
// Make newform 855.2.k.j in Magma, downloaded from the LMFDB on 11 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_855_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_855_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_855_2_k_j();".
// 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_855_2_k_j();".
// 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, 12, 46, 14, 83, 13, 119, -20, 44, -10, 13, -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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-31535764, -3413944, -12566274, -41624620, -19407929, -8355646, -6476307, -2131461, -2066475, 319009, -219232, -9467], [20278716, 1982264, 7227938, -14595442, 10715465, 5464606, 376383, 2267093, -175781, 1240077, -330656, 119083], [4981404, -73592600, 15402354, -136052648, -15993601, -216412954, 39591841, -80346361, 18411181, -24130015, 5628524, -1873019], [12751576, 13838716, 50666432, 17848301, 76473900, 36304389, 17628827, 12727847, 2914321, 4518956, -929709, 489392], [-11971084, 96458496, -67090322, 216199176, -283097945, 348687850, -168306583, 135956727, -62266695, 38772881, -9651444, 2233513], [121494172, 80353640, 294408850, 197090848, 445797105, 208854210, 104402375, 70637065, 21286535, 21441087, -3945300, 2481235], [108328060, 70577144, 258097970, -34697952, 387593873, 187833166, 76101307, 69415229, 8845171, 29254579, -6744800, 2992771], [271084, 1059536, -171038, 2039652, 111321, 3756374, -758053, 1403197, -344149, 429903, -101560, 33067], [2856086, -37251044, 13483075, -60874036, 42613533, -97570459, 34016485, -36937486, 13343636, -10862733, 2635302, -723138], [14863148, -119868512, 67783378, -325565256, 191517271, -521338058, 171226187, -199825235, 68012787, -58052365, 13919486, -3942893]];
    Rf_basisdens := [1, 1, 24043688, 12021844, 24043688, 6010922, 24043688, 24043688, 24043688, 212776, 3005461, 12021844];
    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_855_k();"
function MakeCharacter_855_k()
    N := 855;
    order := 3;
    char_gens := [191, 172, 496];
    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_855_k_Hecke();"
function MakeCharacter_855_k_Hecke(Kf)
    N := 855;
    order := 3;
