
// Make newform 117.2.e.c in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_117_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_117_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_117_2_e_c();".
// 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_117_2_e_c();".
// 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 := [729, -243, -243, -27, -18, 27, 24, 9, -2, -1, -3, -1, 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], [243, 243, 27, 18, -27, -24, -9, 2, 1, 3, 1, -1], [7533, 4374, 405, -738, -963, -582, -120, 62, 79, 60, 13, -7], [3321, 2430, 639, -144, -456, -345, -133, 0, 28, 31, 12, -1], [-2187, 486, 1242, 603, 252, -84, -147, -98, -28, 0, 14, 10], [-32319, -486, 12393, 6948, 3690, 51, -1329, -1046, -364, -87, 116, 103], [33777, -4860, -17253, -8676, -3744, 795, 1941, 1358, 394, 51, -176, -133], [-10044, -2133, 2178, 1692, 1200, 300, -175, -233, -105, -47, 17, 24], [-25029, -1944, 8181, 5058, 2934, 303, -807, -752, -280, -87, 74, 73], [43659, 10260, -9405, -7254, -5352, -1353, 731, 1014, 472, 211, -66, -103], [29403, 8019, -5454, -4689, -3618, -1083, 378, 634, 317, 150, -34, -65]];
    Rf_basisdens := [1, 1, 243, 729, 162, 243, 1458, 1458, 243, 486, 486, 243];
    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_117_e();"
function MakeCharacter_117_e()
    N := 117;
    order := 3;
    char_gens := [92, 28];
    v := [1, 3];
    // 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_117_e_Hecke();"
function MakeCharacter_117_e_Hecke(Kf)
    N := 117;
    order := 3;
    char_gens := [92, 28];
    char_values := [[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 0, 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]];
    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, 1, 0, 0, 0, -1, -1, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, -1, 0, 0, -1, -1, -1, 1, 1, 0], [0, 1, 0, 0, 1, -1, 0, 0, 1, 0, -1, 0], [1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-3, 0, 0, 1, 0, 0, 0, 1, 0, 0, -1, -1], [0, 0, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1], [0, -2, 2, 0, -3, 0, -2, -2, 0, 0, 0, 0], [2, 1, 0, -3, -3, 3, 2, -1, -3, 0, 0, 0], [0, 0, 1, 1, -1, 0, 2, 1, 1, -1, -1, 0], [-2, -2, -2, -1, 0, 1, 0, 4, 1, 3, -1, 0], [0, 2, -1, 0, 3, 0, 0, 3, 0, -1, -1, 1], [-1, 0, -1, 5, 4, -4, -1, 0, 0, -1, 1, -4], [2, -1, 1, -5, -3, 3, 2, 2, 0, 1, -2, 3], [-4, 2, 2, 0, 0, 0, 0, -3, 0, -3, 0, 0], [0, 2, 2, 4, 3, 0, -3, 2, 4, -4, 