
// Make newform 273.2.t.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_273_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_273_t_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_273_2_t_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_273_2_t_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 := [64, -96, 16, 80, -60, -20, 45, -10, -15, 10, 1, -3, 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], [-800, 608, 344, -620, 42, 313, -164, -91, 72, -9, -13, 1], [256, -320, 40, 260, -154, -73, 132, -21, -32, 25, 5, -9], [288, -304, -88, 380, -140, -167, 166, 21, -78, 29, 17, -11], [544, -512, -160, 584, -126, -289, 200, 91, -96, 5, 29, -13], [-464, 216, 260, -340, 7, 194, -101, -42, 51, -9, -13, 8], [-1664, 1184, 720, -1452, 42, 821, -396, -231, 208, -5, -57, 13], [-88, 96, 16, -99, 35, 45, -41, -7, 18, -4, -5, 2], [-48, 102, 3, -82, 56, 43, -44, 0, 20, -6, -4, 3], [464, -664, -64, 634, -273, -271, 269, 21, -121, 30, 20, -15], [1056, -872, -500, 1062, -168, -547, 310, 133, -160, 13, 39, -17]];
    Rf_basisdens := [1, 1, 224, 224, 224, 224, 112, 224, 28, 28, 112, 112];
    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_273_t();"
function MakeCharacter_273_t()
    N := 273;
    order := 6;
    char_gens := [92, 157, 106];
    v := [6, 4, 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_273_t_Hecke();"
function MakeCharacter_273_t_Hecke(Kf)
    N := 273;
    order := 6;
    char_gens := [92, 157, 106];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 1, 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, 1, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, -1, 0, 1, 0, 1, -1, -1, 1, 0], [1, -1, 1, 1, 1, -1, 2, 0, 1, 0, 0, 1], [-1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1, 0], [0, -1, 0, 0, 0, 0, 2, 0, 1, -1, 1, 1], [0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0], [0, 1, -1, -2, -1, 0, -1, 1, -3, 0, 2, -1], [2, 1, -1, 0, -1, 0, -1, 0, 0, 0, 0, -1], [-1, 3, 0, 1, 1, 1, 2, -1, -1, 0, 0, -1], [-2, 1, 1, 0, 0, 0, -1, -1, 1, 0, -2, 1], [1, 2, 1, 2, 3, 1, 4, -1, 3, -2, 0, -1], [-1, -1, -1, 1, -3, 3, -2, 4, 0, 0, 0, -4], [-2, -2, 1, -1, -1, 1, 1, -1, 2, 0, 0, 0], [4, -4, 3, 5, 1, -3, 3, -1, 0, 1, -1, 2], [-2, 2, -1, 0, -4, 2, 0, -1, -1, 2, -1, -1], [5, -2, 0, 1, 4, -3, 1, 3, 0, 1, 0, 0], [-3, 3, -2, 2, -7, 3, 0, -1, -4, 0, 0, -1], [-3, 2, -1, 0, 0, 1, 0, -1, -1, -5, 5, -2], [-1, 1, 1, 4, 0, -1, 3, -3, -2, 