
// Make newform 570.2.u.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_570_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_570_u_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_570_2_u_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_570_2_u_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 := [57, -342, 936, -1584, 1905, -1698, 1225, -678, 318, -110, 33, -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], [1060, -3477, 5358, -5701, 3234, -1278, -252, 438, -324, 127, -33, 6], [-1706, 5080, -6002, 5579, -2069, 3777, -1512, 2301, -745, 544, -89, 36], [-2263, 10469, -20623, 24401, -18150, 9021, -2572, -445, 586, -408, 94, -27], [524, -4712, 12972, -19981, 20118, -12623, 6865, -1590, 449, 187, -34, 26], [555, -4618, 15545, -25822, 31404, -23990, 15780, -6686, 2615, -551, 120, -2], [-83, -1957, 9570, -19635, 26758, -23134, 16285, -8075, 3311, -953, 203, -27], [26, -116, 279, -404, 412, -295, 173, -70, 25, -5, 1, 0], [4350, -19689, 41571, -55710, 52308, -36614, 20190, -8138, 2726, -539, 98, 2], [-2201, 13927, -36296, 56973, -60760, 47327, -28887, 13234, -4892, 1277, -252, 26], [-5194, 25899, -58431, 84604, -84255, 64054, -37373, 17125, -6086, 1634, -307, 36], [-5469, 28743, -71337, 112914, -123498, 99754, -62791, 30405, -11563, 3325, -664, 91]];
    Rf_basisdens := [1, 218, 218, 218, 218, 218, 218, 2, 218, 218, 218, 218];
    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_570_u();"
function MakeCharacter_570_u()
    N := 570;
    order := 9;
    char_gens := [191, 457, 211];
    v := [9, 9, 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_570_u_Hecke();"
function MakeCharacter_570_u_Hecke(Kf)
    N := 570;
    order := 9;
    char_gens := [191, 457, 211];
    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], [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 := 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, -1, 0, 1, 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, -1, 1, 0, 0, 1, -1, 0], [1, -1, -1, 0, -1, 2, 1, 0, -1, -1, -2, 0], [1, -1, 1, 0, -2, 0, 0, 0, 0, 2, 0, -1], [2, 1, 0, 0, 2, 3, -2, 1, 0, -1, 1, 0], [0, 1, 0, -2, 0, 0, 0, 2, 0, 0, -1, 2], [2, -3, -1, -1, -2, 0, 3, -1, 0, 0, -3, -1], [1, -1, 1, -1, 2, -1, -4, 0, -2, 1, 0, 1], [3, -2, -1, 1, -3, 2, 0, -2, -1, -1, 1, -2], [-7, 0, 3, -3, -2, -2, 3, 0, -2, 2, -3, 0], [-1, -2, -1, 1, 1, 1, 0, -1, -1, -3, 0, 0], [-2, 1, 2, -1, -1, 0, -1, -2, 2, 4, 1, -1], [0, -2, 0, 1, 0, 2, -1, -2, -3, -1, 0, -2], [-3, 0, 2, -6, -1, 0, 