
// Make newform 1008.2.t.g in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1008_t();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1008_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_1008_2_t_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_1008_2_t_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 := [3, -12, 19, -15, 10, -3, 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], [6, 0, 1, 5, -1, 1], [0, -3, 2, -5, 1, -1], [6, -21, 19, -16, 5, -2], [-9, 33, -22, 19, -5, 2]];
    Rf_basisdens := [1, 1, 3, 3, 3, 3];
    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_1008_t();"
function MakeCharacter_1008_t()
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    v := [3, 3, 1, 2];
    // 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_1008_t_Hecke();"
function MakeCharacter_1008_t_Hecke(Kf)
    N := 1008;
    order := 3;
    char_gens := [127, 757, 785, 577];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, -1, 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, 0, 0], [-1, 0, 0, 0, 0, 1], [2, 0, 1, 1, 0, 0], [0, 0, 0, 1, -1, 1], [-2, 2, 1, -3, 2, 0], [0, 1, 0, 0, 1, -1], [-2, 0, 0, -2, 2, 4], [1, -1, 0, -1, 1, 0], [-2, -1, -2, 0, -1, 0], [-1, 3, 2, 3, -1, -2], [5, 2, 1, 2, 5, -1], [-3, 2, 0, 2, -3, 0], [-1, 2, 0, -1, 5, 0], [-5, 2, 3, 2, -5, -3], [1, 1, 0, 1, 1, -3], [-1, 5, 0, -1, -1, -3], [0, 5, 4, 5, 0, -4], [3, -8, 0, 3, -5, 2], [-3, -1, -4, -1, -3, 4], [-4, 3, 4, -2, 3, 0], [3, -4, 0, 3, 6, -2], [-4, 1, 0, -4, 4, 7], [-3, 1, -1, 1, -3, 1], [-4, -1, -3, -1, -4, 3], [-10, 0, -2, 0, -10, 2], [0, 4, 1, -7, 4, 0], [1, -3, 0, 6, -3, 0], [6, 0, 3, 0, 6, -3], [1, -3, 0, 1, 1, 1], [3, 1, 0, 3, -1, -7], [9, -5, -1, 9, -5, 0], [0, -4, 2, 10, -4, 0], [-6, 1, 1, -1, 1, 0], [1, -4, 0, 1, -11, 2], [3, -3, 0, 6, -3, 0], [14, -1, 4, 6, -1, 0], [-8, 6, 9, 6, -8, -9], [7, -7, -9, -7, 7, 9], [1, 6, 0, 1, 5, -8], [3, 1, 0, 3, 8, -7], [-3, 6, 0, -3, 3, 0], [12, 5, 4, -6, 5, 0], [-2, 7, 0, -2, 13, -3], [9, 5, 3, 5, 9, -3], [11, -2, -4, 0, -2, 0], [1, -2, 0, 1, -7, 0], [-4, 8, 0, -4, 15, 0], [7, 2, 0, 2, 7, 0], [-13, 5, -3, -13, 5, 0], [2, 2, 8, 4, 2, 0], [9, -1, 4, -1, 9, -4], [4, 4, 0, 4, -5, -12], [3, -8, -12, 4, -8, 0], [19, -2, -3, 1, -2, 0], [4, -8, -3, 13, -8, 0], [14, -4, -3, 5, -4, 0], [-10, 12, 0, -10, 4, 8], [0, 9, 4, 9, 0, -4], [-4, 0, -3, -3, 0, 0], [0, 8, 7, 8, 0, -7], [7, -3, 2, -3, 7, -2], [2, -10, 0, 2, -13, 6], [-2, 1, 4, 2, 1, 0], [-3, -7, -5, -7, -3, 5], [-2, -6, 0, -2, -8, 10], [-9, 15, 0, -9, 15, 3], [-3, 9, 0, -3, -8, -3], [2, 6, 0, 2, 15, -10], [-16, -1, -9, -1, -16, 9], [0, 1, -8, 1, 0, 8], [0, 3, -6, -12, 3, 0], [-4, -2, 7, -2, -4, -7], [-30, 2, 3, -1, 2, 0], [18, 3, 5, -1, 3, 0], [-9, 12, 7, -17, 12, 0], [14, -3, -5, 1, -3, 0], [-3, -8, -5, 11, -8, 0], [-14, -2, -4, -2, -14, 4], [-1, 5, -6, -16, 5, 0], [-10, -5, -3, -5, -10, 3], [4, -14, 0, 4, -2, 6], [14, 8, 14, 8, 14, -14], [2, -10, 0, 2, -1, 6], [12, -8, -3, 13, -8, 0], [-6, 11, 0, -6, 17, 1], [0, 1, 0, 0, 14, -1], [-4, 6, 19, 7, 6, 0], [-1, 10, 0, -1, 20, -8], [-3, 