2, -2, -2], [2, -2, 3, -2, 4, 0, 2, -3, 1, 0, -4, 3], [-2, 2, 0, 0, 0, 0, 0, -2, 0, -2, -2, 2], [2, 0, -1, 0, 3, -1, 4, 1, 3, -1, 0, 1], [2, -3, 3, 0, 1, 0, -2, 0, 1, 1, 0, -3], [-2, 3, -4, -1, -7, 4, -3, -1, -1, 1, -1, -5], [6, -1, -2, 5, -1, -2, 1, 7, -1, 1, 1, -5], [0, 2, -2, 1, 0, -2, 1, 0, -1, 3, 3, 2], [-7, 9, -4, -7, -4, 2, -4, 2, -7, -4, 8, -4], [2, 1, 0, 0, 1, 0, -1, 0, 0, 0, 1, 0], [6, -3, 3, -5, 7, 3, -4, -1, -1, 2, 2, -2], [2, -5, -1, -3, 1, -2, -3, 3, 2, 4, -2, 3], [2, -1, 3, -3, 6, 3, -1, -4, 2, 0, 0, 1], [-7, -2, -2, -5, -10, 1, -4, -1, -1, -5, 0, 2], [-1, 3, -4, -6, -1, -4, -5, 0, -9, 4, 4, 4], [-3, 4, -4, 2, -5, -1, -3, 3, -2, 0, -3, -3], [-1, -3, 5, 0, 2, 5, -2, 0, 5, 0, 0, 0], [6, -5, 3, 4, -3, -6, 1, 3, -1, 0, 0, 3], [-4, 3, -1, 0, -3, 0, -3, 1, -1, 0, 8, -1], [-2, 3, -6, 0, -4, -6, 3, 0, -6, 0, 1, 0], [4, -2, 7, 8, 1, -4, 2, -3, -1, 4, -2, -3], [8, 4, -6, 7, 4, -6, 5, 4, -1, -3, -3, 2], [-4, 4, 0, -2, 0, -2, 5, -2, -7, 0, 0, 0], [0, 8, -3, 5, -4, 0, 8, 3, -8, -10, 5, 3], [3, 1, 0, -3, 0, 0, 1, 3, -1, -3, 3, 3], [7, -2, 0, 4, 2, 3, -1, 3, 4, 0, 2, -3], [2, -5, -2, 5, -2, 2, -4, 2, 7, -2, 4, -2], [2, 6, -2, 1, -1, -2, 8, 4, -3, 0, 0, 4], [-9, -5, 3, 0, -1, -1, 6, -4, 3, 0, -5, 4], [1, -12, 0, -11, 0, -1, -7, 1, 4, -2, 0, 0], [-3, 1, 3, 4, 5, -3, 4, -4, -1, -4, 4, -1], [-9, -7, 2, -7, -12, 2, -5, 3, -3, 2, 2, -5], [4, -2, 7, 3, 5, 2, 1, -2, 5, -3, 0, -7], [-3, 1, 5, 1, -1, 5, -3, -5, 1, 4, 0, -5], [-10, 1, 1, 0, -13, 1, -5, 3, -4, 0, 0, -4], [-13, 0, -3, 1, -3, 2, -4, 2, 2, -3, 6, -3], [11, 0, 1, -1, 9, 1, -4, 2, -3, -2, -2, -3], [9, -5, 6, 2, 6, -3, 2, -3, 6, 1, -2, 6], [-3, -10, 3, -4, -15, -3, -8, 3, -1, 7, 0, -3], [14, -4, 1, 6, 1, -3, 0, -3, 7, 5, -10, 1], [-10, 10, 0, 5, -10, 5, 1, -5, -4, 3, 0, 0], [-9, 7, -1, 8, -8, -1, 3, -1, 2, 2, 2, 2], [-4, 6, -1, -7, 3, 1, -1, 1, 0, 4, -4, 0], [4, 1, -1, -1, 11, 1, 3, -1, 3, 2, 0, 1], [-2, 13, 1, 3, 7, 2, 11, -3, -2, 6, -3, -3], [11, -1, -1, 7, 1, -1, -1, 10, -2, 1, 1, -9], [4, -5, 11, 3, 13, -11, 4, -5, 7, 4, -4, 6], [2, -4, 0, 8, 0, 2, 2, 2, 8, 0, -5, -2], [2, 11, -8, -9, -8, 6, -15, 6, -2, -5, 10, -8], [-1, 9, 2, -11, 2, 0, -6, 0, -3, -2, 4, 2], [-9, 10, -6, -1, -10, -1, -9, 5, -7, 0, -3, -5], [8, -8, 3, 7, 0, -3, 4, 2, -1, 3, -3, 5], [13, -6, 9, 5, 2, -9, 5, -3, 4, -3, 3, 6], [7, -2, -1, 1, 3, -1, 0, 4, -1, -2, -2, -3], [-3, 3, -2, -5, 3, 3, 0, -1, 3, 4, -2, -1], [1, 5, -2, 7, 1, -5, 0, -3, 5, 