
// Make newform 441.2.f.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_f_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_441_2_f_g();".
// 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_441_2_f_g();".
// 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 := [9, 0, -36, 0, 123, 0, -78, 0, 37, 0, -7, 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], [-7266, 0, -252, 0, 861, 0, -1369, 0, 259, 0, -49, 0], [-5076, 0, 17343, 0, -10175, 0, 5217, 0, -987, 0, 148, 0], [0, -43146, 0, 16296, 0, -1224, 0, -3661, 0, 925, 0, -175], [6678, 0, 828, 0, -2829, 0, 3884, 0, -851, 0, 161, 0], [0, -32580, 0, 68325, 0, -31749, 0, 11990, 0, -2036, 0, 269], [1554, 0, -30387, 0, 20350, 0, -10434, 0, 1974, 0, -296, 0], [630, 0, -10034, 0, 8250, 0, -4230, 0, 839, 0, -120, 0], [0, 29142, 0, -71625, 0, 48756, 0, -19204, 0, 3517, 0, -433], [0, 84969, 0, 20292, 0, -14877, 0, 16205, 0, -3182, 0, 602], [0, -46701, 0, 175683, 0, -109806, 0, 50506, 0, -9439, 0, 1321], [0, 23094, 0, -373386, 0, 240393, 0, -124762, 0, 23836, 0, -3580]];
    Rf_basisdens := [1, 4299, 4299, 12897, 4299, 12897, 4299, 1433, 12897, 12897, 12897, 12897];
    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_441_f();"
function MakeCharacter_441_f()
    N := 441;
    order := 3;
    char_gens := [344, 199];
    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_441_f_Hecke();"
function MakeCharacter_441_f_Hecke(Kf)
    N := 441;
    order := 3;
    char_gens := [344, 199];
    char_values := [[-1, 0, -1, 0, 0, 0, 0, 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+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, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, -2, 0, 0, 2, 2, 1, 1], [0, 0, 0, -2, 0, 1, 0, 0, -1, 1, 0, 1], [-1, 0, -1, 0, 0, 0, -1, 2, 0, 0, 0, 0], [0, -1, 4, 0, -2, 0, 1, -2, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0, 0, -2, -2, 0], [-2, 2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 0, 0, 0, -2, -2, -1], [0, 2, 2, 0, -1, 0, -2, -1, 0, 0, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0, -1, -2, 3, -1], [5, -3, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 1, 0, 0, 0, 1, 1, -1], [0, 0, 0, 3, 0, 0, 0, 0, 4, 2, 0, 3], [2, 0, 2, 0, 0, 0, -3, -3, 0, 0, 0, 0], [6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -1, 0, 0, 1, -1, 0, -1], [0, -3, -1, 0, -3, 0, 3, -3, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, -4, -1, -2, -2], [0, 0, 0, 0, 0, -2, 0, 0, 2, 1, 6, 2], [0, 0, 0, -3, 0, 0, 0, 0, 1, -1, 3, 0], [0, 0, 0, 3, 0, 0, 0, 0, -1, 0, -1, -3], [0, 0, 0, 0, 0, 4, 0, 0, 0, -1, -1, 2], [3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [-2, 0, -2, 0, 0, 0, 6, -2, 0, 0, 0, 0], [1, -6, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -3, 0, 0, 0, 1, 1, -1], [0, 4, 12, 0, 3, 0, -4, 3, 0, 0, 0, 0], [0, 0, 0, 3, 0, -7, 0, 0, 0, 1, 1, -2], [3, 0, 3, 0, 0, 0, 2, -3, 0, 0, 0, 0], [0, 1, -8, 0, 4, 0, -1, 4, 0, 0, 0, 0], [0, 0, 0, -7, 0, 3, 0, 0, 0, -1, -1, -2], [-2, -3, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 3, 0, 0, 0, 2, 2, 1], [0, 0, 0, 2, 0, 0, 0, 