4, -1, 4, -3, 1], [-1, -12, 0, -1, 8, 14], [-12, -4, -11, -4, -12, 11], [21, -5, -8, 2, -5, 0], [-3, -6, 0, -3, -5, 12], [13, -8, 0, 13, -8, -18], [17, -7, -5, 9, -7, 0], [-16, 8, -3, -19, 8, 0], [7, -2, 12, 16, -2, 0], [15, -15, 0, 15, 3, -15], [-10, -9, -3, -9, -10, 3], [-16, -7, -16, -7, -16, 16], [-12, -15, -17, -15, -12, 17], [7, -22, 0, 7, -6, 8], [-11, -10, -13, -10, -11, 13], [-6, 0, 0, -6, 9, 12], [-9, -9, -11, -9, -9, 11], [-6, 17, 0, -6, -6, -5], [-14, 11, 11, 11, -14, -11], [21, -6, -12, -6, 21, 12], [29, 5, 9, 5, 29, -9], [-8, -6, -3, -6, -8, 3], [-15, 1, 11, 9, 1, 0], [-1, 12, 0, -1, 6, -10], [1, -9, 0, 1, -1, 7], [3, 3, 7, 1, 3, 0], [15, 5, 15, 5, 5, 0], [-6, -1, -13, -11, -1, 0], [3, 6, 0, 3, -11, -12], [3, -3, 0, 3, 30, -3], [23, -1, 9, 11, -1, 0], [-9, -2, -13, -2, -9, 13], [2, 3, -5, 3, 2, 5], [-9, -4, 0, -4, -9, 0], [3, 6, 0, 3, -12, -12], [-5, -2, 0, -5, 16, 12], [-3, -6, 0, -3, -2, 12], [-29, -1, -8, -6, -1, 0], [-5, 1, 0, -5, -3, 9], [-17, 1, 3, 1, -17, -3], [-14, -18, -19, -18, -14, 19], [-20, 3, -11, -17, 3, 0], [6, 11, 0, 6, 2, -23], [0, 6, 0, 0, -9, -6], [-38, 2, -7, -11, 2, 0], [0, 8, 25, 9, 8, 0], [-19, 4, -13, -21, 4, 0], [-2, 15, 0, -2, 10, -11], [-6, -5, 0, -6, -16, 17], [-13, 2, 7, 2, -13, -7], [5, -1, 0, 5, 11, -9], [1, 8, 0, 1, -15, -10], [17, -13, -20, 6, -13, 0], [3, -7, 0, 3, 7, 1], [8, 7, 20, 7, 8, -20], [5, -7, 0, 14, -7, 0], [13, -18, 0, 13, -11, -8], [-10, -3, 5, -3, -10, -5], [-21, -4, 3, -4, -21, -3], [5, -4, -9, -1, -4, 0], [-23, 0, -15, -15, 0, 0], [8, -11, -8, -11, 8, 8], [-15, 6, 8, -4, 6, 0], [37, -5, 3, 13, -5, 0], [0, 14, 24, -4, 14, 0], [31, 1, 3, 1, 1, 0], [9, 7, 14, 0, 7, 0], [5, 14, 12, 14, 5, -12], [-10, 4, -7, 4, -10, 7], [-5, 1, 0, -5, 16, 9], [21, -14, -21, 7, -14, 0], [10, -4, -10, -4, 10, 10], [-10, -2, 0, -10, 15, 22], [8, -8, 2, 18, -8, 0], [-10, 11, 0, -10, 9, 9], [-13, 2, 6, 2, -13, -6], [-10, -7, -3, -7, -10, 3], [14, -1, -15, -13, -1, 0], [5, -13, 0, 5, -6, 3], [-38, 8, 21, 5, 8, 0], [-9, 19, 15, -23, 19, 0], [1, -8, -18, -2, -8, 0], [4, 9, -4, -22, 9, 0], [13, -24, 0, 13, 4, -2], [9, -9, -24, -6, -9, 0], [-3, 11, -5, -27, 11, 0], [-17, 4, -11, 4, -17, 11], [-22, 30, 0, -22, -2, 14], [1, -20, -8, -20, 1, 8], [-21, 26, 0, -21, 19, 16], [0, -7, -21, -7, -7, 0], [-3, -12, 0, -3, 17, 18], [0, -13, 0, 0, -7, 13], [26, -6, -13, -6, 26, 13], [-10, 19, 0, -10, 38, 1], [-29, -1, -4, -1, -29, 4], [-48, -4, -2, -4, -48, 2], [8, -13, -15, 11, -13, 0], [-8, 8, 0, -8, -13, 8], [-2, -15, -18, 12, -15, 0], [21, -13, -8, 18, -13, 0], [0, 6, 0, 0, -3, -6], [4, 5, 25, 5, 4, -25], [0, -12, 6, -12, 0, -6], [-10, 16, 17, 16, -10, -17], [14, -2, 0, 14, -33, -26], [0, 30, 0, 0, 9, -30], [-11, -9, -20, -9, -11, 20], [-12, 8, 9, 8, -12, -9], [6, -7, 0, 6, -24, -5], [-9, -9, 3, -9, -9, -3], [-19, 5, 0, 5, -19, 0], [17, -10, 3, -10, 17, -3], [-1, -7, 1, -7, -1, -1], [6, -11, 0, 22, -11, 0], [-19, -8, 4, 20, -8, 0], [2, -1, 24, 26, -1, 0], [-10, 32, 0, -10, 29, -12], [-18, -2, -12, -2, -18, 12], [31, 4, 9, 1, 4, 0], [-5, 16, 27, 