0, 2, 3], [12, -3, -1, 4, 5, 0, 3, 1, 3, 0, -9, -1], [-4, 12, -2, 2, -2, 4, 8, -2, -4, -8, 4, -2], [-14, -2, 4, -5, -3, 0, 2, -4, -1, 0, -2, 4], [-3, 3, -7, -3, -1, -1, -4, 1, -8, 8, 0, 7], [6, 3, 3, 9, 3, -3, 1, 3, -5, -5, 0, -3], [1, -5, 0, 1, -11, -1, -4, 1, -1, -10, 5, 1], [-3, -4, 2, -1, -10, -1, -8, 1, -5, -1, 0, -2], [5, -7, 6, 0, 6, -6, -2, -1, 8, 9, -9, 5], [-5, -7, -3, -2, -9, -3, 4, 4, 1, 1, 1, -1], [6, 8, -9, -1, -9, 2, -3, 2, -7, 2, -4, -9], [-4, -1, 9, -3, 2, 9, -5, -6, 4, -1, -1, -3], [-1, 0, 1, 5, 7, -1, 10, 4, -9, -1, 1, 5], [-1, 3, 8, 3, 6, 8, 8, -8, 13, 1, 0, -8], [7, -3, 2, -8, 1, -2, -5, 4, 7, 2, -2, 6], [9, 8, -2, 13, 6, -7, 11, 7, -4, 0, 0, 2], [6, 2, 1, -3, 8, 1, -4, -2, -3, 1, 1, 1], [-10, -1, -9, -6, -3, 9, -11, 5, 2, 5, -5, -4], [2, 4, -3, 2, 5, -1, 3, 1, -2, -7, 0, 3], [-8, -1, -6, -18, -5, 8, -9, -2, 12, -4, 2, -2], [3, -10, 6, 3, 5, -6, -5, -4, 11, -2, 2, 2], [-4, 4, 2, 11, -10, 2, -1, 6, 1, -2, -2, -8], [3, -2, 1, 4, -5, -5, 0, 5, -3, 12, 0, -1], [-5, 3, -2, 4, -1, 5, -2, -3, -6, 0, 0, -3], [4, -13, 4, 12, 4, -3, 3, -3, 13, 3, -6, 4], [10, -15, 3, 3, -5, -3, -10, -1, 13, -6, 6, 2], [5, -2, 10, 2, 10, -10, 9, -10, 3, 4, -8, 10], [6, 3, 7, 1, 3, -7, 6, -5, 1, -2, 2, 2], [-10, 11, -6, -4, -6, -1, -3, -1, -7, 0, 0, -6], [13, -2, -4, 6, -4, 0, -4, 0, 6, 2, -4, -4], [-7, 10, -6, 5, -15, 7, 3, -1, -11, 0, 0, -1], [2, -4, 7, -10, 3, 10, -6, 3, -3, 0, 9, -3], [0, -3, 3, -5, 4, -2, 5, -5, -2, 0, -2, 5], [-4, 8, -7, -3, -1, 7, 7, 9, -14, 0, 0, 2], [-12, 4, -4, -5, 8, 12, -8, -8, 1, -10, 5, -8], [-5, 8, 3, -3, -16, 5, 3, -8, 6, -8, 4, -8], [-3, 12, -3, -2, -2, 3, 9, 0, -1, -10, 5, 0], [-1, 1, -1, -3, 8, 1, 6, 9, -7, -4, 4, 8], [10, -15, 6, -5, 7, -6, -9, 5, 15, 9, -9, 11], [-5, -4, 8, -7, 7, -3, 7, -11, 1, 0, -12, 11], [-13, 12, -15, -3, -17, 15, -6, 0, -9, -2, 2, -15], [-6, -4, 1, 0, 1, 3, -14, 3, 15, 1, -2, 1], [16, -15, 10, 4, 5, -10, 0, 1, 10, 3, -3, 11], [4, -8, 0, 5, 2, 0, 11, 2, 11, -3, -3, -2], [-12, -14, -1, 10, -1, 5, -2, 5, 11, 6, -12, -1], [12, -4, -9, -6, 2, -9, -3, 10, -12, 4, 4, -1], [-2, 15, 7, -4, 14, -7, 16, -2, -9, 2, -2, 5], [7, 3, 9, 14, 8, -7, 10, -2, -5, 2, -1, -2], [13, -1, -1, -1, 14, -1, 2, -1, 1, 5, 5, 2], [-4, -2, 5, 8, -23, -5, -1, 5, -4, -5, 0, -5], [5, -6, 5, 2, -1, -3, -1, 3, 2, 8, 0, -5], [7, -3, 3, 9, 3, -9, 2, -9, 10, -2, 4, 3], [26, -7, 3, 9, 15, 2, 5, -1, 12, 0, -8, 1], [5, 0, 6, 5, 8, -5, 5, -1, 1, -14, 