
// Make newform 570.2.u.i 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_i();".
// 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_i();".
// 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 := [361, 0, 1395, 0, 1770, 0, 905, 0, 216, 0, 24, 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], [-1520, -1177, -2742, -2566, -774, -1304, 38, -277, 28, -27, 2, -1], [1520, -1177, 2742, -2566, 774, -1304, -38, -277, -28, -27, -2, -1], [2109, 0, 4204, 0, 3742, 0, 1387, 0, 211, 0, 11, 0], [171, 417, 3464, -24, 5334, 530, 2185, 315, 333, 55, 17, 3], [2109, -8827, 4204, -16628, 3742, -6690, 1387, -829, 211, -5, 11, 3], [551, 4065, -3028, 5460, -6926, 2078, -2983, 239, -455, -1, -23, -1], [-171, 417, -3464, -24, -5334, 530, -2185, 315, -333, 55, -17, 3], [-3933, 6419, -9384, 10592, -5290, 4686, -1311, 793, -155, 53, -7, 1], [-1805, -1769, -4204, -8996, -3742, -5822, -1387, -1479, -211, -167, -11, -7], [8303, 3515, 23356, 8948, 20994, 6882, 7201, 2109, 1009, 277, 49, 13], [152, 2979, 0, 7304, 0, 5238, 0, 1545, 0, 197, 0, 9]];
    Rf_basisdens := [1, 304, 304, 304, 608, 608, 608, 608, 608, 608, 608, 304];
    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, -1, 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, -1, 0, 0, 1, 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, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, -1, -1, 1, 0, -1, 0, 1, 0, 0, 0, -2], [0, -1, 0, 0, 0, -1, 0, 2, 0, -1, 1, 1], [1, -1, -1, 1, 1, -1, -1, 1, 1, 0, 0, 0], [1, -1, -1, 0, 1, -2, 0, 0, -1, 1, 1, -2], [1, 1, -1, 1, -2, -1, -1, 1, 0, 0, 0, 0], [-1, -1, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1], [-3, -1, 1, -1, 1, 1, 2, -3, 1, -1, -1, 4], [3, 0, 0, -2, 1, 1, 2, -1, 0, -1, -2, 1], [2, 0, 2, 0, 0, 0, 1, -1, 0, -1, -1, -1], [3, -3, -4, 0, 0, -2, 0, 3, -2, 0, 1, -4], [2, 1, 0, -1, -3, 0, 0, 0, -1, 0, 1, -3], [-1, 0, -2, 2, 1, 0, -2, 2, 2, -2, 0, 0], [0, 2, 1, -1, -3, 0, -1, 0, -2, 1, 1, -2], [-2, 3, 0, -2, -1, 3, 2, -2, 1, -1, 0, 6], [0, -2, 3, -1, 5, -1, 1, -4, -1, 0, 1, 4], [-1, 5, 3, 0, 0, 1, 0, -1, 1, 0, -2, 3], [1, 0, 1, -1, 1, 0, -2, 2, 1, -2, 2, 1], [0, 0, 0, -2, 0, 0, -1, -2, 2, 1, 1, 2], [4, 6, 1, 2, -5, -1, -4, 7, -2, 2, 2, -6], [0, 1, 0, 0, -1, 1, -1, 2, 0, 2, -1, -1], [-1, 0, 6, 