    char_gens := [191, 172, 496];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 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, 1, 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, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, -1, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0], [0, -1, 1, 0, 1, 1, 1, 1, 0, 0, -1, 0], [0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0], [0, -1, 1, -1, 0, -1, 0, 0, 1, 1, 0, -1], [2, -2, 2, 0, 0, 1, 1, 1, -1, 1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 1, -1, 2, -1, 1], [3, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -3, 0, 1, 0, 0, 0, 0], [0, -2, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0], [0, -3, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0], [-1, 1, -1, 0, 0, 4, 4, -1, 0, -1, 1, 0], [-3, -1, 1, 0, 4, -1, -1, 0, 1, 2, 0, -1], [0, 0, 0, 0, -2, 0, -2, 0, 0, 0, 1, -1], [2, 1, -1, 0, -3, 1, 1, -2, 1, -2, 2, -1], [-4, 1, -1, 0, 3, -3, -3, -1, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, 0, -1, -3], [0, -1, 0, 0, -1, 0, 1, 0, 0, 0, 1, -2], [0, 1, 0, 0, -3, 0, -3, 0, 0, 0, -2, -1], [5, 0, 3, 3, 0, -2, 0, 1, -2, 0, 0, 0], [2, -2, 2, 0, 0, 2, 2, -3, 1, 0, 3, -1], [0, -2, 0, -1, 6, 0, -1, 0, 0, 1, -3, 3], [0, 2, -2, 0, -2, -2, -2, 1, 1, -4, -1, -1], [0, 0, -1, 1, 0, 2, 0, 0, -3, 0, 0, 0], [7, 0, 1, 3, 0, 4, 0, -1, 2, 0, 0, 0], [0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, -3], [1, 0, -3, -4, 0, 1, 0, 2, 3, 0, 0, 0], [-2, -2, 2, 0, 4, 1, 1, -1, 1, -5, 1, -1], [0, -2, 0, 0, 2, 0, 0, 0, 0, 0, 4, -2], [-3, -1, 1, 0, 4, -3, -3, -2, -1, 2, 2, 1], [5, -2, 2, 0, -3, 5, 5, -4, 2, -1, 4, -2], [0, 6, 0, 6, -4, 0, -4, 0, 0, -6, -3, 1], [-1, 0, 1, -1, 0, 2, 0, 4, 1, 0, 0, 0], [0, -7, 0, -1, 9, 0, 0, 0, 0, 1, 0, 3], [6, 0, -5, -1, 0, 0, 0, 2, -1, 0, 0, 0], [-7, -1, 1, 0, 8, -1, -1, 0, 3, 2, 0, -3], [0, 2, 0, 2, 0, 0, 0, 0, 0, -2, -4, 2], [8, 0, -4, 2, 0, 2, 0, -1, 1, 0, 0, 0], [-1, 1, -1, 0, 0, 5, 5, 2, 1, -2, -2, -1], [-4, 0, 0, 0, 0, 0, 0, -3, 3, 0, 0, 0], [0, 7, 0, -1, -11, 0, 2, 0, 0, 1, 2, -1], [13, 0, 1, 1, 0, 2, 0, 1, 2, 0, 0, 0], [-2, 3, -3, 0, -1, 1, 1, 0, 5, 2, 0, -5], [0, -2, 0, 2, 3, 0, -2, 0, 0, -2, 0, 0], [0, 1, 0, -2, -1, 0, 3, 0, 0, 2, 7, -2], [0, 0, 2, -1, 0, -3, 0, 5, -1, 0, 0, 0], [-1, 0, -6, 2, 0, 0, 0, 4, -2, 0, 0, 0], [0, -1, 0, 2, 10, 0, -1, 0, 0, -2, 4, -1], [-8, 0, -6, 2, 0, -2, 0, -5, -1, 0, 0, 0], [-3, -6, 6, 0, 9, -2, -2, -6, 4, 0, 6, -4], [12, -8, 8, 0, -4, 0, 0, -1, 1, 6, 1, -1], [8, -4, 4, 0, -4, -3, -3, 1, -3, 3, -1, 3], [0, -9, 0, -2, 8, 0, -3, 0, 0, 2, 0, 1], [0, -4, 0, 2, -10, 0, 0, 0, 0, -2, 3, 1], [0, -9, 0, 1, 6, 0, 2, 0, 0, -1, 2, 1], [-6, 0, 0, 6, 0, -2, 0, 0, 4, 0, 0, 0], [-8, 2, -2, 0, 6, 8, 8, -2, -4, -2, 2, 4], [0, 4, 0, -3, -4, 0, -1, 0, 0, 3, -5, -1], [9, 0, 7, 0, 0, -1, 0, -2, 5, 0, 0, 0], [0, 2, 0, -4, 0, 0, 2, 0, 0, 4, 4, 2], [14, 0, 0, -2, 0, 6, 0, 0, 0, 0, 0, 0], [4, 