
// Make newform 8664.2.a.z in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8664_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8664_2_a_z();".
// 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_8664_2_a_z();".
// 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, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 2, 0], [-8, -2, 4]];
    Rf_basisdens := [1, 1, 1];
    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_8664_a();"
function MakeCharacter_8664_a()
    N := 8664;
    order := 1;
    char_gens := [2167, 4333, 5777, 8305];
    v := [1, 1, 1, 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;

function MakeCharacter_8664_a_Hecke(Kf)
    return MakeCharacter_8664_a();
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], [1, 0, 0], [0, -1, 0], [1, 1, 0], [-2, 0, -1], [-1, 0, 0], [0, 1, 1], [0, 0, 0], [0, -1, 0], [-4, -1, 1], [-5, 0, -1], [1, 1, 1], [-4, 2, 0], [3, -1, 0], [-6, 0, 0], [0, -2, 1], [2, -3, 0], [-1, -2, 0], [1, 0, -1], [-2, 3, 1], [-3, 2, 0], [-9, -2, 1], [-4, 3, -1], [-6, 0, -1], [-2, 3, 1], [2, -1, -1], [3, 1, 0], [-2, 1, 1], [-6, 1, 1], [2, 2, -3], [-8, 2, 0], [-2, -1, 1], [-2, -5, 1], [1, 1, -2], [10, -1, 0], [4, 0, -2], [7, -3, -1], [-9, 0, -1], [-2, 1, -2], [4, 1, -1], [-8, 1, 0], [-14, 1, -1], [-8, -2, -3], [-11, -3, -1], [10, 4, -1], [-11, 6, 1], [3, 1, -2], [-9, -2, -5], [0, -6, 1], [9, 2, -2], [6, -1, -1], [-4, 0, 3], [-1, -4, 2], [6, -2, 0], [-8, 6, -1], [-18, -1, 1], [-6, -4, -1], [16, -4, 0], [14, 1, -1], [0, -1, -2], [4, -4, 0], [-14, 4, 0], [-4, -1, -3], [-2, -5, 2], [14, 5, 1], [-12, -3, 2], [5, 1, 0], [3, 3, 3], [16, 4, -3], [7, -8, 0], [-20, -1, -2], [20, -4, 0], [7, 0, 5], [-10, -5, -3], [1, 3, 0], [-14, -2, 3], [4, 0, 1], [-5, -4, 4], [-6, -6, 3], [-10, -1, 3], [-6, 0, 5], [-2, -3, 5], [2, -2, 2], [-15, -5, 3], [5, 5, 2], [-4, -3, -1], [0, 1, -1], [13, 5, -3], [2, -2, 3], [-1, -3, -4], [8, 2, 0], [2, -3, -1], [-16, -6, 0], [-6, 3, 1], [33, 0, -1], [-4, -1, 1], [-10, 2, 2], [-26, -3, 0], [15, -2, -3], [-11, 1, -5], [-23, -3, -4], [6, 5, 3], [10, -1, -1], [16, 2, 4], [13, -8, -3], [18, -5, -1], [10, -3, -2], [-2, 6, 1], [-2, -8, -1], [-3, -5, -1], [-5, 0, 5], [10, 5, -5], [14, 5, 4], [5, -2, 3], [1, -9, -2], [-2, 2, 0], [-15, -6, 1], [-20, 5, -1], [2, 5, 5], [-26, -8, 3], [-26, 0, -4], [9, -5, 3], [-10, -6, 6], [-12, 4, -3], [4, 8, 4], [-12, 9, 4], [-11, 8, 2], [-4, 7, 4], [9, -6, 3], [22, -13, -3], [15, 4, -3], [2, 6, -7], [19, 1, 0], [-15, 4, 2], [10, 8, -5], [27, -2, 2], [-22, 2, -4], [19, 2, 1], [26, -3, -3], [-16, 9, 2], [20, 0, 2], [-20, 1, -1], [-20, 2, -4], [0, 5, 3], [11, 6, 4], [-4, 2, -5], [-25, -3, -3], [-10, -4, -4], [19, -8, -3], [-2, -10, -2], [-3, -3, 1], [28, -1, 2], [11, -3, 2], [2, 7, 0], [4, -4, -4], [-20, 2, -4], [7, -9, 2], [12, 7, 6], [15, -9, -3], [0, -6, 2], [-26, 1, -5], [22, -5, 4], [3, -14, 1], [-22, 14, 2], [2, -10, -4], [6, 6, -1], [-31, 12, 3], [-5, -15, 3], [-42, 2, -2], [-20, 6, -4], [14, 5, -6], [-35, -2, -4], [-30, 7, -1], [13, 3, -5], [11, 7, 6], [-10, 6, 6], [7, -1, 10], [18, 12, -4], [-1, 3, 4], [11, 3, 9], [20, 3, -3], [-36, -1, 3], [23, 11, -1], [24, 6, 1], [-10, 11, 7], [-18, -6, -6], [-34, -3, 5], [17, 2, 1], [10, -1, -1], [10, -2, -3], [-1, -12, 0], [0, -6, 0], [5, 9, 2], [30, 4, 2], [-4, 11, 0], [44, -2, 4], [-10, 1, -5], [21, -1, 5], [-8, 5, 3], [14, 2, -4], [22, 9, 4], [-24, 5, 3], [19, 12, -6], [35, 9, -3], [-12, 9, 2], [-16, 2, -8], [8, -6, 0], [8, 8, 6], [12, -2, 2], [-25, -6, 1], [-1, -1, 7], [-10, -1, -1], [-9, -10, 3], [34, -6, -3], [0, -7, 2], [-27, 2, 4], [-20, -4, 2], [36, -2, 1], [-48, 