
// Make newform 4018.2.a.x in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4018_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_4018_2_a_x();".
// 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_4018_2_a_x();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_4018_a();"
function MakeCharacter_4018_a()
    N := 4018;
    order := 1;
    char_gens := [493, 785];
    v := [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_4018_a_Hecke(Kf)
    return MakeCharacter_4018_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 := [[1, 0], [-2, 0], [-1, -1], [0, 0], [-1, -1], [2, 2], [0, 0], [0, 2], [-2, 0], [1, -5], [2, -2], [-2, 4], [-1, 0], [2, 6], [-4, 2], [-1, 5], [-1, -7], [-5, -1], [-5, 3], [6, -2], [2, 4], [0, 0], [-3, -1], [-6, -4], [-2, 0], [-8, 0], [-2, -2], [4, -4], [-7, -5], [-6, -2], [10, 0], [-3, 7], [0, -2], [-1, -3], [-11, -5], [-14, -2], [-18, -2], [-14, -2], [-8, -4], [9, 1], [-7, 1], [-8, -4], [2, 6], [-10, 0], [-8, 6], [0, 14], [9, -11], [-8, 12], [-12, -2], [12, 4], [-22, -4], [22, 2], [14, 4], [-15, -1], [2, 12], [20, 4], [-13, 3], [-8, 0], [-4, 2], [14, 4], [1, -5], [-18, -2], [-15, 3], [-4, -2], [20, 0], [13, -5], [3, -9], [-2, -12], [27, 3], [-13, 3], [12, 10], [-4, 4], [-12, -8], [-26, -4], [4, -12], [-12, -8], [-6, -12], [16, -12], [10, 10], [22, -4], [9, -17], [3, 9], [2, -16], [-2, -16], [28, 2], [18, 2], [2, -4], [2, -8], [-13, 7], [20, -4], [-9, 17], [16, 10], [4, 4], [-22, 6], [-3, 5], [16, -8], [-16, 0], [12, -12], [9, 3], [2, -12], [-7, -7], [9, -1], [32, -2], [-6, 10], [-21, -1], [12, 8], [30, 4], [14, -12], [12, 12], [-8, -4], [-8, -16], [-24, 6], [-30, 2], [11, 13], [-6, 20], [18, 8], [12, -6], [2, 2], [-15, 3], [-5, -1], [-27, -7], [-32, 2], [-15, 5], [5, -11], [-30, -4], [10, -4], [11, 21], [4, -16], [28, 8], [-15, 1], [18, -2], [0, -16], [14, -10], [1, -21], [-16, 18], [16, -18], [-10, -18], [-13, -3], [-1, -25], [34, 12], [-3, -25], [20, -2], [10, -18], [-15, 9], [3, -21], [-4, 2], [-7, 17], [-10, 0], [-5, 17], [-12, -20], [32, 14], [-32, 6], [31, 11], [-4, -18], [14, -2], [-2, 4], [-10, 6], [-22, 8], [6, 4], [3, 15], [-12, 20], [-6, 14], [16, 12], [-8, 22], [16, 18], [34, 2], [-10, 10], [-52, 0], [-14, -24], [36, -8], [45, 1], [-49, -5], [-18, -22], [-6, -2], [20, -24], [-16, 4], [33, -7], [-10, 16], [20, 6], [-24, -20], [20, -4], [-13, -23], [-39, 3], [12, -10], [0, 32], [23, -13], [-34, 8], [37, 15], [-4, -10], [0, -4], [40, -2], [-41, -9], [-32, 12], [-33, 15], [-36, -4], [26, 16], [28, -18], [-27, 15], [12, 8], [24, 14], [0, -18], [-34, -18], [-16, 4], [-6, -32], [-6, -16], [-8, 16], [-6, 10], [-7, -19], [6, -2], [-13, 17], [-18, -20], [-24, -8], [64, 0], [-27, 23], [60, -4], [20, -30], [16, -4], [6, -28], [-10, 22], [-48, 10], [31, -7], [-12, 12], [0, -38], [-26, 2], [-53, 5], [7, 1], [0, -14], [28, 12], [20, -6], [42, 2], [-26, 12], [-34, 16], [-30, -12], [6, 26], [-4, 38], [-4, 4], [-16, -4], [17, -1], [-2, -18], [-4, 28], [33, 17], [20, -2], [-40, -18], [56, -6], [-8, -26], [-34, -6], [-8, 16], [10, -16], [36, 16], [-42, 0], [8, 26], [10, -12], [-38, -8], [-54, -12], [12, 24], [8, -4], [-51, 15], [-52, 10], [30, -18], [-22, -8], [4, -36], [-57, 11], [8, 36], [52, 4], [-4, -42], [-27, 3], [-9, -27], [-44, 10], [34, 18], [3, 21], [34, 18], [-16, -4], [24, 10], [22, -12], [-16, 12], [40, 24], [9, -27], [17, 15], [-32, -2], [12, -26], [-4, 38], [-8, 4], [18, 22], [-59, 13], [-4, 6], [-4, 20], [-32, -18], [6, 16], [-24, -20], [4, 32], [1, 31], [-3, 15], [12, -2], [-49, 9], [35, -23], [-20, 20], [32, -18], [27, -17], [-42, -8], [-26, -28], [24, -18], [-21, 13], [-22, -12], [27, 27], [62, 6], [-6, -14], [0, -28], [-5, -21], [12, 8], [5, -5], [0, 40], [-15, -13], [6, -14], [18, -10], [-64, 0], [50, 4], [-21, 17], [-6, 36], [26, -28], [30, 8], [39, 21], [-30, 10], [37, 13], [-18, -14], [22, 14], [36, 28], [-23, -3], [48, -2], [-6, 18], [10, -4], [-13, -7], [-30, 16], [22, 30], [-32, -6], [18, -26], [46, -12], [-57, 9], [30, 0], [-16, 4], [50, -6], [11, 9], [-26, -16], [24, -32], [-38, 18], [40, -18], [-64, -12], [-1, -21], [-36, -28], [-20, 28], [39, -9], [22, -20], [-10, 28], [-63, 7], [-26, 2], [49, 23], [-6, 4], [30, 2], [1, -29], [-14, -16], [80, -8], [-14, -40], [60, 4], [6, -14], [18, 6], [0, 22], [40, -2], [-78, -2], [50, -12], [-82, 0], [-25, -17], [34, -8], [-34, 0], [-15, 39], [16, 20], [65, -17], [-13, -27], [-48, 24], [56, -22], [-26, 28], [-34, 12], [-18, 46], [66, 0], [-26, 8], [-30, -8], [-40, -2], [-12, 0], [38, 22], [-50, 18], [-48, 28], [-24, -2], [12, 30], [-18, -26], [21, -31], [-15, 7], [56, 0], [36, 30], [0, 36], [42, 20], [-62, 14], [57, -13], [15, 23], [12, 2], [26, -16], [24, 44], [5, -7], [-4, -22], [30, -12], [38, -8], [-16, -2], [-3, -5], [26, -40], [-12, -22], [25, 41], [13, 5], [-18, 44], [45, -19], [-24, 24], [40, -12], [-62, 8], [-56, 0], [-18, 48], [22, -34], [-66, 4], [30, 32], [-2, 4], [-1, 11], [-39, 25], [-2, 12], [-10, 24], [0, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4018_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4018_2_a_x();".
// 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_4018_2_a_x(:prec:=2)
    chi := MakeCharacter_4018_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(2999) - 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_4018_2_a_x();".
// 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_4018_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-2, 2, 1]>,<11,R![-2, 2, 1]>],Snew);
    return Vf;
end function;