2, -2, 0, -6, 5, 5, -3, -5, 3, 3, 5], [1, -1, 1, 0, 0, -4, -4, 1, -2, -1, -1, 2], [-2, 0, 1, 5, 0, -2, 0, 0, -1, 0, 0, 0], [0, -1, 0, 4, -1, 0, -3, 0, 0, -4, -5, 4], [0, -3, 0, 2, 10, 0, 3, 0, 0, -2, -4, -1], [-3, 0, -2, 1, 0, -3, 0, 2, 2, 0, 0, 0], [-10, 0, 4, -6, 0, -2, 0, 2, -2, 0, 0, 0], [0, 4, 0, -2, -10, 0, -8, 0, 0, 2, -4, -4], [-12, 1, -1, 0, 11, -4, -4, 2, -1, -1, -2, 1], [-12, 0, -2, -6, 0, 4, 0, -4, -2, 0, 0, 0], [-2, 0, 5, 10, 0, 1, 0, -6, -5, 0, 0, 0], [0, -1, 0, -11, -16, 0, 0, 0, 0, 11, -3, 2], [-14, -2, 2, 0, 16, 2, 2, -3, 1, -8, 3, -1], [0, 7, 0, 1, -11, 0, -8, 0, 0, -1, -2, -1], [0, 4, 0, -2, -2, 0, 8, 0, 0, 2, 11, 1], [2, -4, 4, 0, 2, -5, -5, 2, -4, 1, -2, 4], [-10, 0, -12, -2, 0, 4, 0, -1, -7, 0, 0, 0], [0, 12, 0, 5, 2, 0, -3, 0, 0, -5, -2, 0], [-8, 10, -10, 0, -2, 6, 6, 1, 5, -4, -1, -5], [-16, 9, -9, 0, 7, 0, 0, 4, 1, -3, -4, -1], [0, -2, 0, 0, -3, 0, -4, 0, 0, 0, 2, 2], [3, -7, 7, 0, 4, 7, 7, 0, -3, 2, 0, 3], [14, 0, 4, 4, 0, 0, 0, -1, -7, 0, 0, 0], [8, 0, -1, 6, 0, -3, 0, 7, -4, 0, 0, 0], [0, 4, 0, 0, 8, 0, 0, 0, 0, 0, 1, 7], [-2, 0, -1, 3, 0, 4, 0, 2, -5, 0, 0, 0], [-7, 0, -7, -1, 0, 6, 0, 1, -8, 0, 0, 0], [10, -4, 4, 0, -6, 4, 4, 3, -3, 4, -3, 3], [10, 0, 6, -5, 0, 5, 0, -3, -1, 0, 0, 0], [0, -2, 0, -2, 0, 0, 6, 0, 0, 2, 1, 1], [0, -2, 0, 7, 11, 0, -5, 0, 0, -7, 0, -6], [-15, 11, -11, 0, 4, 2, 2, 3, 6, -1, -3, -6], [-4, -6, 6, 0, 10, -2, -2, 2, -8, -2, -2, 8], [18, 0, 8, 0, 0, 2, 0, -3, -1, 0, 0, 0], [-16, -1, 1, 0, 17, -2, -2, 2, 1, -1, -2, -1], [3, 0, 0, 0, -3, -4, -4, 0, -4, 4, 0, 4], [-6, -1, 1, 0, 7, 3, 3, -5, 2, 8, 5, -2], [-6, -2, 2, 0, 8, 1, 1, -7, 3, -5, 7, -3], [19, 0, -5, -4, 0, -3, 0, 4, 1, 0, 0, 0], [-6, 0, -2, -10, 0, -2, 0, -9, 7, 0, 0, 0], [9, 0, -8, 0, 0, -2, 0, -2, 8, 0, 0, 0], [6, 0, -4, 1, 0, 5, 0, 3, -1, 0, 0, 0], [0, 5, 0, 2, -16, 0, -7, 0, 0, -2, 0, -5], [9, -7, 7, 0, -2, 1, 1, 6, -7, 4, -6, 7], [10, 8, -8, 0, -18, -2, -2, 2, -2, -2, -2, 2], [14, 0, 13, 2, 0, -5, 0, 0, 5, 0, 0, 0], [4, 0, -1, -8, 0, -3, 0, 5, -6, 0, 0, 0], [0, -6, 0, -2, 14, 0, 4, 0, 0, 2, -2, 8], [-19, 11, -11, 0, 8, 2, 2, -1, -4, -7, 1, 4], [9, 0, 4, -5, 0, -1, 0, 10, -2, 0, 0, 0], [-3, 2, -2, 0, 1, -4, -4, 6, -6, 8, -6, 6], [0, 0, 0, -2, -14, 0, 2, 0, 0, 2, -3, 3], [0, -9, 0, -12, -13, 0, -1, 0, 0, 12, -3, -2], [-7, 0, -1, -3, 0, 0, 0, -1, 4, 0, 0, 0], [22, 0, 14, 5, 0, -9, 0, 5, 7, 0, 0, 0], [-4, -4, 4, 0, 8, 0, 0, 4, 0, -2, -4, 0], [15, 5, -5, 0, -20, -4, -4, 0, 9, -5, 0, -9], [-6, 0, -9, -5, 0, 6, 0, -2, -5, 0, 0, 0], [13, 0, 5, -4, 0, -5, 0, 0, 1, 0, 0, 0], [11, 0, -7, 