3, 1], [-34, 2, 3], [47, 4, 2], [37, 1, 4], [28, 4, -5], [-7, 15, 6], [-4, -4, 2], [35, 6, 2], [18, 6, 2], [-14, -11, 4], [4, 10, -4], [0, 8, 2], [17, -1, 3], [9, 10, -3], [16, 9, -5], [0, 1, -1], [8, 14, 0], [-26, 2, -9], [13, 11, 3], [-16, 10, 2], [-20, -4, 2], [-20, 14, 1], [20, -10, -3], [36, -2, -4], [-19, -8, 5], [-15, -5, -5], [6, -9, -6], [-44, -1, 3], [0, 8, -8], [42, 3, -2], [3, -20, -1], [-14, 8, -7], [5, 11, -5], [-24, -3, 2], [36, -11, 1], [14, -15, -5], [-14, 1, -6], [8, 11, 5], [-19, -6, -8], [3, -14, -3], [-16, -10, 2], [27, -7, -1], [19, -1, -2], [18, -3, -2], [-23, -6, 0], [17, -1, -1], [14, 4, -3], [53, -2, 1], [-20, -3, -3], [-44, -4, 6], [7, -18, 1], [-64, 0, 2], [17, 6, 4], [-8, 9, -5], [-6, -10, -2], [-16, 4, -4], [31, -7, -5], [29, 12, 3], [-26, -2, 0], [1, 0, 6], [21, -2, -4], [54, 2, 2], [12, 1, 0], [-12, -16, 2], [4, -18, -5], [-23, 9, 5], [-24, 1, 7], [16, 9, 3], [-3, 22, 2], [-36, -3, 4], [15, 10, -7], [6, 17, -3], [30, 2, 8], [-36, -6, 3], [-50, -1, -3], [2, -10, 6], [5, -13, 9], [30, 14, -3], [36, -10, -4], [26, -4, -5], [-10, 14, -3], [11, 6, 1], [46, -7, 3], [14, -8, -8], [-53, -7, 2], [-2, 2, 12], [1, -5, -10], [11, -2, 4], [22, 10, 8], [-21, 14, -2], [-4, 0, 11], [-1, -3, -1], [-34, -7, 5], [2, 2, 3], [-18, -1, -5], [-25, -3, 0], [-8, -15, -4], [-66, -1, -1], [42, -3, 2], [-14, -8, -12], [25, 2, -2], [34, 10, 3], [19, -19, 0], [27, -13, -3], [28, 0, -6], [55, 8, 1], [-16, 25, 4], [-5, 16, 2], [-9, -6, -7], [3, 15, 12], [6, 7, 4], [12, 0, 4], [-3, 5, 3], [8, 14, -4], [-32, -2, -8], [-4, -13, -10], [40, -2, 4], [24, 9, -4], [-3, -17, -1], [-26, -14, 0], [-43, 3, 3], [-35, -5, -8], [-26, 11, 1], [72, 3, 2], [18, 6, -9], [-32, -8, -6], [18, -2, 9], [-28, 18, 1], [-2, 13, -1], [-4, -6, 6], [-40, -10, 1], [0, 4, 7], [11, 5, 0], [-21, 8, 2], [-40, -11, 3], [0, -18, 2], [22, 12, -8], [-18, -16, 6], [-42, 8, 1], [-32, -1, -2], [34, 6, -5], [4, -5, 5], [-14, -7, -15], [40, -8, -3], [34, 13, -4], [-42, -1, -12], [12, 11, 1], [10, -11, -3], [-6, 18, 11], [11, 15, -6], [21, 4, 10], [-16, -6, -1], [-13, -7, -8], [-4, -15, 1], [-34, 12, 2], [27, 8, 5], [17, 15, 3], [-28, 5, -2], [26, 8, 7], [22, 3, -3], [8, 8, 5], [23, 7, -3], [-40, 2, -1], [-34, -6, 4], [-8, -11, -1], [-14, -17, -1], [25, 5, -6], [46, 1, 11], [-9, 12, 1], [14, 9, 11], [-12, -4, 0], [22, 7, 3], [7, 2, -8], [-6, -14, 4], [-10, -10, 12], [33, -15, -4], [44, -1, -6], [14, 15, -7], [15, 0, -7], [32, 5, 0], [-25, 10, 7], [-8, 9, 5], [-55, -5, -1], [48, -6, -5], [-64, -3, 3], [-50, 6, -8], [-2, 5, 1], [7, 3, -10], [-47, 12, 2], [12, 6, 13], [36, -4, -8], [2, -1, -4], [1, 3, -7], [-12, 17, 4], [-8, 15, 6], [27, -1, 8], [22, 7, 11], [42, -1, 2], [-44, 1, 0], [16, 6, -4], [26, 8, 1], [-32, -3, -4], [-8, -3, 8], [-51, 2, 4], [-54, 1, 2], [30, 5, -6], [-47, -10, -8], [10, 15, 2], [14, -12, -7], [58, -14, 0], [29, -25, 0], [-38, 6, 8], [-9, -8, -12], [2, 5, -7], [20, -12, 8], [10, 8, 2], [-22, -3, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8664_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8664_2_a_z();".
// 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_8664_2_a_z(:prec:=3)
    chi := MakeCharacter_8664_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(3037) - 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_8664_2_a_z();".
// 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_8664_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8664_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, -12, 0, 1]>,<7,R![3, -9, -3, 1]>,<13,R![1, 1]>,<29,R![-192, 0, 12, 1]>],Snew);
    return Vf;
end function;