
// Make newform 169.2.e.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_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_169_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_169_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 := [1, 0, -6, 0, 31, 0, -28, 0, 19, 0, -5, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1563, 0, -30, 0, 155, 0, -361, 0, 95, 0, -25], [1004, 0, 30, 0, -155, 0, 361, 0, -95, 0, 25, 0], [-24, 0, 124, 0, -139, 0, 76, 0, -20, 0, 3, 0], [242, 0, 54, 0, -279, 0, 538, 0, -171, 0, 45, 0], [0, 1246, 0, 84, 0, -434, 0, 899, 0, -266, 0, 70], [-95, 0, 3379, 0, -2831, 0, 2071, 0, -545, 0, 114, 0], [0, -95, 0, 3379, 0, -2831, 0, 2071, 0, -545, 0, 114], [852, 0, -4402, 0, 4483, 0, -2698, 0, 710, 0, -128, 0], [0, 1506, 0, -7781, 0, 7314, 0, -4769, 0, 1255, 0, -242], [0, 1848, 0, -9548, 0, 8338, 0, -5852, 0, 1540, 0, -317]];
    Rf_basisdens := [1, 1, 559, 559, 43, 559, 559, 559, 559, 559, 559, 559];
    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_169_e();"
function MakeCharacter_169_e()
    N := 169;
    order := 6;
    char_gens := [2];
    v := [5];
    // 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_169_e_Hecke();"
function MakeCharacter_169_e_Hecke(Kf)
    N := 169;
    order := 6;
    char_gens := [2];
    char_values := [[1, 0, 0, 0, 0, 0, 0, -1, 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, 0, 0, 0, 0, 1, 0, 1, 0], [1, 0, 0, -1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 0, -1, 0, 0, 0, 2, 0, -1, 0, -2, -1], [0, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 1, 0, -2, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -2, 0, 0, 1, 0, 1, 0, 0], [0, 1, -2, 0, 0, 0, -1, 0, 0, 0, 0, 0], [2, 0, 0, -2, 2, 0, 0, -3, 0, 2, 0, 0], [-5, 0, 0, 5, -3, 2, 0, 3, 0, -5, 0, 0], [0, 0, 5, 0, 0, 0, -1, 0, 3, 0, 1, 5], [0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 3, -2], [0, 6, 0, 0, 0, 0, 0, 0, -6, 0, -1, -4], [6, 0, 0, 0, 3, 0, 0, -6, 0, 2, 0, 0], [0, 0, 1, 0, 0, 0, -6, 0, -1, 0, 6, 1], [0, 0, 0, 4, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, -1, 0, 6, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 5, 0, 4, 6], [0, 0, 3, 0, 0, 0, -10, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, 4, 0, 3, 0, -4, -6], [-5, 0, 0, 1, 0, 9, 0, 0, 0, 0, 0, 0], [0, 0, -9, 0, 0, 0, 6, 0, -2, 0, -6, -9], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -6, -7], [0, -7, -1, 0, 0, 0, 5, 0, 0, 0, 0, 0], [-3, 0, 0, 3, -10, -7, 0, 6, 0, -3, 0, 0], [-4, 0, 0, 8, 0, 7, 0, 0, 0, 0, 0, 0], [3, 0, 0, -3, -2, -5, 0, -1, 0, 3, 0, 0], [0, 0, -2, 0, 0, 0, -4, 0, -6, 0, 4, -2], [-8, 0, 0, 0, 2, 0, 0, 8, 0, 1, 0, 0], [-9, 0, 0, 9, -1, 8, 0, -4, 0, -9, 0, 0], [-2, 0, 0, -5, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 0, 0, -8, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -6, 0, 0, 1, 0, -11, 0, 0], [0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 12, 0, 0, 0, -5, 0, 2, 0, 5, 12], [-4, 0, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0], [0, -2, 1, 0, 0, 0, -13, 0, 0, 0, 0, 0], [0, 14, 0, 0, 0, 0, 0, 0, -14, 0, -10, -2], [0, 