
// Make newform 162.2.e.b in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_162_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_162_e_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_162_2_e_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_162_2_e_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 := [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], [11, -30, 41, -23, 14, -3, 1, 0, 0, 0, 0, 0], [1060, -3477, 5358, -5701, 3234, -1278, -252, 438, -324, 127, -33, 6], [-882, 5163, -12275, 16775, -13637, 9493, -4226, 1781, -435, 115, -11, 2], [-2263, 10469, -20623, 24401, -18150, 9021, -2572, -445, 586, -408, 94, -27], [-524, 4712, -12972, 19981, -20118, 12623, -6865, 1590, -449, -187, 34, -26], [-83, -1957, 9570, -19635, 26758, -23134, 16285, -8075, 3311, -953, 203, -27], [-3184, 16878, -42784, 63925, -65711, 44604, -26048, 8816, -3126, 210, -58, -39], [2201, -13927, 36296, -56973, 60760, -47327, 28887, -13234, 4892, -1277, 252, -26], [3256, -18038, 47888, -73416, 78267, -59173, 35787, -15666, 5713, -1365, 268, -19], [-5544, 25228, -58487, 88252, -93068, 77593, -47943, 24960, -9321, 3043, -583, 106], [4454, -23920, 56961, -87816, 92850, -77593, 47943, -24960, 9321, -3043, 583, -106]];
    Rf_basisdens := [1, 2, 218, 218, 218, 218, 218, 218, 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_162_e();"
function MakeCharacter_162_e()
    N := 162;
    order := 9;
    char_gens := [83];
    v := [8];
    // 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_162_e_Hecke();"
function MakeCharacter_162_e_Hecke(Kf)
    N := 162;
    order := 9;
    char_gens := [83];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 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, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, -1, 1, -1, -1, 1, 0, 1, 1, -1], [1, 0, 0, 1, 0, 2, 1, -1, 0, 0, 0, 0], [2, -2, -1, 1, 1, 2, 2, 0, 0, -1, -1, 0], [1, 0, 1, -1, 0, 0, 1, 1, 1, 0, 1, -1], [-2, 0, 2, -1, -1, -1, 1, 0, -2, 0, 0, 0], [-3, -1, 1, 0, -1, 0, 0, 0, -3, 0, -1, 0], [-3, 1, 0, 1, 1, 1, 1, -2, 1, 0, -1, 1], [-2, -1, 3, 1, -3, 1, -1, -1, -1, 0, -1, 1], [0, -1, -3, 2, -1, 5, 3, -1, 3, 0, -1, 1], [1, 0, 1, 1, 2, 0, 0, 0, 2, 1, 0, 0], [-1, 1, 3, 0, 0, 0, 1, -1, 2, 0, 0, -1], [-1, -2, 2, 2, 2, -1, -3, -1, 0, -2, -2, 1], [2, -1, -1, 0, -1, 1, 1, 1, -2, -1, 1, 1], [-2, 1, 0, -1, -1, 0, 0, -2, 2, 1, 0, 0], [-4, 0, 0, 1, 3, 3, -1, 0, 0, -1, -1, 1], [2, 