0, 2, 0, 0, -4, 2], [-2, 3, 0, -1, 1, 2, -1, -1, 0, -3, -1, 0], [0, -1, 2, -4, 1, 1, 1, 2, 0, 0, 1, 3], [6, -4, 0, 3, 5, -1, -5, 1, 2, -3, 0, 1], [1, 0, 1, 0, 3, 0, -1, 2, -2, -3, 1, 3], [-2, 3, 4, 1, 3, -4, 0, 4, 3, 2, -1, 1], [-5, 3, -1, 7, -3, 1, 0, 1, 1, -4, 0, 0], [4, -1, 2, -5, 7, 1, -3, 2, 2, 2, 3, 2], [-5, -5, 0, 0, 1, -1, -6, -1, -3, 3, -3, 0], [0, 2, 0, 2, 8, -1, -7, 6, 1, -2, 6, 1], [-9, 7, 3, 0, 5, -1, -1, -1, 0, -2, 0, -3], [0, 7, 0, -3, -4, -1, 1, 0, 1, -1, 1, 0], [-2, -3, -4, 5, -5, -1, 3, 0, -4, -2, -5, 0], [2, -7, -2, 7, 3, 0, 1, 2, -2, -4, 1, 3], [-10, 0, 2, 2, -6, -2, -2, -2, -2, 6, -2, 0], [-1, -4, -2, 0, 1, 0, -3, 0, -2, 6, -2, 2], [1, 6, 0, 4, 8, 3, -8, 3, 0, -6, 3, 0], [1, 1, 1, 9, 3, -3, -1, 1, 0, 0, 2, -2], [7, -4, -5, 0, -4, -1, 5, -1, -1, -1, -1, 5], [2, 3, 2, 1, -3, -2, 0, 0, 5, 2, 5, -5], [-3, 0, -5, 3, 4, 1, -3, 3, 1, -4, 5, 0], [-1, 0, -1, 0, -2, 0, 0, 4, -4, -3, -6, -2], [-1, 4, 1, -9, -7, -3, -3, 0, 2, 3, 3, -1], [5, 1, -3, -1, 3, 1, -1, -3, 0, 0, 7, -2], [-5, 4, 0, 1, -2, -2, 2, -4, 0, -4, -4, 0], [-1, -4, 1, 0, 5, 3, -2, 0, -4, 2, -3, 3], [12, -12, -2, 8, 8, 4, -9, -2, -1, -8, 0, -4], [-7, 0, -3, -4, -2, 0, 4, -5, 0, 2, 3, 0], [-5, 5, -1, 2, -6, 1, 0, -4, -4, 3, 0, 0], [3, -6, 1, -1, 5, -4, 3, 4, 1, 4, -3, 4], [4, -4, 0, -4, 2, 2, -1, -2, 1, 4, 0, -2], [-7, 2, 0, 1, -12, -5, 11, -4, -1, -2, -4, -1], [3, -1, -2, 1, -7, 0, 8, -1, 1, -2, -6, -7], [-2, 0, -4, -1, -8, 1, 1, 2, 1, 8, 4, 0], [9, 0, -3, 5, 6, 5, -5, -1, 5, -6, 3, 0], [4, 6, 0, -6, 4, 0, -10, -2, 2, -2, 6, 4], [-2, -3, 2, -7, -7, -1, -2, 0, -1, 2, 1, -2], [0, -1, 2, 0, 0, 1, -3, 1, 1, -6, 1, -2], [6, -10, -6, 5, -10, 3, 10, -6, 0, 0, 2, -3], [1, 1, 1, 3, -4, 3, 6, -4, 4, -3, 0, -3], [6, -2, -6, -4, -4, 6, 0, -3, -4, 2, -1, 1], [1, 3, 1, 1, 1, -1, 0, 1, 3, 2, 2, -2], [12, 1, 0, -1, 4, 0, -5, -3, 3, -9, 7, 4], [-4, -7, 4, -4, -2, -2, 0, 0, -2, 0, 2, 2], [2, -5, 1, 5, -2, -3, 5, 1, 0, 0, 2, -2], [10, -11, 0, 3, -6, 2, 0, -1, -6, 11, -1, -6], [6, 2, -6, 4, 1, 0, 7, 0, 6, -7, 0, 4], [1, 4, 1, -2, -4, -1, 0, 1, 0, 6, -1, 1], [6, -9, 0, 1, -11, -3, 7, -10, 0, 6, -3, -10], [1, -9, -1, -7, 12, -3, -8, 4, -4, 7, 0, 3], [10, -5, 1, 0, -6, -2, 0, -2, 7, 15, 7, -1], [-2, 14, 2, 7, 1, 0, 7, 0, -2, -7, 0, 1], [14, -10, -4, 10, -2, 0, -2, -4, 4, -10, 2, -2], [1, 2, -9, -2, -5, 0, 13, 0, 0, -1, -5, -5], [0, 3, 0, 7, -4, 3, 10, 3, 0, 3, 3, 3], [2, -8, -2, -10, -10, 2, -2, 0, 0, 2, -2, -2], [10, 1, 0, 14, 4, -2, -5, -3, -1, -1, -3, -1], [-17, 9, 7, 0, 