7, -1], [14, -10, 12, 9, 12, -11, 20, -11, 1, 6, -12, 12], [11, -18, 2, 7, -3, -2, -5, 4, 7, -6, 6, 6], [16, -2, 8, 8, -4, -8, 10, -4, -2, -8, 8, 4], [9, -11, 9, 14, 11, 5, 6, -4, 23, 0, -13, 4], [-1, 6, -2, -9, -2, 5, -3, 5, -8, 5, -10, -2], [-6, 13, 0, 5, -7, 6, 7, -6, -5, -2, 1, -6], [4, -4, 11, 6, 16, -6, 10, -17, 17, 0, 2, 17], [-11, 12, -2, -18, -6, -3, -9, -1, -20, 0, 4, 1], [-8, -4, 3, 7, -35, -8, -8, 8, -12, 7, 0, -3], [1, -17, -4, -18, 0, -3, -12, 3, 2, 0, 0, 4], [2, 4, -7, 2, -1, -7, 2, 3, -5, 3, 3, 4], [-10, -7, 8, -11, -1, 8, -3, -9, 5, 9, 9, 1], [7, 5, -3, -5, 31, 7, -1, -7, 1, -2, 0, 3], [4, -9, -5, -19, 23, 5, -8, -5, 6, -6, 0, 5], [4, -14, -4, 15, -4, 3, 5, 3, 6, -3, 6, -4], [-7, 6, -7, 0, -9, -1, 7, 1, 0, -9, 0, 7], [12, 11, -10, 9, 0, -10, -4, 12, -14, 9, 9, -2], [-16, 6, -5, 0, 7, 5, 2, 1, -7, -10, 10, -4], [-11, 3, -4, -1, -18, 0, 1, 0, -2, 7, 0, 4], [6, -2, -5, 4, 0, -6, 4, 11, -9, 8, -4, 11], [-5, 6, 6, 5, 0, 6, -2, -5, 4, 0, 0, -1], [6, -5, 2, 14, -7, -2, 6, -5, -4, 10, -10, -3], [-25, 0, 3, -4, -5, -9, 9, -12, -1, 0, -8, 12], [2, 14, -7, 2, 9, -2, 16, 9, -9, -14, 7, 9], [10, -13, -3, 8, -3, 8, -9, 8, 14, -3, 6, -3], [14, 9, -4, -11, -4, 6, -9, 6, -6, -2, 4, -4], [-21, -8, 4, -14, -13, 4, -2, -8, 2, -2, -2, 4], [-7, 5, -6, -3, -6, 4, 7, 4, -16, 1, -2, -6], [13, -4, 8, 8, 5, -13, 9, 5, 0, -8, 4, 5], [-6, 6, 4, -9, 6, -10, 4, -14, -5, 0, 6, 14], [-10, 9, -14, 6, -14, -1, 0, -1, -8, -1, 2, -14], [7, 17, -1, -17, 6, -7, -10, -6, -18, 0, 9, 6], [22, -5, 3, 0, 37, -2, 0, 2, 3, 8, 0, -3], [-4, -5, 1, 11, -9, 1, 10, 5, 11, 2, 2, -6], [-17, -5, 2, -14, -14, 11, -6, -11, 10, 1, 0, -2], [13, -2, 1, 11, 17, -13, 11, 12, -10, 0, 0, 12], [-2, 4, -9, -5, -13, 9, -7, 3, -2, -2, 2, -6], [14, -4, 1, -2, 7, 2, 2, 1, -1, 0, 1, -1], [2, -9, 6, -6, 11, 6, 6, -9, 12, 0, 0, 3], [-9, 8, -4, -11, -4, 7, 6, 7, -21, -5, 10, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_273_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_273_2_t_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_273_2_t_c(:prec:=12)
    chi := MakeCharacter_273_t();
    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_273_2_t_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_273_2_t_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_273_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![49, 0, 262, 0, 435, 0, 312, 0, 107, 0, 17, 0, 1]>],Snew);
    return Vf;
end function;