4, -1, -3, 1, 2, 2, 0, -4, 1, 2, 0], [-2, 2, -1, -1, 4, -1, 3, 0, 3, 1, 0, 1], [-3, 0, 2, 3, 2, 3, 2, -1, 0, -1, 0, 1], [3, 0, -1, -2, -1, -3, -5, -1, -1, 0, -1, -1], [2, -1, -2, -1, -5, -3, 1, 3, -5, 0, 2, -1], [-2, -2, -1, 4, 6, 7, -1, -1, 7, -3, -2, 3], [4, -1, -1, -2, 0, 1, 4, 3, 2, 0, 0, 1], [8, 1, -3, -1, -1, -5, 3, 1, 0, 1, 1, -1], [-4, 1, 2, 2, 5, -2, 0, -1, 2, 1, -3, 0], [-1, 5, 2, 0, 2, 4, -2, -2, -2, 2, 1, 1], [-1, 2, 1, 0, 3, -3, -3, -1, 4, 1, 3, 3], [4, -1, -2, 2, -3, -3, 0, -1, -2, -1, -1, 0], [1, -2, -2, 3, -2, -3, -1, 1, -1, -1, -2, 1], [1, -2, 5, 1, -5, 1, -2, -1, 2, 1, -1, 1], [-1, -3, 7, -3, -4, 4, 2, 1, -4, 0, -2, -3], [-3, 0, -4, 2, 4, 2, 0, -2, -1, 0, 0, 2], [-10, 4, 4, -5, -8, -2, -2, 0, -1, 1, 2, -2], [0, -1, 4, 4, 0, 3, 5, -3, 3, 1, 0, 1], [8, -3, -4, 2, -7, -2, 0, -1, -4, -3, 1, 0], [0, 0, -1, 0, -8, 4, 8, 1, -5, -1, 0, 0], [4, -1, 1, 1, -2, -6, 0, 1, -5, -1, 0, 0], [-2, 1, 8, 0, 0, -1, 5, -1, 3, 3, 0, -1], [1, 0, -7, -2, -7, 4, -2, 1, -2, -1, 1, -2], [1, 3, 2, -5, -1, -1, 1, 0, -2, 3, 3, -3], [2, 1, -5, 3, -2, -2, 0, 1, -2, 0, 2, 2], [1, 0, 3, -5, -3, -5, -1, 5, -4, 4, 4, -5], [2, 0, 0, -2, -4, -6, -10, 2, 6, 2, 0, -2], [-4, 6, 3, -5, 2, -1, 3, 2, 3, 3, 2, 1], [2, -2, 0, 0, 2, 7, 0, 4, -2, -2, 2, 0], [8, -1, -1, 1, -3, -7, -1, 1, 0, -1, -1, -1], [13, -3, -9, 4, 0, 14, 7, -1, 7, 1, 0, 3], [2, -4, 3, 1, 4, -5, -5, 0, 4, -2, -2, -1], [6, -5, -2, 1, 6, 0, 7, 1, 7, -3, 1, -4], [5, -2, -7, 3, 7, 3, -3, -3, 8, -5, -7, 3], [3, 4, 5, 0, 2, 5, -7, -2, 2, 0, 2, 2], [7, 2, -2, -4, 2, -1, 0, 3, 2, 5, 2, -5], [-8, 4, 8, 4, 5, -7, -7, 0, 12, 3, 2, 5], [4, 0, -2, -1, 11, -1, 1, 2, 0, 1, 1, -3], [-1, -7, -4, 0, 2, 2, -2, 4, -6, -4, 0, 0], [-1, -2, -1, 0, -4, -8, 4, 1, 7, -1, -2, -2], [-1, 5, 5, -5, -3, -4, -4, 4, 0, 5, 5, -4], [-10, 3, 0, -4, 0, -11, -13, 1, -5, 4, 0, -3], [-6, -6, -4, 5, -9, 0, 0, 0, 1, -2, -3, 1], [0, -1, 7, -3, -7, -3, 3, 3, -3, 0, -1, -3], [-1, -2, -3, 3, 10, -7, 0, -3, 0, -2, -3, 1], [0, -4, -10, 3, -6, 4, 4, 0, -7, -5, -2, 4], [-7, -2, 0, 0, 0, -9, -5, 2, -9, -5, 0, 2], [-3, -6, -2, 3, -9, 8, 6, -7, 0, -3, -3, 4], [5, 1, -7, -3, -10, 4, 10, 0, 0, 3, 3, -2], [-11, 0, 13, -4, -1, -7, -7, 0, 9, -1, 0, -3], [-12, 3, 9, -3, -9, -3, -3, 3, -15, 3, 6, -3], [0, -5, -7, 1, 4, -6, 2, -1, 4, 0, -6, -2], [5, -3, -11, -5, -2, -16, -1, 5, -2, 0, 2, -4], [-4, -3, 0, 6, -1, 0, -4, -1, 3, -4, -3, 4], [-4, -1, -2, 0, 0, -5, -5, 1, 8, 6, 0, 1], [-1, 1, 