0, 0, 0, 8, 0, 0, 0, 0, 16, 0, 0], [5, 0, 0, -5, 4, -1, 0, 2, 0, 5, 0, 0], [-6, 0, 0, -1, 0, -9, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -9, 0, 0, 0, 0, 4, 0, 0], [0, -10, 0, 0, 0, 0, 0, 0, 10, 0, 8, 8], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, -8], [-11, 0, 0, 0, -11, 0, 0, 11, 0, -6, 0, 0], [5, 0, 0, -5, -5, -10, 0, -2, 0, 5, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 3, 0, 9, 3], [0, -3, -8, 0, 0, 0, 2, 0, 0, 0, 0, 0], [0, 0, -8, 0, 0, 0, -8, 0, -7, 0, 8, -8], [5, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 3, 0, 0, 0, 13, 0, 6, 0, -13, 3], [0, 11, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 16, 0, 0, -6, 0, 2, 0, 0], [-13, 0, 0, 13, 5, 18, 0, 11, 0, -13, 0, 0], [6, 0, 0, -6, 13, 7, 0, -9, 0, 6, 0, 0], [6, 0, 0, 0, -1, 0, 0, -6, 0, 3, 0, 0], [0, 9, 0, 0, 0, 0, 0, 0, -9, 0, -16, -14], [0, 0, 0, 0, -5, 0, 0, 0, 0, -10, 0, 0], [0, 0, 5, 0, 0, 0, -8, 0, 6, 0, 8, 5], [20, 0, 0, -20, 14, -6, 0, -12, 0, 20, 0, 0], [0, -16, -7, 0, 0, 0, 17, 0, 0, 0, 0, 0], [0, 0, -2, 0, 0, 0, 2, 0, 10, 0, -2, -2], [14, 0, 0, -15, 0, -18, 0, 0, 0, 0, 0, 0], [-13, 0, 0, 4, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 8, 0, 0, 0, -22, 0, 18, 0, 22, 8], [0, 11, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0], [9, 0, 0, -5, 0, 7, 0, 0, 0, 0, 0, 0], [12, 0, 0, 0, 7, 0, 0, -12, 0, 2, 0, 0], [0, -3, 0, 0, 0, 0, 0, 0, 3, 0, -7, 4], [0, 7, 0, 0, 0, 0, 0, 0, -7, 0, -5, 1], [0, 0, 10, 0, 0, 0, -17, 0, 18, 0, 17, 10], [-11, 0, 0, 11, 6, 17, 0, -7, 0, -11, 0, 0], [-10, 0, 0, 0, -5, 0, 0, 10, 0, -16, 0, 0], [0, 12, 0, 0, 0, 0, 0, 0, -12, 0, -25, -13], [0, -16, -8, 0, 0, 0, 2, 0, 0, 0, 0, 0], [-14, 0, 0, 2, 0, 18, 0, 0, 0, 0, 0, 0], [0, -10, -11, 0, 0, 0, 6, 0, 0, 0, 0, 0], [0, 9, 0, 0, 0, 0, 0, 0, -9, 0, 10, -1], [0, 21, 15, 0, 0, 0, -16, 0, 0, 0, 0, 0], [9, 0, 0, -9, 2, -7, 0, 22, 0, 9, 0, 0], [0, 0, 6, 0, 0, 0, 18, 0, -5, 0, -18, 6], [0, -10, 0, 0, 0, 0, 0, 0, 10, 0, 11, -6], [-4, 0, 0, 0, 7, 0, 0, 4, 0, -5, 0, 0], [-9, 0, 0, 9, -14, -5, 0, 9, 0, -9, 0, 0], [12, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0], [0, -11, -6, 0, 0, 0, 10, 0, 0, 0, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0, 8, 0, 16, 17], [0, 7, -6, 0, 0, 0, -12, 0, 0, 0, 0, 0], [0, 0, 8, 0, 0, 0, -3, 0, -5, 0, 3, 8], [10, 0, 0, 19, 0, -11, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, -2, 0, -27, 3], [0, -31, -11, 0, 0, 0, 23, 0, 0, 0, 0, 0], [11, 0, 0, -11, 8, -3, 0, 12, 0, 11, 0, 0], [0, 0, -15, 0, 0, 0, -4, 0, -6, 0, 4, -15], [15, 0, 0, 0, 24, 0, 0, -15, 0, 21, 0, 0], [0, 9, 0, 0, 0, 0, 0, 0, -9, 0, 0, -2], [-29, 0, 0, 10, 0, 12, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 8, -5, 3, 0, 24, 0, -8, 0, 0], [0, 0, 0, 0, 0, 0, 27, 0, -6, 0, -27, 0], [21, 0, 0, -12, 0, -17, 0, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 15, 9], [-18, 0, 0, 0, -3, 0, 0, 18, 0, -7, 0, 0], [10, 0, 0, -10, 3, -7, 0, -20, 0, 10, 0, 0], [3, 0, 0, 7, 0, 11, 0, 0, 0, 0, 0, 0], [0, 0, -8, 0, 0, 0, -27, 0, 3, 0, 27, -8], [0, -21, 0, 0, 0, 0, 0, 0, 21, 0, -10, -9], [0, 0, 14, 0, 0, 0, -6, 0, -14, 0, 6, 