, 0, 2, 0, 8, -2, 0, 4, 0, 0, 0, 0, 0], [22, 14, -3, -3, 0, 4, 0, 0, 4, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 2, 0, 0, -1, 0, -1, 3, 1, 3, -3], [-2, -9, -2, 11, 0, -9, 0, -1, -2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, -2, 0, 0, 2, 0, -4, -2, 2, 0, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 1, 0, 0, 1, -4, 1, -1], [0, 0, 0, 0, -6, 0, 2, 0, 0, -4, 0, 1, 1, -1, -1, 0], [-2, -1, -2, 1, 0, -1, 0, 6, -3, 0, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, -4, 0, 0, 0, 0, 8, 2, 2, -1, -1], [0, 0, 0, 0, -3, 0, 0, 0, 0, -1, 0, 1, 1, 1, -1, -3], [0, 0, 0, 0, 1, 0, -1, 0, 0, -1, 0, -4, 0, -1, -1, 2], [12, -5, 11, 11, 0, 1, 0, 0, 1, 0, 19, 0, 0, 0, 0, 0], [0, -1, 6, -6, 0, -6, 0, -4, 6, 0, -2, 0, 0, 0, 0, 0], [-12, -13, 7, 7, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 8, 0, 0, -4, 0, -2, 0, -3, 0, 0], [0, 0, 0, 0, 6, 0, -4, 0, 0, 4, 0, 6, 4, -6, -2, 0], [8, -9, 4, 9, 0, -9, 0, 6, -5, 0, 12, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, -2, 0, 0, 1, 0, -3, 4, -1, 4, -4], [-10, 11, -9, -1, 0, 10, 0, -20, -9, 0, -20, 0, 0, 0, 0, 0], [-8, -13, -4, -3, 0, -3, 0, 4, 7, 0, 0, 0, 0, 0, 0, 0], [-8, -5, 0, -3, 0, -3, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 1, 0, -2], [0, 0, 0, 0, -1, 0, 1, 0, 0, -2, 0, 3, 3, -3, -6, -3], [3, 13, 9, -7, 0, -7, 0, -1, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 0, 0, 0, -5, 0, 5, 4, -5, -4, 6], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, -2, 0], [20, 10, 0, 0, 0, 7, 0, 0, 7, 0, 8, 0, 0, 0, 0, 0], [0, 6, 8, -8, 0, 5, 0, -16, -5, 0, -8, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, 0, -9, 0, 0, 9, 0, -3, -1, 2, 1, 0], [0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 7, -1, -1, -4, 5], [-9, -13, 3, -5, 0, -5, 0, -5, 2, 0, 0, 0, 0, 0, 0, 0], [25, -6, 9, 6, 0, -6, 0, 7, 3, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 7, -1, -3, -2, 6], [-7, -15, 1, 8, 0, 7, 0, 1, -1, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -6, 0, 1, 0, 0, -1, 0, -3, -1, 0, -1, 0], [-18, -16, 7, 7, 0, -16, 0, 0, -16, 0, 11, 0, 0, 0, 0, 0], [-3, -3, -10, 6, 0, 4, 0, -10, -10, 0, -10, 0, 0, 0, 0, 0], [0, 10, 3, -3, 0, -6, 0, 14, 6, 0, 7, 0, 0, 0, 0, 0], [-4, -1, 0, 1, 0, -1, 0, 4, -1, 0, 8, 0, 0, 0, 0, 0], [-12, 11, -1, 1, 0, 0, 0, 4, -1, 0, 4, 0, 0, 0, 0, 0], [7, 8, 7, 6, 0, 6, 0, -5, -13, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 6, 0, 0, 0, 0, -6, 4, 4, 1, -5], [10, 11, -8, -1, 0, 7, 0, 5, 8, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, -2, 0, 0, 4, 0, -4, -4, 4, 6, 2], [0, 0, 0, 0, 1, 0, -5, 0, 0, -5, 0, 8, 0, -1, -1, 2], [12, 7, -1, -1, 0, 1, 0, 0, 1, 0, 13, 0, 0, 0, 0, 0], [-7, -11, -1, 4, 0, 5, 0, 5, 1, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 2, 0, 0, 2, 0, -5, -7, -3, 4, 6], [-7, 1, -4, 6, 0, -2, 0, 14, -4, 0, 14, 0, 0, 0, 0, 0], [0, 4, -15, 15, 0, 6, 0, -10, -6, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 8, 0, 0, -8, 0, 6, 4, -6, -2, 0], [0, 0, 0, 0, 6, 0, 3, 0, 0, 0, 0, 10, -2, -2, -3, 5], [14, -7, 2, -7, 0, 5, 0, 4, 2, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 6, 0, 5, 0, 0, -10, 0, -6, -6, 6, 2, -4], [1, -5, 7, 6, 0, -1, 0, -6, -7, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, 0, 0, 1, 0, -3, -7, 3, 7, 