3, 4, 1, 4, 0, -4, -2, 1, -5, 0], [-9, 2, -3, 0, 2, 5, -2, 3, -8, -3, 6, 6], [-22, 2, 9, -1, 21, -6, 0, 0, 13, 2, -1, 0], [3, 4, -7, -1, 0, 6, -8, -3, -9, 3, 0, -4], [-3, 0, 2, 3, 4, 5, -3, -1, 7, -1, 0, 1], [8, -2, -13, 2, 13, 2, 14, -2, 10, 2, 0, 2], [11, -3, -1, -5, 1, -5, 5, 5, 6, 5, 2, -5], [0, 4, 14, -7, 6, 1, 1, 0, 7, 5, 2, -8], [2, 2, 8, 2, 0, 6, 8, -4, -4, 6, 0, -2], [1, 4, 1, 0, 7, -1, -7, -1, 2, 1, 5, 5], [5, -5, -4, 0, 7, -9, -7, 4, 5, -4, -2, -2], [3, -8, -4, 10, 6, -1, -5, -1, 0, -10, -10, 3], [-3, 1, 2, -4, 0, -6, -2, 3, -7, -7, 0, -1], [7, 0, 15, -3, 1, -16, -8, 7, -8, 4, 7, -3], [-9, 2, 14, -8, -9, 6, 9, 5, -9, 0, 7, -3], [15, 1, 3, -1, 9, 2, -14, 2, 9, 0, 3, 0], [10, -2, -8, -10, 8, -10, 0, 10, 0, 2, 0, -10], [6, -2, -10, 5, -7, -5, 13, -2, -20, -4, -2, 4], [-4, 3, -15, 1, 7, 15, 11, 3, 0, -1, -1, 1], [10, -8, 0, 0, 2, 12, -2, 0, -12, 0, -4, -4], [4, -2, 0, 7, -3, 12, 0, -6, -4, -2, -5, 0], [-9, 12, 15, -6, 3, 5, 5, 0, 9, 9, 6, -3], [1, -3, -6, 3, -20, 14, -8, -2, -8, -2, -2, 0], [13, -4, -2, -7, 2, -7, 17, 7, 6, 6, 2, -7], [8, 4, -4, -5, -2, -2, -15, 0, 13, 4, 4, -4], [4, -6, -2, 3, 4, 5, 7, -2, 7, -5, -2, -3], [0, -3, -1, 0, 4, -6, -4, 1, 5, -1, 4, 4], [3, 3, 12, -6, 8, -12, -6, 0, 0, 6, 6, -3], [-8, 11, 11, -10, 7, -7, 3, 3, 3, 4, 3, 1], [-1, 0, 0, 2, 0, 2, -4, -2, 1, -4, -4, 2], [-7, 1, 10, 7, -3, 17, 12, 2, -3, 0, 3, 4], [13, -4, 9, -4, -9, -4, 0, 4, 9, 2, -2, -4], [-4, -5, -30, 9, -5, 5, -9, -9, -9, -5, -9, 4], [-1, -7, -1, 6, -6, 3, 4, -9, 0, -6, -6, 8], [4, -3, 2, 0, 11, -13, -11, -2, 15, 2, -3, -3], [-10, 3, 4, -6, 3, -18, 0, -1, 6, 3, 3, 0], [-10, 6, 4, -4, -8, 4, -3, 3, 0, 4, 4, -1], [2, -6, -5, 1, 1, 12, 12, 0, -4, -9, -3, 4], [-15, -2, 10, 0, -2, 1, 3, -3, -5, -5, -2, 5], [-5, 2, -5, 8, 5, 8, -4, -8, 3, -9, -7, 8], [-11, 4, -4, 10, -15, 6, 5, -9, -15, 0, -5, 7], [-12, 9, 5, -4, -5, 3, -5, -3, -5, 2, -3, 5], [9, -4, -15, 2, 2, 8, 11, 2, 0, -2, -2, -4], [-18, 0, 0, 0, 15, -2, 0, -3, 18, 0, 0, 0], [6, -2, 2, 0, -13, 3, 13, -2, -1, 2, -1, -1], [9, 5, 8, -13, -12, -12, 0, 10, -17, 5, 8, 0], [-5, 7, 23, -3, 0, -1, 11, -4, 4, 13, 0, -7], [0, -8, -22, 5, -18, 6, 6, 0, -17, -7, -4, 4], [-9, 12, -7, -6, 6, -13, -5, -5, 6, 0, 7, 3], [-7, 7, 4, -1, 23, 3, -3, -6, 23, 0, 1, 3], [-19, 2, 16, 0, -9, -4, -5, -5, -4, -3, 2, 3], [-7, -2, 3, -3, 0, -12, -2, 5, -10, -3, 0, 2], [-7, 7, 0, 0, -9, -13, 9, 0, 13, 0, 7, 7], [13, -9, -9, 3, 5, 2, 2, 4, 