
// Make newform 4018.2.a.bi in Magma, downloaded from the LMFDB on 23 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_bi();".
// 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_bi();".
// 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 := [12, 3, -9, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-5, -2, 1, 0], [6, -5, -3, 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, 0, 0], [-1, 0, 0, 1], [0, 0, 0, 0], [0, 1, 1, 0], [1, 0, 1, -1], [2, -2, 0, -1], [-2, 2, -1, 0], [-2, -1, 0, 1], [-5, 0, -1, 0], [3, 2, -1, 1], [2, -3, 0, 0], [1, 0, 0, 0], [1, 0, -1, -1], [1, -1, 1, 0], [2, -1, 0, 0], [-1, 3, -1, -1], [3, -2, 2, -3], [3, 1, 1, -1], [-3, 3, 2, 1], [7, -2, 1, 0], [-6, 0, 1, -2], [6, -5, 0, -2], [-3, 2, -1, 0], [-1, -1, 1, -3], [6, -2, 0, 1], [-3, 3, 3, -2], [-9, 3, -1, 1], [-7, 0, 1, 2], [-7, -1, 0, 0], [-8, 1, 0, -1], [-3, -1, -3, 2], [-3, 4, -3, 1], [-4, 1, -4, 1], [-3, -3, -3, -3], [-2, -4, 0, -3], [-6, 1, 2, 0], [0, -4, 0, -5], [-14, 0, 0, -1], [-3, -2, -6, 4], [-3, 2, 0, -1], [4, 0, 0, 3], [1, -1, -1, -1], [-4, 2, -2, 4], [-10, 3, -3, -2], [-4, 1, 2, 1], [6, 1, 5, -5], [2, 0, -2, -2], [-7, 5, 4, -1], [10, -4, 4, 1], [-10, -5, 2, -3], [0, 1, -3, -2], [2, 0, 0, 2], [4, 0, -4, 3], [-6, 8, -2, -2], [3, 2, -1, 0], [-12, -1, -3, 5], [-13, -1, -1, 2], [-12, 2, -3, 7], [2, -1, 2, 4], [11, -9, -1, -3], [3, 7, 1, -3], [-7, 8, 3, 1], [7, 2, -3, 1], [-1, 1, -1, 4], [-1, 4, -1, 0], [-2, 5, -4, 7], [-2, -4, 5, -5], [-2, -6, 1, 4], [11, -9, -1, -3], [-13, -2, 2, -3], [-9, 2, -3, 2], [-9, -5, 1, -2], [4, -7, 1, 2], [-1, 2, -7, 6], [-6, -2, 1, 6], [-12, 0, -3, -3], [0, 1, -2, 4], [-12, 0, -1, 7], [-11, -1, -8, 1], [9, -4, 3, -7], [12, -5, 0, -3], [-9, -5, 3, -2], [-8, -6, -3, 3], [-13, 5, -4, 2], [-9, -4, -3, -1], [-12, 2, 3, 2], [17, 0, 3, -3], [-23, 2, -6, -2], [2, 6, -1, -5], [4, 1, 6, -3], [1, -4, -4, 2], [-25, 7, 3, 1], [-11, 9, 3, 2], [17, -11, 3, 1], [13, 2, 8, 2], [6, -1, -2, 4], [-3, -4, -5, 4], [17, -2, -1, 1], [17, 0, -3, -3], [6, -3, 0, -7], [8, -7, 4, -7], [-6, 2, 1, 1], [0, 9, -6, 6], [-14, 1, -1, 3], [-10, 5, 0, 10], [16, -6, 3, -10], [12, -9, 0, -6], [-18, 1, 0, 7], [-2, 6, -2, -1], [7, 6, 1, 8], [-1, 1, -4, 8], [10, -9, -3, -4], [-29, 0, -1, 7], [21, -3, 1, 1], [2, -4, -2, -7], [17, -2, 6, 0], [-10, 6, 0, 4], [18, -3, 6, 0], [1, 2, 5, 2], [20, 2, 5, 5], [-1, 6, -1, 1], [10, -7, -1, -7], [-17, 4, -3, 9], [-7, 9, 0, -4], [-9, 0, 2, 4], [3, 4, 1, -7], [-7, 11, 2, 4], [-18, 1, -4, 9], [24, 3, -2, 6], [-4, 2, 2, 6], [-18, -2, -8, -3], [-2, 11, 3, 0], [15, 3, 3, 8], [-18, -5, 4, 0], [17, 7, 0, 0], [-11, 2, -1, -4], [-1, -3, 1, -5], [8, -7, -4, 7], [5, 3, -1, 5], [5, 9, 1, 7], [-12, -12, 3, 