0, -4], [1, -4, 4, -3, -2, 2, 1, 1, 1, 4, -1, 3], [0, 0, -2, 1, 2, 0, -2, 1, 1, 2, -1, 0], [-2, -1, -1, 7, 0, -6, 0, -3, -2, -2, 1, -5], [0, 0, 0, -2, 0, 1, 0, -2, 0, -1, 1, 2], [0, -4, 3, 2, -1, 1, 0, -2, -2, 3, 3, -1], [5, 1, -3, 8, 6, -6, 5, 0, 3, -3, 2, -3], [4, -1, -1, -3, 0, 0, 0, -4, 1, -2, 2, 2], [-1, 2, -3, 9, 5, -5, -1, -3, -1, -3, 4, -6], [0, 0, -3, 3, 3, -3, 0, 3, 4, -3, 0, 1], [0, -2, 2, -1, -4, 0, 2, 4, -1, 0, -6, 7], [-10, -2, -2, 0, 0, 3, 0, 4, -1, 2, -2, 1], [3, 0, 0, -2, 0, 1, 0, -2, 3, -4, -2, -1], [0, -1, 4, -2, -6, 0, -1, -7, -2, -3, 6, -4], [-3, 0, 0, 6, 0, -4, 0, 1, -2, 1, 0, -4], [0, -2, 3, -2, 0, 0, -4, -3, -2, -1, 1, 1], [5, -1, 4, -2, 0, 0, 5, -1, 2, 4, -2, 2], [0, 1, -3, 2, 4, 0, 4, 6, 2, 2, -5, 3], [0, 3, -5, 1, 3, 0, 4, 7, 1, 2, -4, 3], [-3, 2, -1, -7, -3, 3, -3, 3, 5, -1, -4, 8], [0, -1, -4, -4, 0, 0, -1, 0, -4, 5, -1, 5], [1, 1, 1, 2, 0, 1, 0, 1, -2, 0, -1, 0], [0, -5, 6, 3, 3, 0, 5, -3, 3, -1, -2, -1], [2, 2, -2, 2, -3, 3, 2, -5, -5, -2, 5, -2], [-1, 1, 1, -1, 0, 6, 0, 5, -2, 2, -3, 3], [-4, 4, 4, 3, 0, -3, 0, -5, 3, -8, -3, -6], [0, -6, 0, 2, 3, -3, 0, 7, 4, 0, -1, 1], [0, -1, -3, -5, -4, 0, 3, -6, -5, 4, 5, 0], [2, -1, -1, -6, 0, 3, 0, 3, 3, 3, 0, 3], [-2, 1, 1, -2, 0, 2, 0, -2, -1, -1, 1, 3], [0, 4, -5, -1, 3, 0, 3, -1, -1, 1, 5, -4], [4, -3, 3, -8, -9, 9, 4, -1, -2, 3, 1, 7], [1, -1, 3, -1, 0, 0, 1, -1, -3, 3, -1, -3], [0, 1, 3, 0, 1, 0, -8, -7, 0, -4, 8, -8], [-5, 4, 4, 9, 0, 0, 0, 3, -6, 0, -3, -3], [0, 4, -7, -6, -3, 0, 4, -2, -6, 3, 6, 0], [3, 3, -1, -11, -2, 2, 3, 7, 8, -1, -9, 10], [-6, 6, 6, -4, 0, 0, 0, -2, 5, -3, -1, -1], [0, -1, 2, -5, -6, 0, 9, -1, -5, -1, 0, 5], [12, 5, -1, -4, 0, 0, 12, 0, 2, -1, -4, 2], [6, -1, -1, 2, 0, -9, 0, -9, -1, -1, 8, -1], [4, -5, -5, -5, 0, 5, 0, 7, -4, 11, 4, 9], [5, -3, 0, -12, -6, 6, 5, 9, 3, 0, -6, 9], [-8, 4, -1, 11, 6, -6, -8, -8, -3, -1, 5, -9], [0, 1, 4, 3, 0, 0, -3, -5, 3, -5, 6, -9], [0, 3, -4, -2, 0, 0, -4, 3, -2, 1, 0, 2], [-5, 5, 5, 7, 0, 2, 0, 2, -4, -3, -5, -3], [0, -2, -1, -1, -6, 0, -5, 1, -1, 3, -3, 4], [-9, 0, 0, 4, -3, 3, -9, -7, -11, 0, 7, -8], [-9, 7, -5, 7, 0, 0, -9, -9, -8, -5, 7, -8], [0, 0, 1, -4, -4, 4, 0, -1, 1, 1, 0, 5], [0, 8, -3, -1, 2, 0, -14, 2, -1, -5, 6, -5], [0, 0, -3, 3, 12, 0, -12, -1, 3, 3, 1, -4], [2, -10, 0, 3, 2, -2, 2, 9, 2, 0, 1, 0], [4, 4, -1, 0, 6, -6, 4, 3, 9, -1, -6, 3], [-6, 5, 5, 4, 0, -3, 0, 4, 4, -1, -5, -8], [-15, 3, 3, -7, -3, 3, -15, -2, -4, 3, -4, -1], [0, -5, 