
// Make newform 494.2.g.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_494_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_494_g_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_494_2_g_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_494_2_g_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, 4, 15, 6, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-20, 4, 15, -25, 5, -1], [216, 20, 75, -46, 25, -5], [5, 315, 95, 105, -21, 20], [128, 480, 299, 160, -32, 38]];
    Rf_basisdens := [1, 1, 79, 79, 79, 79];
    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_494_g();"
function MakeCharacter_494_g()
    N := 494;
    order := 3;
    char_gens := [457, 287];
    v := [2, 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_494_g_Hecke();"
function MakeCharacter_494_g_Hecke(Kf)
    N := 494;
    order := 3;
    char_gens := [457, 287];
    char_values := [[-1, 0, 0, 0, -1, 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, -1, 0, 0, -1, 0], [0, 0, 1, 1, 0, 0], [0, -1, 1, 0, 0, 0], [-1, 0, 1, 1, -3, -1], [1, 1, 0, -1, 1, 2], [0, -2, 2, 0, 0, 0], [-1, 0, 0, 0, -1, 0], [-2, -3, 2, 2, -1, -2], [3, 2, -3, -3, -3, 3], [-4, 0, 4, 3, 0, 0], [-4, -2, 4, 4, -2, -4], [2, 1, -2, -2, 6, 2], [1, -2, 2, 0, 1, 0], [-3, 0, -3, 1, 0, 0], [4, 0, -4, -3, 0, 0], [5, 6, -6, 0, 5, 5], [-2, -3, 3, 0, -2, -1], [4, 2, -4, -4, 4, 4], [3, 0, 0, 0, 3, 0], [9, 0, -4, -8, 0, 0], [1, 0, -5, 0, 0, 0], [6, 0, -4, -9, 0, 0], [3, 3, -3, -3, -7, 3], [6, -2, 2, 0, 6, 0], [4, 6, -4, -4, 1, 4], [-2, 0, 4, -5, 0, 0], [-2, -1, 2, 2, -4, -2], [-3, 0, 6, 0, 0, 0], [-9, 4, -4, 0, -9, -2], [-5, -1, 5, 5, -9, -5], [-13, 0, 1, -1, 0, 0], [3, 7, -7, 0, 3, 6], [3, -5, 5, 0, 3, -3], [13, -3, 3, 0, 13, -2], [-11, 0, 3, 0, 0, 0], [-6, 0, 3, 3, 0, 0], [0, 0, 0, 0, 0, -1], [3, -2, -3, -3, -4, 3], [-1, 9, -9, 0, -1, 2], [7, 1, -7, -7, 11, 7], [-3, 0, -7, 3, 0, 0], [-6, -3, 3, 0, -6, -4], [9, 7, -9, -9, 6, 9], [6, 3, -6, -6, 16, 6], [-6, 8, -8, 0, -6, 11], [3, -6, -3, -3, -10, 3], [2, 4, -2, -2, -5, 2], [10, 4, -4, 0, 10, 5], [-2, 0, 9, 8, 0, 0], [-4, 0, -4, 1, 0, 0], [-10, 0, 11, 14, 0, 0], [-4, 2, -2, 0, -4, 1], [4, 3, -3, 0, 4, 2], [5, -3, -5, -5, 2, 5], [-1, 5, 1, 1, -16, -1], [0, 7, -7, 0, 0, 10], [-7, -10, 7, 7, -9, -7], [12, 1, -1, 0, 12, -4], [16, 0, -5, 3, 0, 0], [-6, 5, 6, 6, -9, -6], [-11, -6, 6, 0, -11, -15], [5, 0, 6, 2, 0, 0], [5, 0, 5, 2, 0, 0], [-15, 0, 7, 2, 0, 0], [9, 0, -3, -16, 0, 0], [-1, 2, -2, 0, -1, -5], [-1, 0, 3, 2, 0, 0], [3, -3, 3, 0, 3, -11], [-11, 6, 11, 11, -3, -11], [6, -3, -6, -6, 16, 6], [-11, 0, -1, -3, 0, 0], [-2, -6, 2, 2, -18, -2], [-16, 2, -2, 0, -16, -13], [-13, -18, 13, 13, -12, -13], [-16, -13, 13, 0, -16, 1], [20, 0, 2, -7, 0, 0], [-20, -18, 18, 0, -20, -9], [2, 