
// Make newform 4018.2.a.bm in Magma, downloaded from the LMFDB on 20 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_bm();".
// 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_bm();".
// 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 := [8, 5, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, 0, 1, 0], [0, -5, 0, 1]];
    Rf_basisdens := [1, 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_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, 0, 0], [1, 0, 1, 0], [1, -1, 0, 0], [0, 0, 0, 0], [0, 1, 1, 0], [2, 0, 0, 1], [-1, 2, -1, 0], [4, 0, 0, -1], [0, -2, 0, -1], [-1, -1, 0, 1], [1, 0, -1, -1], [-4, 0, -2, 0], [1, 0, 0, 0], [1, -2, -1, -1], [0, -2, -2, -1], [3, 3, 2, -1], [-2, 3, -1, 1], [3, 3, 0, 0], [2, -1, 3, 0], [3, 0, -1, 1], [0, 0, 0, 2], [3, 2, -3, -1], [4, -3, -1, -1], [9, 0, -1, -1], [-3, 2, 1, 3], [6, -4, 0, 1], [5, 0, 1, -3], [-7, 4, -3, -3], [-6, -1, -3, 2], [1, 2, -5, 0], [-4, 2, 0, -3], [2, -3, -1, -3], [0, 2, -4, 0], [2, 3, 1, -3], [7, -5, 0, -1], [1, -6, 1, -3], [0, 6, -4, -1], [-4, 2, -2, 0], [-4, 6, -2, 2], [17, 1, 0, 2], [2, 5, -3, 4], [-8, -4, 6, 1], [-5, 4, -1, 5], [6, 2, 0, 4], [0, -2, 0, 0], [-4, -2, -6, -1], [-12, 3, -5, -2], [-15, -2, -1, 1], [-3, -8, -3, 0], [6, -4, 4, 1], [4, -4, 6, 0], [0, -8, -2, 4], [8, 4, -2, 0], [-2, 5, 1, 5], [11, 4, 1, -3], [0, -4, 4, -2], [-2, -3, -3, -1], [4, 8, -6, 2], [-6, 2, -10, 2], [-6, 4, -8, -2], [10, -3, 7, 5], [6, -10, 2, 5], [-12, 1, 1, 3], [10, -2, 0, -1], [10, -4, 0, 1], [6, 1, 1, 0], [6, -5, -5, -6], [1, 6, 1, -1], [-12, 9, -3, -4], [-10, 5, -3, -5], [8, -6, 0, 0], [-6, 6, 0, 8], [3, 2, 3, 1], [0, -6, 4, -6], [-13, 2, 1, 1], [-12, -2, 2, 2], [-10, 0, 4, 0], [10, 0, 4, -9], [7, 2, 9, 0], [0, -4, 2, -6], [10, 1, -9, -3], [5, -11, 0, 5], [8, 2, 0, 1], [-8, -4, 6, 4], [-6, 2, 8, 1], [13, -6, 11, 5], [-5, 0, -3, 1], [4, -2, 2, 0], [1, -5, 8, -4], [8, -4, 0, -6], [-14, 3, -13, -1], [-4, -2, 2, -7], [18, -6, 4, 0], [-21, 2, 1, -3], [2, 1, -3, -2], [16, 4, 8, -4], [-16, 4, -6, 3], [-2, -4, 4, -5], [8, 5, 3, 1], [-2, -12, 0, 0], [2, -9, 9, 0], [-11, 1, -2, 5], [12, 2, 2, 1], [-13, -2, 5, -4], [12, -1, -3, -2], [6, 0, 4, -3], [23, -6, 1, 4], [-1, 6, -1, -1], [-10, 6, 12, 4], [-1, -6, 11, -2], [-7, -6, -7, 7], [-10, 0, -4, 0], [-2, 16, 2, -1], [-22, -1, -5, 5], [0, 0, -6, 5], [8, -10, 2, -6], [5, -4, -3, 6], [-3, -4, -11, 3], [-13, 7, -12, -5], [8, 7, 1, -8], [26, -9, 3, 3], [0, -10, 0, 8], [22, -5, 9, 1], [-8, 13, 3, -4], [-7, -10, -1, -4], [-28, 6, -8, -4], [-5, -1, 10, 5], [-14, -2, 8, 4], [-18, 6, -4, 0], [3, -7, -4, 0], [25, 2, 15, -5], [24, -8, 4, 4], [-16, 0, -6, -4], [19, -5, 16, 3], [0, -6, 6, 2], [-22, 10, 0, 8], [4, 4, -2, 7], [-4, -9, 13, 3], [15, -1, -4, -4], [24, -2, 14, -2], [-10, 5, -17, -5], [6, 