1, 0, -1, -2, -3, 1, 0, 0, -5], [0, 2, 0, -3, 3, 4, 3, -4, 0, -2, -4, 1], [0, 6, 4, -1, -3, 1, 0, -7, 1, 0, 1, 3], [-4, -6, -3, 2, 1, -1, -2, -3, 0, 2, 0, 2], [1, -3, 0, 0, 0, 2, 0, 1, 2, 0, -1, 0], [6, 0, 2, 4, 2, 0, -2, -2, 2, 0, 0, 0], [-3, -3, 0, 2, -1, 2, 2, 6, 0, 2, -2, 4], [0, -9, -5, -1, 7, -1, 1, 4, -3, 2, 2, 3], [0, 5, 3, 3, -11, -2, -3, 3, -5, 5, 6, 3], [7, -5, 0, 4, 4, -4, -3, 6, 0, 3, 4, -6], [-3, 0, -3, 3, -1, 0, 2, 2, -3, 4, -2, -5], [3, -4, 3, -4, 4, 2, 5, -4, -1, -2, -4, 2], [0, -2, -6, 8, -4, -1, -4, 4, 3, 4, 1, -10], [0, 9, -3, -2, -9, 2, 0, -6, -2, -1, -3, 5], [3, -10, -4, 1, 6, -2, -1, 1, -2, 2, 1, -12], [-5, 2, 9, 0, -2, 0, 0, -4, 0, 0, 0, -4], [0, 10, 6, 0, -3, 0, 0, -9, 0, -1, -1, 1], [-4, -1, 1, 1, 7, 2, -2, 1, 1, 0, -1, 9], [-2, 5, -4, -7, 2, 3, 5, -2, 1, -3, -6, 3], [3, 0, 3, 2, -1, 0, -4, 3, -2, -3, 4, 1], [9, 12, 7, 1, -7, -1, 0, 4, 1, 1, -1, -10], [0, 5, 3, 5, 7, 0, 0, -3, 7, 0, -5, 7], [-4, 6, 5, -1, -4, 3, 4, -5, 1, -2, -2, 1], [6, -5, -5, -6, 3, 2, 3, 3, -1, -3, -3, -2], [17, -3, -1, 1, 0, -3, -2, 0, -4, 3, 6, -3], [9, 2, 3, 6, 0, 0, -5, 0, 5, 0, 0, 0], [8, -12, -4, 4, -2, 0, -2, 10, 2, 2, 0, -6], [0, -1, -10, 1, -8, -1, 0, 2, 1, -2, -1, 1], [-1, -2, 0, -4, -2, -1, -1, 5, 0, -4, 1, 7], [7, 5, -2, 0, -7, -3, 0, 0, 2, -2, -5, 2], [-7, -11, 10, -2, 15, -2, 2, -12, 3, 0, 2, 13], [-4, 0, -4, -1, 8, 0, 3, 0, 1, -4, -3, 5], [4, 0, 4, 0, 6, 0, 7, -4, 0, -1, -7, -6], [4, 0, -10, -6, 3, 2, 3, 1, -1, -3, -4, -7], [0, 8, 0, -4, -6, 4, 0, -6, 2, -4, -2, 13], [-1, 1, -1, -3, 0, 2, 3, -3, -1, 1, 0, 3], [-12, -5, 11, -3, 6, 4, -1, 2, -2, -1, -1, 2], [0, 13, 2, 2, -13, -2, 0, -15, -2, 4, 2, -2], [-13, 0, -13, -1, -11, 0, 3, 4, 1, 3, -3, 3], [2, 8, -1, -4, 7, 3, 6, 7, 2, -4, -2, 2], [1, -5, -10, -2, 13, -4, 4, 6, -7, 0, 2, 9], [2, -10, 0, -5, 6, 3, 4, 4, 0, -6, -3, 13], [0, 12, 3, -2, -5, 2, 0, -8, -3, -1, -4, 8], [0, -6, -4, 8, -4, -4, -4, 4, 0, 4, 2, -8], [-1, -13, -5, -4, 2, -1, 2, -3, -3, -2, 6, -3], [-3, -6, -14, 7, 9, -5, -5, 6, 2, 7, -2, -4], [0, 2, -2, 4, -10, -4, 0, -8, 0, 6, 6, 2], [-12, -1, 3, 0, 3, 2, -1, 10, 2, -2, -2, 11], [14, -14, 0, -3, 11, -2, -3, 3, 0, -2, 2, 3], [-1, 10, 9, 1, -14, 1, -5, 5, -4, 1, 4, 0], [-6, -13, 9, 5, -8, -5, -6, 8, -4, 5, 10, -5], [7, 0, 7, -1, 8, 0, 