14], [-5, 0, 0, 14, 0, 9, 0, 0, 0, 0, 0, 0], [6, 0, 0, -6, 0, -6, 0, -9, 0, 6, 0, 0], [0, 0, 0, 0, -22, 0, 0, 0, 0, -9, 0, 0], [0, 7, 0, 0, 0, 0, 0, 0, -7, 0, 8, -12], [0, 20, 20, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, -6, 0, 0, 0, 4, 0, 3, 0, -4, -6], [0, -18, 3, 0, 0, 0, -14, 0, 0, 0, 0, 0], [29, 0, 0, 0, -1, 0, 0, -29, 0, -4, 0, 0], [0, 5, 17, 0, 0, 0, 9, 0, 0, 0, 0, 0], [-12, 0, 0, 12, -16, -4, 0, 4, 0, -12, 0, 0], [-15, 0, 0, 15, -14, 1, 0, -5, 0, -15, 0, 0], [-25, 0, 0, 0, 1, 0, 0, 25, 0, -7, 0, 0], [0, -17, 0, 0, 0, 0, 0, 0, 17, 0, 10, 25], [-24, 0, 0, 24, -11, 13, 0, 23, 0, -24, 0, 0], [-22, 0, 0, -4, 0, -10, 0, 0, 0, 0, 0, 0], [0, 17, 10, 0, 0, 0, -11, 0, 0, 0, 0, 0], [0, 13, 0, 0, 0, 0, 0, 0, -13, 0, -29, -8], [15, 0, 0, -4, 0, 16, 0, 0, 0, 0, 0, 0], [0, 5, 31, 0, 0, 0, -2, 0, 0, 0, 0, 0], [-6, 0, 0, 0, -13, 0, 0, 6, 0, -9, 0, 0], [8, 0, 0, -3, 0, -14, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, 0, -6, 0, 27, 0, 6, 2], [0, 2, 0, 0, 0, 0, 0, 0, -2, 0, 12, -10], [0, 7, 0, 0, 0, 0, 0, 0, -7, 0, 21, 1], [10, 0, 0, -10, 16, 6, 0, 6, 0, 10, 0, 0], [0, 0, 0, 0, 22, 22, 0, 7, 0, 0, 0, 0], [0, 9, 24, 0, 0, 0, -5, 0, 0, 0, 0, 0], [0, -38, 0, 0, 0, 0, 0, 0, 38, 0, 32, 23], [0, -1, 8, 0, 0, 0, -32, 0, 0, 0, 0, 0], [0, -24, -19, 0, 0, 0, 23, 0, 0, 0, 0, 0], [-9, 0, 0, 0, 9, 0, 0, 9, 0, -15, 0, 0], [-16, 0, 0, 16, -6, 10, 0, 28, 0, -16, 0, 0], [0, 0, -17, 0, 0, 0, 23, 0, -3, 0, -23, -17], [0, 21, 0, 0, 0, 0, 0, 0, -21, 0, -9, 6], [-17, 0, 0, 0, -6, 0, 0, 17, 0, -18, 0, 0], [0, 0, -23, 0, 0, 0, 32, 0, -30, 0, -32, -23], [-2, 0, 0, 2, -5, -3, 0, 25, 0, -2, 0, 0], [0, 14, -14, 0, 0, 0, 14, 0, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, -36, 0, 0, 0, 36, -4], [-30, 0, 0, 8, 0, 7, 0, 0, 0, 0, 0, 0], [6, 0, 0, -8, 0, -20, 0, 0, 0, 0, 0, 0], [0, 0, 8, 0, 0, 0, 17, 0, -8, 0, -17, 8], [0, -11, -33, 0, 0, 0, 21, 0, 0, 0, 0, 0], [12, 0, 0, -12, 13, 1, 0, -34, 0, 12, 0, 0], [-9, 0, 0, -21, 0, -7, 0, 0, 0, 0, 0, 0], [5, 0, 0, -5, 8, 3, 0, -15, 0, 5, 0, 0], [-7, 0, 0, 7, 9, 16, 0, -5, 0, -7, 0, 0], [22, 0, 0, 12, 0, -9, 0, 0, 0, 0, 0, 0], [-14, 0, 0, 0, 7, 0, 0, 14, 0, 2, 0, 0], [0, 4, 9, 0, 0, 0, 13, 0, 0, 0, 0, 0], [17, 0, 0, 9, 0, 3, 0, 0, 0, 0, 0, 0], [0, 0, 12, 0, 0, 0, -11, 0, 12, 0, 11, 12], [0, 8, 0, 0, 0, 0, 0, 0, -8, 0, 1, 20], [3, 0, 0, 0, -27, 0, 0, -3, 0, 9, 0, 0], [0, 0, 17, 0, 0, 0, -7, 0, 24, 0, 7, 17], [3, 0, 0, 0, 8, 0, 0, -3, 0, -16, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 1, 0, -25, 0], [0, 0, 13, 0, 0, 0, -38, 0, 33, 0, 38, 13], [-12, 0, 0, 12, -25, -13, 0, 26, 0, -12, 0, 0], [-3, 0, 0, 0, 13, 0, 0, 3, 0, 11, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_169_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_169_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_169_2_e_b(:prec:=12)
    chi := MakeCharacter_169_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_169_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_169_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 0, -5, 0, 19, 0, -28, 0, 31, 0, -6, 0, 1]>],Snew);
    return Vf;
end function;