5, 0, 4, 0, -5, -2, 0, 0, 0], [1, 0, 3, -3, 0, 6, 0, -2, 5, 0, 0, 0], [0, -14, 0, 0, 16, 0, 0, 0, 0, 0, -4, 6], [-7, 2, -2, 0, 5, -8, -8, 2, -4, 2, -2, 4], [0, 4, 0, -10, -4, 0, -2, 0, 0, 10, -1, 1], [0, -7, 0, -3, 7, 0, 4, 0, 0, 3, 4, -5], [2, 0, 0, -3, 0, -7, 0, 1, -3, 0, 0, 0], [0, 4, 0, 2, -3, 0, 4, 0, 0, -2, 6, 0], [0, -15, 0, -13, 14, 0, 2, 0, 0, 13, 7, 2], [11, -16, 16, 0, 5, -6, -6, -4, 2, 0, 4, -2], [0, -9, 0, 6, 3, 0, -5, 0, 0, -6, 3, -8], [-8, 0, 6, -8, 0, 2, 0, 0, -2, 0, 0, 0], [-5, -2, 2, 0, 7, -2, -2, 10, 0, 0, -10, 0], [-10, 6, -6, 0, 4, -15, -15, -3, -1, -1, 3, 1], [0, 0, -13, -8, 0, 5, 0, 3, 0, 0, 0, 0], [-9, 0, -1, 4, 0, -7, 0, 4, 7, 0, 0, 0], [-16, 0, -18, -2, 0, 2, 0, 3, 3, 0, 0, 0], [15, -13, 13, 0, -2, 0, 0, 8, -7, 5, -8, 7], [-10, 10, -10, 0, 0, -6, -6, -7, 5, -6, 7, -5], [18, 2, -2, 0, -20, 1, 1, -1, -3, 7, 1, 3], [-5, -1, 1, 0, 6, 2, 2, 1, 4, 5, -1, -4], [-7, 0, 0, -3, 0, -1, 0, -4, 2, 0, 0, 0], [0, -12, 0, 0, 0, 0, 6, 0, 0, 0, 10, -2], [0, 7, 0, 1, 9, 0, 12, 0, 0, -1, 8, -3], [0, -1, 0, 5, 16, 0, 2, 0, 0, -5, -5, 2], [-5, 2, -2, 0, 3, -4, -4, 2, 6, -12, -2, -6], [8, 0, 16, -2, 0, 6, 0, -4, 0, 0, 0, 0], [0, -1, 0, -8, 5, 0, -7, 0, 0, 8, -3, -6], [6, 0, -8, 0, 0, 2, 0, -3, -1, 0, 0, 0], [-4, 5, -5, 0, -1, -8, -8, -6, 3, 3, 6, -3], [-5, -1, 1, 0, 6, 9, 9, -2, -3, -2, 2, 3], [0, -12, 0, 0, -4, 0, -4, 0, 0, 0, 0, 4], [-2, 0, 4, 10, 0, -6, 0, 1, -5, 0, 0, 0], [-17, 0, -12, -1, 0, -5, 0, -4, 4, 0, 0, 0], [0, 10, 0, -8, -16, 0, 6, 0, 0, 8, -9, 7], [-4, 17, -17, 0, -13, 4, 4, -4, 1, -3, 4, -1], [4, -6, 6, 0, 2, -10, -10, 3, -1, -4, -3, 1], [0, -4, 0, -9, 26, 0, -1, 0, 0, 9, 5, 1], [0, 11, 0, -8, -36, 0, 5, 0, 0, 8, 4, -1], [0, 3, 0, 10, -23, 0, 9, 0, 0, -10, -7, -2], [0, -10, 0, -4, -10, 0, 4, 0, 0, 4, -4, -2], [23, 0, 9, -2, 0, 7, 0, 2, 3, 0, 0, 0], [0, -3, 0, 0, -8, 0, -17, 0, 0, 0, -2, -3], [0, 0, 0, 11, 9, 0, 1, 0, 0, -11, 4, -4], [8, -7, 7, 0, -1, -1, -1, 13, 2, 0, -13, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_855_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_855_2_k_j();".
// 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_855_2_k_j(:prec:=12)
    chi := MakeCharacter_855_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_855_2_k_j();".
// 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_855_2_k_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_855_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 24, 146, 48, 189, 117, 189, 78, 62, 18, 13, 3, 1]>,<7,R![-64, -256, 180, 44, -27, -2, 1]>],Snew);
    return Vf;
end function;