0, -3, -3, -10], [8, -10, -11, 9, -12, 9, -5, -2, -5, -3, -2, -1], [9, 0, -3, -6, -17, -9, -3, 9, -17, 0, 9, -3], [-2, -3, -7, 11, 7, 11, -9, -11, 9, -4, -7, 11], [-7, 5, 11, 0, -4, 20, 14, -2, 14, 3, -2, 5], [-8, 4, -13, -2, -10, 5, 5, 0, -15, -9, 2, 11], [-9, -2, -1, 2, 6, 16, 0, -5, 10, -2, 0, 0], [11, 2, -5, 0, 4, 0, -4, 5, -5, -5, -2, -2], [-4, 4, 2, -2, 6, 18, 0, 4, 2, 4, -2, 0], [20, -4, -11, 12, 16, 12, 12, 0, 1, 1, -2, 7], [-4, 5, -14, -3, 3, -6, -2, 1, -2, 3, 1, 2], [6, 5, -2, -7, 2, -7, -24, 7, -1, 1, 6, -7], [-4, -4, -11, -1, -14, -11, -11, 0, -12, -6, -2, 1], [16, -7, -3, 0, -18, 8, 18, 3, -11, -3, -5, -5], [-10, 13, 5, 0, 16, -3, -16, -5, 8, 5, 1, 1], [4, -4, -8, 8, 0, 8, 0, -4, 0, -6, 0, 4], [4, 2, 7, -8, -1, -10, -10, 0, -1, 6, 1, -12], [3, 5, 26, -7, -6, 6, 8, 8, 8, 6, 8, -2], [-18, 3, 26, -13, -17, -22, -12, 5, -5, 8, 3, -8], [5, 2, -5, 4, -7, -1, -5, -1, -7, 0, 1, 3], [19, -1, -11, -1, 0, 17, 9, 2, 10, 2, 0, 1], [-11, 5, 13, -2, -14, -5, -15, 4, 0, 2, 2, -1], [5, -1, 7, 0, 0, -8, 0, 5, -12, -1, 1, 0], [1, -2, 1, 0, 4, -12, -4, -1, 13, 1, -2, -2], [-2, 2, 5, 6, 17, 10, 10, 0, 11, -6, 1, 14], [0, -4, 0, 1, 8, -5, -1, -4, -1, -7, -4, -3], [0, -3, -4, 2, -5, -4, 9, -1, -14, -4, -3, 4], [12, 9, 3, -7, -3, -7, -7, 7, 5, 10, 19, -7], [11, -3, -2, 1, 0, 9, 12, 2, 2, 0, 0, 3], [8, 3, -9, -2, -14, 0, 14, -3, 0, 2, 2, 4], [-4, 1, 5, 0, -19, 9, 19, -5, -4, 5, -1, -1], [-14, -8, 2, 6, 0, -16, -4, 2, -24, -6, 0, 8], [10, 2, -3, 1, 0, -5, -9, 5, 9, 7, 2, -7], [4, 4, 5, 0, -5, 0, 12, 0, 4, -1, 3, 0], [1, -11, -4, 1, 8, -3, 14, 9, 8, 0, -2, -5], [-14, 9, 24, -9, -8, -9, -2, 1, -6, 10, 9, -10], [11, -2, -19, -2, -6, -26, 0, 7, 8, -2, 4, 0], [-21, -8, -1, 0, -1, 13, 1, 1, -14, -1, -5, -5], [3, -1, 2, 0, -10, 18, 0, -7, -5, -1, 1, 0], [-5, 10, 11, -15, 0, -10, -4, 5, 10, 10, 0, -10], [10, -1, -18, 15, 7, 14, 8, -7, -1, -8, -1, 8], [-35, 1, 23, 7, -9, 30, 26, -12, -9, 0, -11, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_162_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_162_2_e_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_162_2_e_b(:prec:=12)
    chi := MakeCharacter_162_e();
    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_162_2_e_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_162_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![5184, 7776, 2592, 3672, 3564, -5184, 1053, 27, 162, -24, 9, -3, 1]>],Snew);
    return Vf;
end function;