0, 3, 2, -1, 4], [11, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1, 0, 0, 1, -2, 5, 0], [0, -2, 6, 0, 3, 0, 2, 3, 0, 0, 0, 0], [-6, 0, -6, 0, 0, 0, -3, 6, 0, 0, 0, 0], [-8, -3, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 5, 1], [-11, 0, -11, 0, 0, 0, 2, -2, 0, 0, 0, 0], [0, 0, 0, -6, 0, 0, 0, 0, 3, -3, 9, -3], [0, 0, 0, -1, 0, 0, 0, 0, 1, -1, 3, -2], [0, 0, 0, -3, 0, -3, 0, 0, 0, 0, 0, -3], [-4, -4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [-7, 0, -7, 0, 0, 0, -6, -4, 0, 0, 0, 0], [0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 3], [0, 0, 0, 3, 0, -2, 0, 0, 2, 4, -9, -1], [0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 4, 2], [0, 3, 11, 0, -1, 0, -3, -1, 0, 0, 0, 0], [0, 0, 0, -6, 0, 5, 0, 0, -5, 2, -6, 1], [0, 0, 0, 6, 0, -8, 0, 0, 8, 3, 8, 2], [0, -3, 8, 0, -3, 0, 3, -3, 0, 0, 0, 0], [0, -7, -6, 0, 3, 0, 7, 3, 0, 0, 0, 0], [0, 0, 0, 9, 0, -1, 0, 0, 0, -2, -2, 4], [0, 0, 0, -5, 0, -1, 0, 0, 0, 3, 3, -3], [0, 0, 0, 3, 0, 4, 0, 0, -4, -6, -13, -7], [0, 0, 0, -8, 0, -6, 0, 0, 0, -2, -2, -7], [0, 0, 0, 3, 0, 0, 0, 0, 1, -1, 3, -6], [0, -4, 1, 0, 1, 0, 4, 1, 0, 0, 0, 0], [0, 9, 0, 0, -3, 0, -9, -3, 0, 0, 0, 0], [8, 0, 8, 0, 0, 0, 1, 8, 0, 0, 0, 0], [3, 0, 3, 0, 0, 0, -7, -9, 0, 0, 0, 0], [0, 0, 0, -4, 0, 0, 0, 0, -8, -2, -4, -2], [0, 0, 0, -1, 0, 0, 0, 0, 4, 2, 0, 7], [14, -3, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0], [-16, 0, -16, 0, 0, 0, 6, -3, 0, 0, 0, 0], [2, -11, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -7, 0, 7, 0, 0, 0, -6, -6, 0], [0, -11, 11, 0, -1, 0, 11, -1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0, -1, 2, -5, 0], [-17, 0, -17, 0, 0, 0, -6, 1, 0, 0, 0, 0], [0, 0, 0, -6, 0, 2, 0, 0, 0, -2, -2, -2], [0, 0, 0, 6, 0, -2, 0, 0, 0, 1, 1, 2], [0, -3, 13, 0, 3, 0, 3, 3, 0, 0, 0, 0], [10, -11, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, -9, 0, 0, 9, 1, 13, 2], [0, 0, 0, -3, 0, 0, 0, 0, -5, -1, -3, 0], [0, 10, 14, 0, -4, 0, -10, -4, 0, 0, 0, 0], [-12, 5, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0], [0, 1, 1, 0, -11, 0, -1, -11, 0, 0, 0, 0], [0, 0, 0, -11, 0, 0, 0, 0, -4, -5, 6, -4], [13, 0, 13, 0, 0, 0, -3, -6, 0, 0, 0, 0], [0, 0, 0, 5, 0, 1, 0, 0, -1, -6, -7, -6], [0, 0, 0, 3, 0, 0, 0, 0, -1, 0, -1, -3], [-2, -7, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0], [-4, 0, -4, 0, 0, 0, 2, 5, 0, 0, 0, 0], [6, 0, 6, 0, 0, 0, 12, -6, 0, 0, 0, 0], [0, 0, 0, -2, 0, 2, 0, 0, -2, -3, 4, 0], [0, 0, 0, 12, 0, -4, 0, 0, 0, -1, -1, 4], [0, 0, 0, 5, 0, -1, 0, 0, 1, -5, -1, -4], [0, 0, 0, -8, 0, 7, 0, 0, -7, -8, 12, 1], [-1, -12, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 3, 5, 0, 6, 0, -3, 6, 0, 0, 0, 0], [-2, -9, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 6, 0, 0, 0, -6, -6, 3], [0, 0, 1, 0, 4, 0, 0, 4, 0, 0, 0, 0], [-10, 0, -10, 0, 0, 0, -3, 0, 0, 0, 0, 0], [0, 0, 0, 15, 0, -16, 0, 0, 16, 9, 1, 1], [0, 0, 0, -4, 0, 0, 0, 0, 0, -1, 2, 1], [0, 0, 0, -6, 0, 7, 0, 0, -7, 1, -12, -1], [-6, 0, -6, 0, 0, 0, 10, 12, 0, 0, 0, 0], [0, 0, 0, -8, 0, 0, 0, 0, -4, -1, -2, 5], [0, 0, 0, 1, 0, 13, 0, 0, 0, 0, 