0], [6, -7, 5, -7], [-22, -6, 2, -4], [-19, -3, -5, -2], [11, -3, -2, -6], [0, 5, -1, -5], [4, 13, -1, -5], [-35, 1, -7, -1], [2, -2, 6, -7], [38, -3, -1, -2], [7, -2, -4, -2], [-3, 1, -1, 7], [-8, -1, -5, 9], [-7, -1, -5, 10], [-9, 12, -5, 2], [6, -4, 1, 0], [19, -8, -3, -6], [-19, 0, 5, 0], [1, -7, -6, -2], [-6, -7, 0, -7], [-4, 4, 12, -7], [2, 8, -3, -4], [-26, 11, -4, -1], [17, -10, 1, -13], [25, -10, 5, 2], [1, 4, -8, 1], [-7, -1, -3, 1], [31, 5, 3, -3], [-16, 2, -6, 0], [6, -9, 7, -4], [-1, -10, -3, -8], [-9, 4, -2, 3], [-5, -6, 2, -10], [-17, 6, 6, 6], [13, -5, 5, -2], [-8, -7, 1, -7], [-5, -5, -10, 10], [6, -1, 1, 2], [39, 1, 1, -2], [-7, -2, 5, -1], [35, 4, 1, 1], [-14, 8, -2, 5], [-25, 5, -1, 1], [18, 2, 4, 7], [9, -3, 6, -12], [21, 1, 3, -12], [19, 11, -1, 9], [-13, -9, -1, 7], [14, 2, 1, -1], [25, 5, 7, -2], [2, -6, 5, -1], [-7, 2, -7, 3], [3, -13, 6, 2], [-16, 4, 4, 7], [-30, 7, 0, 10], [-2, 7, -4, -9], [-23, 14, -3, 6], [1, 3, 1, -11], [-10, -7, -11, 5], [4, 2, -13, 8], [-16, -6, -6, -4], [-14, -1, 6, -10], [-3, 8, 5, -5], [-10, 4, -2, 4], [4, -9, 4, -5], [-1, 6, 5, 12], [-36, 0, 3, -6], [-16, 10, 1, -2], [5, 5, 7, 3], [6, -14, 4, -4], [-6, 2, -6, 5], [-13, -6, 7, -1], [-5, 5, 13, -9], [5, -4, 2, 1], [-26, -7, -1, 0], [18, -18, -1, -9], [-6, 9, 0, -6], [10, 1, 2, -2], [34, 6, 2, 3], [-7, 11, -5, 1], [22, -11, 6, 4], [31, -8, 3, 0], [-29, -9, 0, 4], [-4, -8, -2, 1], [31, 5, -7, 7], [8, -9, 5, -13], [-20, 16, -2, 11], [-25, -1, 4, 3], [23, 8, -5, 2], [-28, -2, 7, 0], [7, 14, -3, 11], [24, -7, -10, 2], [10, -18, 5, -9], [-24, 16, 5, 6], [-25, 2, -2, 20], [24, -7, -4, 1], [-10, -1, 6, -3], [-24, -1, 6, -7], [3, -3, 5, -11], [-17, 10, 5, -5], [-4, 7, -7, 16], [32, -6, 13, 4], [0, -7, 0, 1], [9, 8, -3, -7], [-4, -5, -10, 3], [-20, 6, -8, 11], [28, 6, -3, 2], [-19, 2, -5, -5], [-13, 2, 1, -10], [-22, 4, 3, 7], [-37, 1, -8, 4], [-28, 17, 0, 3], [-50, 3, 3, 0], [-3, -9, 1, -7], [-22, 21, 4, 6], [-17, 4, 3, 14], [20, -8, -2, -1], [10, -1, 6, 7], [29, 1, -7, 4], [-5, -8, -5, 13], [-16, 8, -3, 15], [2, 8, 2, 5], [16, -12, 12, 0], [-14, 0, -5, 7], [11, 6, 3, -1], [2, 2, 4, -4], [-47, 12, 2, 9], [5, -8, 9, -2], [-34, 1, -4, 9], [-14, 3, 2, 4], [13, -5, 1, -8], [12, 0, 1, 0], [15, 6, 11, -6], [0, -8, 10, -11], [21, 3, 7, -9], [-24, 5, 6, -1], [-21, 5, 1, -6], [-3, -7, -1, 11], [-13, -12, -2, 6], [12, 20, -8, 16], [-13, -1, 0, 11], [5, 9, 1, 0], [-27, 10, -6, 14], [28, -24, 0, 0], [-2, 18, 5, -6], [-19, 1, -9, 4], [-10, 13, 1, 13], [-12, -3, -8, 0], [-9, 10, 3, 1], [37, -1, 1, -6], [18, -8, 6, -11], [-8, 4, 8, -2], [1, -10, 11, -13], [-19, -17, 3, 4], [30, -5, 8, 2], [1, 3, 8, -2], [27, 1, -8, 0], [48, 0, 9, -15], [-9, -19, 