16, -5, -27], [-30, 10, -18, 10, -30, 18], [10, -6, -18, -6, 10, 18], [13, -13, 0, 13, 30, -13], [-5, 16, 10, 16, -5, -10], [-1, -2, 0, -1, -6, 4], [-34, -1, -6, -1, -34, 6], [9, -12, 0, 9, -16, -6], [-8, -15, -7, -15, -8, 7], [25, 18, 14, -22, 18, 0], [-18, -8, -19, -8, -18, 19], [-5, 15, 0, -5, -7, -5], [-2, 10, -4, -24, 10, 0], [-10, 12, 0, -10, -1, 8], [-33, 0, -12, -12, 0, 0], [18, 2, 17, 2, 18, -17], [-17, 25, 0, -17, 31, 9], [-35, 3, -3, 3, -35, 3], [-16, 16, 0, -16, 30, 16], [-9, 20, 0, -9, 37, -2], [-4, 22, 0, -4, 13, -14], [-12, 21, 0, -12, 9, 3], [23, -1, -16, -1, 23, 16], [-19, 21, 0, -19, 26, 17], [13, -31, 0, 13, -35, 5], [4, -5, -27, -5, 4, 27], [-2, 20, 18, 20, -2, -18], [23, 2, 30, 26, 2, 0], [-15, 15, 14, -16, 15, 0], [-24, 9, 21, 9, -24, -21], [-6, 9, 0, -6, -15, 3], [11, 7, -1, -15, 7, 0], [12, -2, 22, 26, -2, 0], [19, -4, -12, -4, 19, 12], [-7, 3, 21, 3, -7, -21], [30, -6, -13, -6, 30, 13], [9, -16, 0, 9, -38, -2], [-1, 14, 0, -1, 35, -12], [-13, 18, 5, -31, 18, 0], [6, -2, 2, -2, 6, -2], [8, 15, 15, 15, 8, -15], [36, -15, 15, 45, -15, 0], [-2, -30, 0, -2, -5, 34], [-5, -8, -33, -8, -5, 33], [-22, -1, -21, -1, -22, 21], [1, 5, 16, 5, 1, -16], [36, -4, 20, 28, -4, 0], [-38, 8, -13, 8, -38, 13], [-21, 33, 0, -21, 26, 9], [30, -2, 2, 6, -2, 0], [41, 6, -5, -17, 6, 0], [8, -14, -33, -14, 8, 33], [3, -24, -23, 25, -24, 0], [27, 2, 25, 2, 27, -25], [29, 24, 28, 24, 29, -28], [8, -4, 26, 34, -4, 0], [-14, 19, 9, 19, -14, -9], [-19, 21, 0, -19, 11, 17], [-2, 13, 15, -11, 13, 0], [31, 5, 20, 5, 31, -20], [10, -22, 0, 10, 16, 2], [-6, 6, 3, 6, -6, -3], [-33, 8, 1, -15, 8, 0], [-4, -13, 0, -4, 24, 21], [20, -26, -11, 41, -26, 0], [30, -1, -3, -1, -1, 0], [-4, 16, 0, -4, -34, -8], [-33, -15, -5, -15, -33, 5], [-8, 17, 10, -24, 17, 0], [-17, 18, -5, -41, 18, 0], [-16, 0, -2, 0, -16, 2], [8, 11, 0, 8, -46, -27], [-39, 9, 18, 9, -39, -18], [4, 1, -14, 1, 4, 14], [57, -6, 0, -6, 57, 0], [48, 8, -3, 8, 48, 3], [-13, 9, 35, 17, 9, 0], [-18, 16, 29, 16, -18, -29], [1, -1, 0, 1, 56, -1], [2, -23, 0, 2, -30, 19], [5, -20, -39, 1, -20, 0], [28, -8, 0, 16, -8, 0], [27, 0, 12, 12, 0, 0], [-10, 20, 0, -10, 14, 0], [21, -9, -15, 3, -9, 0], [7, 12, 0, 7, -4, -26], [-18, 2, 15, 2, -18, -15], [-1, 29, 27, 29, -1, -27], [-3, 3, 0, -3, 46, 3], [-44, -14, 0, -14, -44, 0], [-3, -3, -12, -3, -3, 12], [43, 10, 4, 10, 43, -4], [-19, -13, -34, -8, -13, 0], [-18, -6, 3, -6, -18, -3], [11, 15, 0, 11, 2, -37]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1008_t_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1008_2_t_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_1008_2_t_g(:prec:=6)
    chi := MakeCharacter_1008_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(1999) - 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_1008_2_t_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_1008_2_t_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1008_t();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 4, -5, 1]>,<11,R![33, -26, 1, 1]>],Snew);
    return Vf;
end function;