0, -2, 1, 4, 0, -9], [-3, -6, -6, -2, 5, -1, 1, 5, -7, 0, 2, 4], [3, -7, 10, 0, 5, -2, -3, -10, -2, 2, 2, -15], [13, 1, -1, -1, -1, 1, -1, -11, -1, 0, 0, -12], [0, 8, -4, 6, -16, 2, -2, 6, 5, 0, -6, -14], [16, -4, -4, -1, 2, -4, 0, -2, -8, 4, 8, -4], [13, 0, 13, 3, 5, 0, -1, -2, -3, -1, 1, -9], [-4, 2, 0, 2, -8, -6, -4, 2, 0, 0, 6, 0], [3, 11, -9, 0, -11, -3, 0, 0, 0, 0, -3, 8], [17, 3, -11, 2, -8, -7, -3, -3, -3, 5, 5, -11], [-1, 4, 0, 7, -1, -1, 0, -4, 0, 6, 1, -10], [-4, 9, -2, 2, -2, 2, -4, 13, -2, 2, 2, 2], [5, 5, -9, 1, -4, -5, -6, 4, -4, 5, 10, -5], [-10, -12, 13, 4, -2, -3, -2, -8, -1, 2, 3, 11], [-18, -3, -2, 0, 8, 8, 6, -7, 6, 0, -6, 18], [0, 5, -12, 5, -14, -5, 0, -2, -7, 4, -3, -7], [9, 7, 7, -2, -17, 7, 2, -2, 4, -4, 1, 14], [-1, 11, 19, 1, -19, 6, 0, -9, 1, 1, -1, 0], [10, 0, 10, 1, -7, 0, -10, 5, -1, 4, 10, -9], [-16, 1, 1, -2, 1, 5, 1, -17, 4, -1, -2, 2], [-8, -3, -9, 3, 8, 2, -2, 11, 6, 0, -3, 10], [6, 8, 4, 2, -16, -4, -2, 2, -4, 4, 2, 4], [0, 16, 15, -5, -4, 5, 0, -19, 4, -1, 3, -9], [-5, -10, -6, -2, -1, -3, 3, 3, -7, 0, 2, -4], [2, 2, 1, -7, -3, 7, -7, 6, -1, 0, 7, 4], [1, -5, -6, 3, -1, -4, -3, 3, -6, 6, 5, -9], [6, -2, 0, -3, 7, 3, 1, -11, 0, -1, -3, -1], [-10, 6, 7, -4, 3, 9, 10, -7, 6, -4, -6, 14], [5, 0, -1, -9, 22, 2, 5, -22, -1, -2, -4, 2], [4, 6, -5, 10, -12, 2, 3, 12, 1, -2, -4, 2], [3, -4, -2, -5, 1, 3, -1, 4, -6, -5, 6, 2], [-1, 8, -8, 0, -11, -3, -2, 11, -3, 3, 3, 6], [2, -5, 15, -3, 3, 4, 3, -3, 1, -1, 0, -1], [-2, 0, 3, -1, -5, 3, -3, 0, 5, 0, 1, -2], [0, -3, 8, 0, -10, -10, -2, -9, -2, 0, 2, 0], [6, -2, 7, 5, -10, 1, -3, 10, 5, -1, -2, 1], [5, -16, -7, -8, 4, -2, 4, 1, -6, -4, 6, -3], [-11, -4, -8, 4, 9, 3, -3, 11, 11, 0, -4, 12], [0, -19, -17, -1, -2, 1, 0, 15, -1, 5, 4, 6], [-5, 5, -1, 2, -3, 3, -2, 2, 2, -2, 3, 8], [14, 15, -23, 0, 0, -3, 0, 14, -3, 0, -5, -23], [-10, 0, -10, 8, -4, 0, -1, 12, -8, 9, 1, -10], [-8, 15, -6, 3, 9, 3, -3, -9, 9, -3, -6, 3], [7, 9, -2, -3, 6, -3, 4, 10, 1, -3, -1, -4], [16, 0, 0, 1, 16, -5, 0, -12, 0, -4, 5, -17], [-9, 10, 12, 7, -8, -1, -7, 7, -1, 1, 7, 9], [0, 13, 2, -7, -11, 7, 0, -13, 8, -8, 0, 16], [-5, -13, -13, -5, 14, -1, 1, 4, -4, -5, 4, 10], [1, -7, 10, 5, 8, 2, 1, -8, 3, -2, -4, 2], [-3, -2, 1, -3, -3, -8, -2, 2, -5, -3, 5, 6], [1, 