8, -14], [28, -12, 1, -4], [-24, -1, -8, 1], [-11, 12, -5, -9], [9, 6, 15, 3], [6, -5, -8, 11], [-10, 16, -10, 5], [2, -23, 4, 1], [7, -3, 0, 3], [-27, -7, 2, 0], [-8, 5, 4, -7], [7, 6, -1, -6], [-14, 4, -4, 10], [20, -22, -1, 1], [47, -8, 15, 4], [-11, -3, 11, 0], [8, -12, 1, -5], [-14, -3, 10, -2], [-25, -3, 5, -1], [20, -18, 4, 4], [-9, -7, -9, -7], [23, 12, 1, -5], [-6, -1, -3, -2], [-13, 18, -4, 2], [-49, -5, -6, 16], [2, 16, 10, -11], [-17, -14, 1, -3], [38, -5, 7, -12], [-21, 13, -10, 6], [-14, -13, -5, 3], [-27, -5, -4, -10], [12, 23, -6, 10], [-44, 10, 8, 1], [15, -12, 1, -3], [27, 3, 18, -15], [-18, -7, -2, -9], [36, 11, 0, 2], [-2, 13, 6, -1], [35, 0, 13, -12], [10, 5, -4, -6], [43, 8, 1, -11], [-17, 3, -3, 7], [-1, -5, -14, 13], [44, -6, 10, 6], [9, 16, -2, -7], [-19, 20, -11, -1], [50, 1, 10, 1], [31, -11, -4, 8], [32, -12, 8, -5], [-38, 5, -4, 11], [-10, -4, -8, 8], [-36, 7, -4, 0], [39, -1, 11, -1], [22, 11, -8, 9], [7, 1, -17, 6], [8, 13, -8, 3], [49, -8, 11, 1], [-23, 23, -7, 5], [15, 6, -9, -3], [54, -2, 12, -8], [-9, 25, -3, 4], [4, -7, 1, -16], [-10, 17, -2, 8], [-26, -5, 3, -6], [39, -15, 13, 8], [-38, 5, -12, 5], [16, -2, 4, 1], [-3, 0, 9, -6], [-32, -7, -8, -7], [-3, 6, 0, 14], [59, 1, 2, 0], [-56, 1, 1, 0], [-12, 8, 0, 14], [-14, 2, 10, 3], [37, -11, 9, -11], [0, 15, 4, 4], [-36, 10, 0, -2], [-38, 4, -16, 7], [-34, -7, -1, 11], [-7, -1, -5, -1], [2, 10, 14, -14], [-7, 7, -5, -8], [-55, 13, 4, -2], [-54, 1, -6, 0], [10, 1, 7, 2], [-1, 7, -10, 7], [-2, 11, 12, -11], [15, -13, 2, -7], [-62, 9, 2, 1], [-21, -7, -3, -5], [1, 1, 4, -10], [-5, 4, -3, 11], [5, -17, 17, -21], [7, 0, -2, -2], [11, 1, -9, -9], [28, -8, 2, -11], [-8, 24, -6, 16], [39, -13, -8, -10], [-4, -22, 1, 8], [55, -7, 9, -9], [37, -12, -3, 2], [-30, -7, 3, -9], [-6, 11, -4, 3], [25, -26, 7, -5], [6, 5, 0, 8], [31, -24, 1, -14], [20, 11, 0, 4], [43, 9, -8, 6], [9, 3, -13, 7], [7, 0, -10, -8], [1, -8, -1, 10], [-14, -11, -3, -14], [-14, -12, -13, 0], [-12, -18, 1, -13], [14, -15, -1, -2], [9, -15, -1, 0], [-58, 0, -4, 8], [-30, -1, -18, 11], [-26, 15, -9, -3], [20, 13, 2, 9], [-37, 7, 7, 4], [-32, -10, 6, -2], [3, 8, -17, 11], [5, -4, 7, -4], [5, 2, 18, 0], [3, -2, 2, 9], [-9, -15, 10, -1], [7, -5, -4, 1], [-12, 5, -2, 13], [-23, -6, 16, -13], [12, -4, -9, 14]];
    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_bi();".
// 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_bi(: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_bi();".
// 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_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<5,R![-27, -50, -15, 3, 1]>,<11,R![15, -49, -33, 0, 1]>],Snew);
    return Vf;
end function;