7, 5, -3, 0, 1, 2, 5, -2, -7, 2], [7, 6, 6, -7, 0, 0, 0, -2, 8, -3, -1, -1], [0, -6, 10, 2, -3, 0, 1, 1, 2, -4, -7, 5], [2, 7, -3, -2, 0, 0, 2, -2, -2, -3, -2, -2], [-1, 1, 1, 5, 0, 3, 0, 5, -7, 4, -1, 2], [0, -7, -2, -6, -9, 0, 11, -3, -6, 9, -4, 10], [0, 4, -2, 3, 3, 0, 15, 13, 3, -2, -9, 6], [0, 2, 3, 5, 3, 0, -17, 4, 5, -5, -2, -3], [-5, -2, -3, 3, 1, -1, -5, 3, -2, -3, 2, -3], [2, -2, -2, -8, 0, 3, 0, 2, 2, 5, 3, 6], [-1, 0, 0, 1, 0, -1, 0, 0, 5, -5, -5, -6], [6, 9, -6, -1, 6, -6, 6, 4, 2, -6, -7, -4], [14, 6, -1, 1, -3, 3, 14, -9, -8, -1, 4, -5], [-10, -9, -9, 15, 0, -9, 0, 0, -7, 1, 1, -8], [11, 11, -5, 10, 10, -10, 11, -6, 5, -5, 0, -5], [-3, -3, 0, 8, 6, -6, -3, 1, 6, 0, 2, 0], [0, -3, -6, -4, 0, 0, 2, 1, -4, 9, -4, 8], [-16, -8, -8, 4, 0, -3, 0, 1, -2, 2, 1, -2], [-8, 2, 0, 7, 0, 0, -8, -9, -12, 0, 7, -12], [-6, 7, 7, -4, 0, 8, 0, 6, 0, 2, -4, 4], [0, -1, -4, 5, 3, 0, 3, 4, 5, 5, -5, 0], [0, 1, 3, 6, 10, 0, -9, 14, 6, -4, -13, 7], [-2, 1, 1, -9, 0, 0, 0, 2, 8, 3, 1, 1], [0, 3, 6, -2, -3, 0, 3, -2, -2, -9, 5, -3], [5, -15, -15, -6, 0, 0, 0, 8, -5, 19, 11, 11], [2, -3, -3, 1, 0, 5, 0, 9, 2, 1, -8, -3], [1, 6, 6, 2, 0, 0, 0, 4, 5, -3, -7, -7], [18, -6, -1, 7, 4, -4, 18, 4, 2, -1, 3, -2], [-1, 4, 4, 7, 0, 6, 0, 6, -6, -1, -7, -1], [0, 12, -4, 7, 15, 0, -10, 11, 7, -8, 1, -8], [-17, 10, -1, -1, 3, -3, -17, -5, -1, -1, -4, -4], [0, 2, -1, -4, -6, 0, 7, -9, -4, -1, 11, -7], [5, -5, -5, -6, 0, 9, 0, 4, -3, 4, 0, 9], [9, -2, -2, 14, 3, -3, 9, -7, -10, -2, 11, -13], [15, -5, -5, -7, 0, 12, 0, 2, -4, 1, -1, 11], [0, 12, -10, 7, 6, 0, -1, 6, 7, -2, 6, -13], [-4, 16, -3, -6, 1, -1, -4, -6, -5, -3, -7, -6], [0, 3, -15, 2, 9, 0, 5, -2, 2, 12, 5, -7], [0, -11, -11, 0, 0, 4, 0, 6, -4, 6, 0, 4], [0, -8, 8, 4, 3, 0, -11, -11, 4, 0, 3, -7], [23, -8, 0, -6, -5, 5, 23, 9, 4, 0, -1, 9], [3, -6, -6, 9, 0, -7, 0, 1, -5, 4, 3, -4], [1, 8, 6, -9, -3, 3, 1, -8, 2, 6, -6, 5], [0, -12, -2, -16, -16, 0, -4, -12, -16, 14, 0, 16], [4, -8, -8, -12, 0, 3, 0, 4, 1, 12, 8, 11], [0, 0, 1, -3, 6, 0, -5, 7, -3, -1, -7, 10], [19, -5, -1, 5, 0, 0, 19, 1, 3, -1, 5, 3], [0, -11, 7, -9, -21, 0, -12, -17, -9, 4, 6, 3], [-12, -10, 9, -17, -19, 19, -12, -1, -9, 9, 2, 10], [-3, -2, 4, -14, -3, 3, -3, 9, 11, 4, -11, 14], [-17, 7, 7, 5, 0, -6, 0, -14, -4, -9, 5, -1], [-13, -9, 10, -7, 2, -2, -13, 8, 7, 10, -9, 5], [2, -1, -1, -8, 0, 0, 0, -7, 10, -9, -2, -2], [0, -2, 3, 0, -2, 2, 0, -3, 4, 3, 2, 6], [5, 5, 5, 6, 0, -9, 0, -12, 