4, -2, -2, 11, 2], [6, 5, -5, 0, 6, 16], [-8, 4, 8, 8, -15, -8], [16, 0, 1, -4, 0, 0], [9, 7, -9, -9, 6, 9], [21, 10, -10, 0, 21, 3], [1, 10, -1, -1, -10, 1], [-24, 0, 10, 1, 0, 0], [6, -5, 5, 0, 6, 10], [7, -9, -7, -7, -9, 7], [-5, 13, -13, 0, -5, 17], [-12, 0, 9, 8, 0, 0], [3, 0, -9, 7, 0, 0], [-2, 2, 2, 2, -12, -2], [-1, -10, 10, 0, -1, -6], [-16, -5, 16, 16, -5, -16], [1, 0, 2, 0, 0, 0], [6, 14, -14, 0, 6, -6], [-5, -17, 5, 5, -19, -5], [-7, 0, 2, 3, 0, 0], [6, -11, -6, -6, -8, 6], [21, 0, -11, 0, 0, 0], [-15, 0, -13, -12, 0, 0], [6, 3, -6, -6, 19, 6], [-9, 0, 0, 0, -9, -19], [-3, -13, 3, 3, -19, -3], [1, 0, 5, 7, 0, 0], [-9, 0, -6, 13, 0, 0], [-2, 15, 2, 2, 15, -2], [20, 0, -4, 6, 0, 0], [-21, 0, 14, 16, 0, 0], [19, -1, -19, -19, 11, 19], [24, 1, -1, 0, 24, -1], [-14, 5, 14, 14, -7, -14], [14, 7, -7, 0, 14, -6], [-34, 0, 1, 9, 0, 0], [-25, -4, 4, 0, -25, -2], [13, -5, 5, 0, 13, -4], [-31, 5, -5, 0, -31, -1], [-17, -3, 17, 17, -9, -17], [-16, -5, 16, 16, 0, -16], [-5, 7, -7, 0, -5, -7], [15, -1, -15, -15, 20, 15], [-12, -6, 12, 12, 11, -12], [5, 0, -8, -12, 0, 0], [28, 3, -3, 0, 28, -2], [6, 3, -6, -6, 10, 6], [22, 0, -14, -1, 0, 0], [-2, -8, 8, 0, -2, -11], [-10, -25, 25, 0, -10, -24], [-17, 0, -6, 7, 0, 0], [33, 0, -12, -4, 0, 0], [13, 17, -13, -13, 14, 13], [4, 8, -4, -4, 18, 4], [-10, -13, 10, 10, -29, -10], [-1, -1, 1, 1, -27, -1], [22, -7, 7, 0, 22, 8], [25, 0, -25, -25, 15, 25], [-9, -10, 10, 0, -9, 2], [0, 16, -16, 0, 0, 15], [0, -9, 9, 0, 0, 12], [-3, 0, 3, 3, 8, -3], [18, 0, -4, 1, 0, 0], [15, 14, -15, -15, 12, 15], [27, 2, -2, 0, 27, -4], [-24, 0, 3, 8, 0, 0], [1, -6, -1, -1, -2, 1], [0, 17, -17, 0, 0, 14], [14, 0, -20, -24, 0, 0], [14, 0, -18, 4, 0, 0], [-11, 0, 14, 14, 0, 0], [16, 0, 11, 5, 0, 0], [-31, -16, 16, 0, -31, -5], [9, 20, -9, -9, 14, 9], [-3, 0, 2, 9, 0, 0], [7, 7, -7, -7, 31, 7], [10, 0, -10, -10, -9, 10], [18, 0, -25, -10, 0, 0], [13, -18, 18, 0, 13, -16], [23, 17, -17, 0, 23, 15], [-11, 0, 5, 20, 0, 0], [25, 0, 4, 9, 0, 0], [-2, 18, 2, 2, -7, -2], [36, 13, -13, 0, 36, 8], [15, 0, 0, 5, 0, 0], [-20, -5, 5, 0, -20, 0], [-22, -25, 22, 22, -14, -22], [2, 0, -15, -8, 0, 0], [15, 8, -15, -15, -1, 15], [19, 2, -2, 0, 19, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_494_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_494_2_g_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_494_2_g_b(:prec:=6)
    chi := MakeCharacter_494_g();
    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_494_2_g_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_494_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_494_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, -15, 19, -4, 7, -2, 1]>,<5,R![5, -3, -2, 1]>],Snew);
    return Vf;
end function;