11, -7, 3, -13, -5, -7, 13, 1, 2, 2, 4], [-2, 11, 0, -6, -4, 5, -2, -5, 0, 1, -5, -3], [-9, 0, -9, -5, 13, 0, -2, 0, 5, -4, 2, 15], [-3, 5, -4, 2, -1, 2, -1, -2, 3, 1, 1, -5], [0, -11, -4, 8, 1, -8, 0, 5, -2, 8, 6, 2], [6, -10, -6, -3, 0, -7, -2, 2, -13, 10, 10, -10], [-2, 14, 5, 0, -8, 6, 4, -7, 6, -6, -6, 3], [-12, -7, -6, 4, 8, 1, -1, 7, 2, 0, -4, 9], [13, 0, 13, -1, 5, 0, -4, 1, 1, -2, 4, -8], [-8, 0, -3, -3, -13, -1, 1, 13, -3, 1, 2, -1], [4, -2, -24, -1, 13, -5, 5, 19, -5, 0, 1, 8], [-16, 7, 0, 5, -12, 1, 4, 0, 0, 2, -1, 4], [0, 7, 16, 1, 5, -1, 0, -11, 2, 2, 4, -14], [15, 19, -16, -10, -3, 8, 13, 3, 3, -8, -16, 8], [0, 6, -13, -7, -13, 7, 0, 0, -5, -1, -6, 5], [-7, 10, 0, 13, -7, -2, 0, 1, 0, 11, 2, -16], [2, -1, -11, 3, 1, -8, -3, 3, -3, 3, -2, -9], [-2, 7, 0, -5, 12, 13, 14, 6, 0, -6, -13, 0], [-8, 6, 1, 9, -5, 5, -5, 4, 1, 0, -9, 0], [-4, 16, 14, 0, -10, 4, 4, -2, 4, 0, -4, 4], [15, 0, 15, -5, -9, 0, -6, 4, 5, -8, 6, -5], [-3, -6, 17, 0, -1, -7, -3, -11, -7, 7, 7, -21], [-18, -12, 10, 1, 15, 2, 4, 4, 4, -3, -3, 11], [4, 5, 16, 1, -16, 2, -1, 1, -4, 4, 7, 7], [0, 15, 12, 0, 1, 0, 0, -11, -1, -3, -4, 2], [-9, 0, -9, 2, -24, 0, -5, 6, -2, -3, 5, 11], [27, 2, -3, 1, -4, -15, -7, 11, -6, 1, 6, -28], [-1, 7, -8, 14, 9, -6, 6, 2, 11, 0, -14, 3], [2, -9, 0, -1, 5, 5, 3, -10, 0, 1, -5, 8], [-5, -3, 2, -4, 2, 0, 2, -7, -2, -2, 3, 4], [-1, 0, 11, 4, 8, 4, -3, -8, 11, -4, -8, 4], [-3, -21, -8, -1, 8, 2, 0, -12, -1, -1, 1, 4], [4, 25, -2, 0, -29, -4, -3, 1, -4, 4, 4, -6], [13, -31, 0, 5, 7, -4, -6, -9, 0, 7, 4, 13], [4, -11, 2, -8, 20, -7, 7, -9, -10, 0, 8, 13], [-15, 0, -15, -5, 7, 0, 1, 20, 5, -5, -1, -1], [-9, 0, -9, 1, 19, 0, -2, 12, -1, 4, 2, -5], [-4, 14, 8, 3, 1, 12, 8, -12, 18, -15, -15, 11], [-5, 14, -1, -3, -8, 9, 9, -3, 3, -6, -6, 12], [0, 2, -3, -1, 3, 1, 0, 6, 1, -9, -8, -18], [-2, 2, -9, 16, -8, 2, -8, 6, 10, 8, -6, -17], [14, 0, 14, -5, -10, 0, -2, -15, 5, 9, 2, -6], [-7, -6, 4, -6, 2, 7, -7, 3, -1, 0, 6, 9]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_570_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![5041, -4047, 7296, -1153, 3951, -270, 1503, 180, 261, 17, 24, 3, 1]>,<11,R![1, 39, 1602, -3077, 8145, 2160, 2547, 882, 675, 217, 66, 9, 1]>],Snew);
    return Vf;
end function;