4, -4, 6], [-5, 4, 7, 1], [-18, -3, 9, -2], [-11, -1, -12, -2], [6, 18, -4, 1], [-11, 9, 0, -2], [18, -4, 16, -2], [-10, 7, 9, 1], [2, 14, 0, -4], [-20, 6, -4, 8], [10, -2, -2, -10], [10, 13, -5, 4], [4, -6, 10, 5], [-23, -2, 3, 1], [-26, -2, 2, 7], [9, 0, 1, 7], [-30, 6, -6, -10], [1, -6, 5, -11], [30, 1, 1, 3], [16, -4, -8, -6], [-2, -10, 4, -1], [11, 8, -3, 1], [1, 2, 11, 6], [-6, 22, -2, -2], [9, -12, 15, 3], [16, -6, 2, -4], [-12, 10, -8, 5], [1, -2, -11, -9], [0, -8, 6, -3], [8, -13, 7, 0], [25, -1, 12, 4], [12, -10, 6, -10], [-13, 8, 11, 8], [-34, -6, 4, -4], [27, -2, -1, 2], [8, -3, -9, 0], [-4, -6, -4, 8], [-18, 6, 0, -1], [-12, 6, -12, 5], [16, 10, -2, -2], [24, -5, -1, 1], [-6, -1, -3, 6], [12, -22, -4, 2], [-4, 12, 4, 0], [13, 7, -4, 2], [6, 16, -16, 0], [-18, -15, 11, -5], [0, -2, 12, -4], [8, -14, 2, -6], [16, 2, 12, -8], [-2, -1, -13, 2], [-8, -18, -2, 2], [-17, 15, -12, -6], [-9, 0, -5, 3], [10, 4, -10, 0], [-14, -8, 2, 4], [10, 15, -7, -4], [23, -14, -1, 6], [0, -10, 6, -13], [4, 2, 4, 0], [-16, -14, 6, -2], [10, 14, 6, 3], [10, -4, -8, -2], [13, 16, -11, 2], [0, 4, -18, -5], [1, -20, 1, -3], [-22, 5, 1, 8], [-18, 8, 6, 15], [2, 19, 7, -1], [6, 10, 4, 6], [0, -2, -12, -5], [0, 4, 4, 8], [4, -7, 5, -11], [10, 24, -6, -2], [-12, -18, 2, 2], [-13, -10, -11, -1], [28, 0, 2, -8], [-31, 4, -7, 15], [10, 10, 6, -4], [-29, 15, 6, 5], [25, -2, -7, -5], [32, -14, -8, -6], [16, 8, -10, -2], [48, -1, 7, -9], [11, -1, 14, -7], [22, 6, 2, 4], [32, 6, 2, 2], [-26, 10, 4, 1], [8, -16, 8, 0], [-12, 16, -6, -4], [-23, -2, -1, 4], [-2, 0, -4, -1], [-25, 8, 3, 8], [-20, 18, 2, 11], [18, 0, -10, -2], [5, -6, -3, 0], [30, -3, 1, -10], [15, -4, -9, -7], [-16, 8, 8, 8], [14, -5, 1, 4], [-13, 6, -15, -2], [-12, 10, -22, 3], [-22, 4, -16, 2], [16, -2, 16, -4], [-21, 8, 13, 7], [-16, -4, -6, 6], [35, 4, -5, 8], [-1, 10, -3, -9], [8, 16, 2, 5], [-12, -8, 14, -4], [-6, -16, -14, 0], [-18, -8, 4, 9], [6, -4, 14, 4], [26, 6, 10, 3], [11, -14, 9, -5], [5, 9, 6, 5], [36, 2, -2, 1], [30, -4, -12, -7], [35, -12, -19, -9], [-2, 12, -2, -6], [-20, 7, -11, 8], [16, -12, -2, 9], [8, -6, -4, 7], [30, 10, 4, -8], [2, 25, 3, -7], [27, 11, 6, -3], [-4, 26, 8, -2], [-12, 8, 4, 12], [-18, -9, 3, -6], [2, -2, 18, 1], [-27, 2, -5, -7], [14, 2, -6, -10], [22, 6, 18, -5], [3, -10, -1, -14], [0, -20, -6, 4], [32, -3, 11, -2], [-14, -5, 13, -4], [16, 0, 8, -14], [-24, -2, 14, 3], [18, 2, -12, -1], [-11, -10, -1, -1], [13, -16, 7, 9], [-58, 7, 5, -1], [31, 8, 3, -6], [-27, -12, -1, 3], [8, -6, 0, 4], [26, 4, 8, -12], [-8, 8, -8, 6], [23, -10, 23, -2], [25, -11, 14, -9], [20, -3, 7, -1], [-10, -20, 22, -2], [10, 15, -23, 1], [-14, 3, 11, 8], [30, -4, 4, -11], [-8, -18, 14, -1], [-46, -3, -5, -1], [15, -8, 3, 10], [-28, 12, -8, -15], [8, 