0, 7], [-11, 6, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [31, 0, 31, 0, 0, 0, -2, 1, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, 4, -4, 12, -3], [-5, 12, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0], [0, -1, 9, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -7, 0, 5, 0, 0, 0, 3, 3, -1], [0, 0, 0, -10, 0, 11, 0, 0, -11, -5, -4, -1], [0, 0, 0, 0, 0, 0, -5, 3, 0, 0, 0, 0], [0, -17, -9, 0, -9, 0, 17, -9, 0, 0, 0, 0], [0, 0, 0, -5, 0, -1, 0, 0, 0, 5, 5, -3], [0, -2, -29, 0, -5, 0, 2, -5, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, 1, -3, 7, 0], [-18, 20, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -8, 0, 0, 0, 0, 6, 1, 4, 11], [-24, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -2, -2, 0, -2, 0, 2, -2, 0, 0, 0, 0], [0, 0, 0, 4, 0, 7, 0, 0, -7, -10, -20, -11], [0, 0, 0, 16, 0, 0, 0, 0, -3, 4, -11, -4], [0, 0, 0, 9, 0, 1, 0, 0, 0, 8, 8, 5], [20, 3, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, -3, 0, 0, 0, 0], [-5, 0, -5, 0, 0, 0, 5, -5, 0, 0, 0, 0], [6, 3, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, 2, 0, 0, 0, 6, 6, 6], [0, 0, 0, -8, 0, 12, 0, 0, 0, 1, 1, 2], [0, 0, 0, -5, 0, -3, 0, 0, 3, 7, 13, 8], [0, 0, 0, 9, 0, -13, 0, 0, 0, 4, 4, -2], [0, 0, 0, -6, 0, 8, 0, 0, 0, -7, -7, 1], [18, 7, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 2, 0, 0, -2, 1, 5, 3], [0, 21, 13, 0, 1, 0, -21, 1, 0, 0, 0, 0], [-8, 0, -8, 0, 0, 0, -6, 0, 0, 0, 0, 0], [-28, 3, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 10, 0, -3, 0, 0, 3, 1, -20, -7], [0, 0, 0, 6, 0, 0, 0, 0, -7, 3, -13, 3], [0, 0, 0, 13, 0, 0, 0, 0, 5, 8, -11, 11], [0, 0, 0, 1, 0, 0, 0, 0, -6, 4, -14, 11], [0, 0, 0, 5, 0, -3, 0, 0, 0, 2, 2, 1], [-10, 3, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0], [16, 0, 16, 0, 0, 0, -15, 3, 0, 0, 0, 0], [0, 0, 0, 11, 0, -11, 0, 0, 11, 6, -1, 0], [8, 10, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, -16, 0, 0, 0, 1, 1, -10], [0, 2, -25, 0, 2, 0, -2, 2, 0, 0, 0, 0], [0, 1, 14, 0, -10, 0, -1, -10, 0, 0, 0, 0], [17, 10, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 19, 0, 0, 0, 0, -3, 7, -17, 2], [0, 0, 0, 11, 0, 0, 0, 0, 0, -10, -13, -11], [0, 0, 0, 11, 0, -15, 0, 0, 0, 2, 2, -2], [0, 7, 29, 0, -10, 0, -7, -10, 0, 0, 0, 0], [-10, 20, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [-26, 0, -26, 0, 0, 0, 21, -9, 0, 0, 0, 0], [0, 0, 0, -3, 0, 6, 0, 0, -6, -6, -3, -3], [-11, 0, -11, 0, 0, 0, 22, -8, 0, 0, 0, 0], [0, 0, 0, -9, 0, 0, 0, 0, 8, 0, 8, 9], [-35, -1, 0, 0, -7, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, 0, 0, 0, 6, 6, -6, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_2_f_g();".
// 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_441_2_f_g(:prec:=12)
    chi := MakeCharacter_441_f();
    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_441_2_f_g();".
// 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_441_2_f_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 4, 17, -2, 5, 1, 1]>,<5,R![6561, 0, 8748, 0, 9963, 0, 2106, 0, 333, 0, 21, 0, 1]>],Snew);
    return Vf;
end function;