7, -1, -5, -1, -2, -1, -2, -7], [14, -17, -6, -4, -1, 3, -4, -5, -6, 0, -3, -5], [-9, 0, 6, 1, 6, 3, -1, 6, 3, -6, -6, 0], [0, -1, -4, 5, -14, 4, 0, -4, 1, -6, 5, -5], [-13, -1, 5, -22, 3, -2, 2, -3, -9, 22, 0, 2], [-3, -9, 0, -7, -12, 2, 5, -5, -7, 9, -5, -7], [-8, 5, 0, -6, -12, -1, 16, 3, 4, -5, 3, 4], [0, -6, -5, 15, 5, 1, -7, 4, -8, -15, 0, -1], [-7, 0, 3, 13, -1, -1, -13, 0, -1, 1, -3, 0], [-15, 8, 11, 8, 8, -11, 0, 3, 4, 0, 1, -1], [0, 8, -2, 0, 0, 0, 10, 0, -2, 0, -2, 2], [-8, 12, 2, -6, 6, -3, -12, 2, 0, 0, 10, -1], [-6, 3, 0, -7, 10, -1, -5, -3, 5, -3, -3, 5], [3, 4, -7, 0, 7, 0, 4, 0, 3, 13, 3, 7], [-2, 8, 8, 2, -6, -2, 3, -4, 8, 1, 6, -4], [6, 0, -2, -7, 10, 1, 7, 6, 1, -10, 2, 0], [-5, 15, 9, -15, -6, 0, -2, -4, 4, 9, -2, -6], [0, 4, 2, -12, 12, 2, 3, 0, 2, 15, 4, 0], [6, -13, -6, -9, -3, -1, -2, 0, 7, 2, 1, 4], [-16, 14, 8, 2, 6, -11, -14, 8, 0, 0, 2, -3], [5, -5, 9, -5, 7, -9, -5, 2, 9, 0, 0, 2], [3, 1, 0, 8, 7, 0, 0, 12, 11, -7, -1, 1], [12, -1, -3, 1, 1, 0, -20, 1, -1, -13, 0, 1], [-10, 1, -1, 7, -4, 3, 2, -2, -11, -7, 0, -3], [10, -18, -6, 2, -14, 2, 18, -6, 0, 0, -2, -4], [0, -7, 0, -4, 5, -1, -2, 0, 1, 2, 1, 7], [-7, 1, 1, 10, 5, -6, -10, 5, -11, -10, 0, 6], [16, 2, -2, 3, -9, 7, -13, -5, -4, -3, 0, -7], [-15, 10, 11, 9, 6, -13, -10, 11, 0, 0, -3, -2], [-6, -13, -6, 0, 15, 5, -22, 5, -2, -5, -2, 6], [8, -15, -6, 11, -7, -1, 13, 4, -6, 2, -7, 4], [20, 0, -6, 2, -3, 0, -2, 1, 0, 3, 6, 0], [-14, 0, -3, 9, -2, 1, -9, 3, 1, 2, 3, 0], [-22, 27, 4, 5, 2, 1, 2, 7, 4, -3, 5, 7], [0, 16, 0, 11, 14, 3, -12, 0, 2, -16, 0, 2], [15, -15, -5, -13, -5, 9, 15, -5, 0, 0, -6, 4], [-16, 2, 6, -9, 7, -9, 13, 3, 6, 9, 0, 9], [6, -16, -2, 2, -8, -8, 3, -6, -2, 1, -10, -6], [-15, 0, -5, -15, -12, -2, 15, -13, -2, 12, 5, 0], [5, -3, -8, 3, -4, 0, -6, -2, 2, -3, -2, -4], [7, -15, -3, 12, 19, -4, -2, 7, 9, -12, 0, 4], [-1, -5, 1, -18, -1, 4, -8, 0, -5, 8, -4, 9], [-16, 4, -4, -12, -16, 3, -4, -4, 0, 0, 6, -1], [5, -9, 3, 9, -12, 0, -3, 0, 0, -5, -12, -12], [11, -17, -5, -9, -9, 5, 0, -9, -3, -8, 6, -6], [-21, 0, 9, -12, 3, 0, 12, -1, 0, -3, -9, 0], [15, -18, -9, -2, -4, 6, -10, -6, -9, -8, -3, -6], [-8, 11, -1, 0, 16, 6, -4, 6, 1, 15, 1, 1], [3, -1, 5, -17, 7, 4, 1, 5, 0, 0, 6, 9], [-11, 7, 11, -3, 4, 0, -5, 0, -11, 5, 0, 2], [6, -5, 0, 26, -19, 1, 5, 7, 0, -21, 1, 7], [2, 0, -4, 13, 15, 4, -13, 3, 4, -15, 4, 0], [1, -5, 3, -7, 1, -7, 12, -6, 3, 19, -4, -6], [-10, -1, 0, 4, 8, -9, -14, -1, -6, 1, -1, -6], [15, 5, 1, 0, -4, 2, 8, 2, 10, -15, 10, -1], [14, -17, 