10, 6, 4], [38, -9, -13, -2], [2, -20, -4, 11], [22, -15, -13, -8], [21, 16, -7, -5], [9, -20, 13, 0], [-20, 8, -4, 20], [14, 5, -1, 13], [29, 8, -9, 7], [11, 27, -12, 9], [26, -14, 4, 18], [-17, -21, 0, -1], [11, -4, 35, 4], [23, -10, 5, 5], [28, -20, -8, -4], [22, -16, -2, 12], [14, 7, 13, -7], [-36, -20, 6, 1], [-33, 4, -7, 13], [-10, 12, 16, 10], [38, -9, 11, -9], [18, 2, -8, 1], [-18, 3, -3, 19], [-29, 4, -5, -9], [-17, 2, 13, 0], [-6, 12, 12, -5], [6, -31, 1, 2], [-21, -4, -13, 12], [29, 18, 1, 5], [0, -4, 22, 10], [25, -11, 8, -5], [-8, 20, -26, 4], [-29, -16, 17, -1], [-15, 10, -13, 0], [-43, -8, -15, 5], [-20, -2, -22, 1], [3, -25, -6, 3], [18, -4, -2, -16], [31, -10, 11, 2], [0, -22, -2, -6], [31, -17, 2, -15], [35, -8, 9, -15], [14, -2, -14, -3], [71, -8, -9, -5], [-42, -18, -2, 2], [-1, 4, 3, -7], [-6, 21, -17, -7], [4, 2, -6, -12], [2, -4, 2, -6], [38, 17, -13, 1], [-25, 4, 11, -8], [35, 10, 3, -3], [37, -15, 14, 1], [-16, 10, 10, 8], [-33, -1, 16, 1], [-22, -10, 10, 4], [-23, 12, -3, 13], [20, 17, 13, -5], [-34, -4, 12, 0], [22, -24, 18, 10], [-4, 2, 8, 10], [-23, -6, -7, -2], [8, 16, -10, 2], [-11, 22, 7, -9], [-7, 2, -5, 10], [12, -8, 12, 0], [38, -4, 20, -13], [-30, -2, -18, 17], [19, -16, 25, 5], [14, -1, -17, 4], [25, -26, -5, -6], [34, -16, 8, -5], [-42, 11, 13, 10], [0, 8, -2, -14], [13, -7, 2, -5], [20, -5, 11, -19], [-40, 10, -14, 2], [30, -2, 34, 0], [10, 6, 6, -10], [-5, 4, -19, -1], [12, 14, -12, -13], [-29, -10, -5, 1], [-20, -12, 2, 9], [47, -2, -9, 1], [23, -14, -7, 12], [1, -14, 27, 3], [1, 24, -3, 5], [-32, 8, -2, 1], [-53, -8, 7, 1], [6, 6, -8, 5], [32, 20, -4, -6], [-40, -2, -8, 3], [12, -9, 11, 2], [30, -3, 1, -3], [-16, 30, -6, -6], [12, 6, -20, -10], [-24, -6, 2, -14], [-8, 4, -14, -2], [43, -10, 9, 5], [-19, 25, -2, 7], [-39, 11, -8, 8], [22, -26, 14, 6], [-24, -20, 18, 3], [26, -4, -8, -3], [-29, 17, 4, 6], [-36, 6, 10, 15], [44, -30, -12, -2], [40, -22, 6, 8], [-59, -10, 7, -2], [24, -3, 11, -13], [-34, -20, 2, -4], [10, 0, 0, 16], [-6, -13, 9, -16], [2, 21, 9, 2], [26, 8, -12, 0], [-15, 17, -24, -10], [26, 4, 6, -14], [10, 4, 10, -20], [19, 28, 1, -1], [-7, 2, -9, 3], [-6, -30, 2, 6], [24, -8, -6, -1], [14, -12, -4, -7], [28, 32, 0, -5], [26, 12, 16, -2], [-30, 5, -31, -1], [-20, 11, -25, -10], [-34, 4, -16, -10], [-36, -14, -18, 11], [18, -6, 4, 15]];
    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_bm();".
// 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_bm(:prec:=4)
    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_bm();".
// 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_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 16, -6, -3, 1]>,<5,R![6, 8, -4, -3, 1]>,<11,R![8, -10, -20, 0, 1]>],Snew);
    return Vf;
end function;