3, -12, 0, -9], [1, 0, 7, -18, -10, 10, 1, 1, -7, 7, -8, 3], [0, -8, 8, 4, 5, 0, 7, -10, 4, 0, 2, -6], [-7, 10, 10, 3, 0, -2, 0, -6, 2, -9, -3, -5], [0, -1, -3, 2, 6, 0, -5, 3, 2, 4, -4, 2], [0, -3, 3, -9, -12, 0, -5, -15, -9, 0, 12, -3], [0, -11, -11, -8, 0, 3, 0, 6, -5, 16, 10, 13], [0, -6, 0, 3, 9, 0, 3, -12, 3, 6, 6, -9], [0, 0, -6, -1, 3, 0, -3, 7, -1, 6, -7, 8], [3, 5, 5, -16, 0, 12, 0, 2, 13, -7, -9, 3], [0, -5, 0, 6, 4, 0, 2, -4, 6, 5, -1, -5], [0, 2, 0, 3, -3, 0, -5, 7, 3, -2, -5, 2], [-4, 0, 0, 0, 0, -6, 0, 6, 11, 1, -5, -11], [-7, 5, 5, 8, 0, -5, 0, -10, -5, -8, 2, -3], [-9, -4, -10, 12, 9, -9, -9, 11, 4, -10, 3, -5], [0, 5, -4, -4, 15, 0, -10, 0, -4, -1, 5, -1], [2, -8, -8, -8, 0, -15, 0, -20, 4, -1, 19, 4], [6, -4, -4, -10, 0, 11, 0, 13, 9, 3, -10, 1], [-6, 13, -8, 3, 15, -15, -6, 7, 11, -8, -12, -4], [-1, -5, 8, -2, -10, 10, -1, -11, -21, 8, 8, -11], [-5, -5, 8, 12, 0, 0, -5, -15, -16, 8, 12, -16], [0, -4, 9, 6, 8, 0, -18, 1, 6, -5, -5, -1], [15, 3, 3, -10, 0, -3, 0, -3, 10, 0, 3, 0], [0, 9, -2, 19, 23, 0, -1, 19, 19, -7, -10, -9], [20, 2, 2, 0, 0, 9, 0, 4, -5, 0, -4, 5], [-15, -2, -2, -6, 0, 11, 0, 17, 4, 8, -9, 2], [0, 7, 3, 0, -6, 0, -4, -2, 0, -10, 9, -9], [-1, 3, -15, 15, 9, -9, -1, 6, 6, -15, 6, -3], [-16, 0, 2, 7, -3, 3, -16, -12, -14, 2, 10, -11], [-2, 14, 14, 18, 0, -8, 0, -14, -7, -17, -3, -11], [-18, -15, 3, -3, -15, 15, -18, 0, -22, 3, 12, -7], [11, 8, 8, -6, 0, -13, 0, -15, 13, -9, 6, -7], [0, 10, -11, 1, 15, 0, 24, 10, 1, 1, 0, -1], [4, -1, 3, 13, 21, -21, 4, 6, 16, 3, -8, -5], [-12, -12, -12, 7, 0, 6, 0, 12, -19, 18, 6, 12], [0, -6, 2, -15, -23, 0, 7, -20, -15, 4, 14, 1], [-12, -1, -1, -1, 0, 18, 0, 23, -2, 8, -15, 3], [0, -7, -9, -2, 0, 0, -11, -7, -2, 16, 0, 2], [-5, -4, 0, -5, -9, 9, -5, 0, 0, 0, 4, 9], [7, -9, -9, 10, 0, -13, 0, -12, -10, 1, 13, 0], [-3, 15, -2, -11, 5, -5, -3, 3, 12, -2, -16, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_117_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_117_2_e_c();".
// 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_117_2_e_c(:prec:=12)
    chi := MakeCharacter_117_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_117_2_e_c();".
// 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_117_2_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_117_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, -22, 101, -158, 246, -38, 196, -55, 63, -10, 11, -2, 1]>],Snew);
    return Vf;
end function;