0, -23, 5, 0, 0, 0, -9, 6, -9, 9], [16, 2, 0, -2, -7, 0, 2, -9, 9, -7, 2, -7], [10, 4, -10, -6, -13, 10, 5, 0, 0, -5, -10, -2], [13, 7, 0, 10, -6, 7, 9, 0, 3, -7, 0, 3], [-4, -9, 0, -9, -2, 0, -2, 4, -4, 9, 4, -4], [20, -6, -2, -7, -1, 7, 4, -5, 17, 7, 0, -7], [18, 6, -2, -13, 1, 2, 0, -4, 3, 19, 7, -7], [-13, 0, 1, 3, -9, 6, -3, 1, 6, 9, -1, 0], [14, 0, 0, -30, -3, 3, -13, -3, -2, 30, 0, -3], [6, -4, 1, 0, 6, 12, -11, 12, 1, 5, 1, -1], [9, -1, -9, 11, -2, 6, 19, 0, 3, -19, -6, 6], [-5, 0, 3, 6, -13, -5, -6, 7, -5, 13, -3, 0], [-1, -2, 1, 10, 0, 2, -5, 0, -3, 5, -2, -15], [8, 12, 6, 0, -13, 0, 19, 0, 1, 6, 1, -6], [-13, -5, 9, -9, -9, -7, 5, 9, 0, 0, -16, 2], [10, 8, 3, 0, -6, 2, 11, 2, 5, -1, 5, -3], [-4, -10, 2, 7, 7, 3, -3, -5, -10, -7, 0, -3], [20, -24, -8, -2, 6, 4, 4, 4, -8, 6, -4, 4], [22, -9, -16, -3, 7, 16, 0, 0, 3, -6, 3, -3], [6, -15, 0, -11, -19, 9, 12, -5, -7, 15, -5, -7], [2, -18, 0, -13, -14, 8, 3, -16, -11, 18, -16, -11], [-16, 4, -7, -3, -19, 3, -4, -7, 0, 0, -5, -4], [2, 8, -2, 8, 5, 4, 0, 0, -2, 0, -4, -3], [28, -10, -19, -12, -7, 19, 0, -3, -3, 2, 0, 0], [14, -9, -2, -10, 13, 7, -2, 3, -2, 8, 5, 3], [-1, -5, 1, -3, 10, -5, 12, 4, 16, 3, 0, 5], [-14, 16, 2, 0, 13, 3, 1, 3, -2, 6, -2, -2], [-4, -3, 16, 3, 7, 0, -11, 8, -8, -4, -1, 7], [-15, 0, 13, 19, 20, -7, -19, 13, -7, -20, -13, 0], [-11, 11, 1, 2, -20, -1, 9, -18, 1, 7, 0, -18], [16, 0, 0, 11, 0, 0, 5, 3, 5, 0, 3, 5], [-4, -7, -8, 0, 13, -8, -12, -8, 9, -22, 9, 8], [4, -14, -2, -11, 15, -1, -20, 3, -15, 11, 0, 1], [9, 16, -7, 2, 4, 7, 0, -5, 4, 14, 9, -9], [-4, -15, -4, 3, -7, 4, 0, 0, -5, -18, -5, 5], [-6, 0, 0, -1, -4, -10, 9, 8, 5, 0, 8, 5], [6, -15, 0, -4, 27, -7, -25, 15, 2, 15, 15, 2], [13, -17, -13, -8, -4, 10, 3, 0, 3, -3, -10, 7], [8, -7, -14, 7, -4, 0, 8, -10, 10, 2, 6, -4], [11, -8, -9, -7, -9, 9, 0, -4, -5, -1, -1, 1], [-9, 16, 7, -4, -14, -2, 2, -5, -5, 4, 0, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_570_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_570_2_u_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_570_2_u_j(:prec:=12)
    chi := MakeCharacter_570_u();
    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_570_2_u_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_570_2_u_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![361, 1425, 3972, 5879, 6579, 3672, 1857, 342, 189, 11, 24, -3, 1]>,<11,R![927369, 78003, 483246, 104355, 207765, 32724, 25731, -486, 1755, -15, 54, -3, 1]>],Snew);
    return Vf;
end function;