1], [-1, -5, -11, 3, 0, 3, 0, -7, 8, 0, 0, 0, 0, 0, 0, 0], [-7, 5, 2, -12, 0, -14, 0, 5, -2, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 7, 0, 0, -7, 0, 6, 4, -7, -3, 0], [0, 0, 0, 0, 11, 0, 6, 0, 0, -3, 0, 11, 1, -1, 1, -1], [0, 0, 0, 0, 7, 0, 10, 0, 0, -10, 0, 1, -5, 5, 0, 0], [0, 0, 0, 0, 6, 0, -2, 0, 0, 1, 0, 6, -3, -2, -3, 3], [0, 0, 0, 0, -6, 0, -6, 0, 0, 12, 0, 1, 1, -1, 2, 3], [-2, -7, 10, 7, 0, -7, 0, 2, 3, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -2, 0, 0, 0, 0, 4, 2, 2, 4, -6], [0, 0, 0, 0, 1, 0, -4, 0, 0, 8, 0, -4, -4, 4, 5, 1], [0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, -8, -4, 0, 4, 0], [-38, -12, -7, -7, 0, 2, 0, 0, 2, 0, -14, 0, 0, 0, 0, 0], [0, 12, -2, 2, 0, -1, 0, -10, 1, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, -1, 0, 0, 1, 0, -2, 0, 5, 5, 0], [0, 5, 10, -10, 0, -6, 0, 2, 6, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 3, -1, -1, 5, -4], [-14, -29, 6, 1, 0, 1, 0, 10, -7, 0, 0, 0, 0, 0, 0, 0], [-11, 1, -3, -1, 0, 1, 0, -11, -2, 0, -22, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, -2, 0, -11, -7, 6, 7, -5], [0, 0, 0, 0, 5, 0, 1, 0, 0, -2, 0, 5, 5, -5, 0, 5], [-20, -36, 10, -4, 0, -4, 0, 2, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 1, 0, 0, 1, 0, -3, -1, -4, -3, 8], [0, 0, 0, 0, -1, 0, 0, 0, 0, -4, 0, -7, -10, 4, 10, 2], [0, 0, 0, 0, -10, 0, -12, 0, 0, 6, 0, -10, 0, 7, 0, 0], [15, -15, 9, 0, 0, -9, 0, 26, 9, 0, 26, 0, 0, 0, 0, 0], [0, 2, 1, -1, 0, 4, 0, -18, -4, 0, -9, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, -12, 0, 0, 12, 0, 3, 3, -3, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, -1, -1, 1, 2, 1], [-1, 6, -1, -8, 0, -8, 0, 13, 9, 0, 0, 0, 0, 0, 0, 0], [1, 9, -19, -9, 0, 9, 0, -7, -10, 0, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 0, 1, 0, 0, 0, 0, 11, -7, -7, -8, 15], [2, 5, -3, -3, 0, 0, 0, 6, 3, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, -15, 0, -1, 4, -1, -4, -2], [0, 0, 0, 0, 3, 0, 6, 0, 0, 6, 0, 11, 1, -3, -4, 6], [-9, 17, 1, -8, 0, 7, 0, -10, 1, 0, -10, 0, 0, 0, 0, 0], [-1, -4, -1, 4, 0, -4, 0, -7, -5, 0, -14, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 3, 0, 0, 0, 0, -6, -2, -2, -3, 5], [-1, -5, -23, 6, 0, 17, 0, -11, -23, 0, -11, 0, 0, 0, 0, 0], [14, 25, -10, 3, 0, 3, 0, -2, 7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, 0, 7, 0, 0, -14, 0, 1, 1, -1, -3, -2], [-19, -32, 5, -6, 0, -6, 0, 23, 1, 0, 0, 0, 0, 0, 0, 0], [-14, -11, 4, 4, 0, -6, 0, 0, -6, 0, 16, 0, 0, 0, 0, 0], [0, 0, 0, 0, -8, 0, 0, 0, 0, 1, 0, 4, -1, 2, 1, -3], [0, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, -10, -6, 2, 8, -4], [0, 0, 0, 0, 3, 0, 18, 0, 0, -9, 0, 3, -1, 1, -1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_864_p_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_864_2_p_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_864_2_p_b(:prec:=16)
    chi := MakeCharacter_864_p();
    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_864_2_p_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_864_2_p_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_864_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![266256, 0, 339012, 0, 312453, 0, 123903, 0, 35106, 0, 4923, 0, 498, 0, 27, 0, 1]>],Snew);
    return Vf;
end function;