
// Make newform 4018.2.a.be 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_be();".
// 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_be();".
// 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, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, -1, 1]];
    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_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, 1], [1, -1, 0], [0, 0, 0], [1, 0, -1], [0, -1, -1], [4, 1, -1], [2, -1, -1], [-2, 1, -3], [1, -4, 1], [4, 0, -2], [6, 2, -2], [-1, 0, 0], [-4, -2, 2], [8, -1, -1], [-3, -2, 1], [3, 3, 0], [-7, 1, -4], [3, -2, -3], [-2, -2, 4], [4, -2, 2], [4, -4, 0], [-7, -1, -2], [0, 2, 4], [-6, 4, -4], [8, 3, -1], [-6, 4, 2], [6, 0, -2], [-1, -2, 7], [-10, 1, -1], [14, -1, 3], [-3, -5, 4], [8, -6, 0], [-1, -1, 2], [-11, 2, 3], [-6, 4, -4], [-2, -5, 5], [-6, -8, 2], [10, -4, 2], [3, 5, -6], [1, 2, -1], [0, 1, -3], [-6, 6, 4], [2, -4, 2], [-12, 10, 4], [-8, 7, -1], [-5, -2, -5], [-4, -8, 4], [8, 3, 1], [-16, 3, -1], [2, 10, -6], [2, 4, -4], [-18, 6, 2], [1, -5, -6], [-4, -2, -8], [0, -2, -2], [-3, 1, 0], [6, 8, 0], [4, 2, 4], [6, -6, 2], [13, 3, 4], [10, -1, 1], [9, 11, -4], [16, -7, 5], [4, 3, 5], [-7, 8, -3], [-5, 0, -13], [-18, 8, 4], [1, 0, 7], [-11, 5, 4], [16, 6, -8], [10, 0, -2], [2, -12, 8], [0, -12, 6], [0, 6, -10], [-2, 12, -6], [-6, -4, 8], [0, -7, 5], [-2, 11, -7], [-18, 4, -4], [-7, -1, 8], [-1, 2, 1], [10, -1, 3], [14, 2, -2], [-4, -9, 3], [-4, 0, -4], [-20, 2, -4], [14, -6, -4], [13, 11, -4], [-8, 6, 10], [-13, 5, 6], [20, -15, -3], [-2, 8, -6], [0, 2, 2], [-11, -8, 9], [24, 0, 4], [-4, -5, -1], [-8, -3, 11], [1, 5, -8], [18, -8, -8], [-33, 2, -1], [3, -6, 11], [28, 1, 3], [14, -1, 1], [-25, 2, 5], [-4, -5, -11], [-2, 1, 13], [-10, -2, -2], [-14, 12, -6], [-4, -21, 9], [-10, 0, 0], [-6, -16, 8], [30, 3, -3], [-13, -1, 8], [0, 1, -5], [10, -12, 14], [24, 5, 3], [10, -6, 8], [-3, -10, 11], [27, -4, -1], [-29, -3, 6], [-16, 10, 8], [15, -13, -10], [17, -8, 9], [-22, 11, -1], [0, -6, -4], [17, -4, -5], [-22, 0, 2], [2, -8, 10], [-13, -7, 4], [8, -6, -2], [-4, 16, 0], [-14, 8, 12], [-15, 14, -1], [-26, -2, 0], [-18, 2, 12], [24, -13, 7], [-41, 3, 0], [-3, -15, 4], [-14, -16, 6], [1, -7, -2], [-46, 2, 2], [-18, 10, 4], [-11, -20, 9], [-31, -7, -2], [20, -13, -5], [-1, 25, -10], [6, 2, -10], [11, -1, -10], [-6, 8, -10], [-8, 10, -12], [0, 6, -8], [-15, -2, 5], [-12, 13, 5], [16, 0, -8], [10, 15, -5], [-26, -2, 0], [-40, 0, 2], [14, -14, 2], [5, -15, 0], [4, -10, -14], [8, -19, -3], [12, -6, 12], [-30, -3, 9], [-8, 14, -16], [4, 8, -14], [2, 12, -6], [-22, -7, -1], [26, -12, -8], [4, -11, 9], [-9, -8, -3], [-7, 11, -8], [10, -22, 12], [4, 1, -3], [2, 4, -14], [4, 3, 5], [-3, 20, -11], [20, 10, 0], [-20, 1, 13], [22, -23, 3], [6, 4, -6], [-21, -3, 8], [-1, 18, -13], [-12, -12, 2], [0, 8, 4], [-11, 17, -14], [-26, 12, -4], [5, 17, -22], [16, 22, -8], [26, 16, -6], [0, -14, 16], [-5, 0, -9], [6, -10, 20], [21, -1, 14], [30, 24, -14], [8, -10, 14], [4, 12, -8], [-17, 4, -3], [0, 3, -15], [8, 19, -9], [0, 18, -4], [-2, -6, -12], [2, 1, 3], [2, -10, 10], [12, -15, -13], [-12, -9, -5], [14, -16, -6], [29, 6, 3], [-6, -13, 5], [3, 17, 6], [-30, -2, 0], [2, -17, 1], [-4, 0, -4], [13, -15, 14], [4, -4, -16], [-10, 6, -4], [-16, 12, -8], [34, -10, 2], [-6, 6, -16], [-24, -12, 6], [5, -12, 11], [-2, -8, 20], [12, 12, -6], [-2, -2, -14], [-1, 23, -2], [21, 8, -1], [0, -12, -2], [-6, -12, 14], [28, 19, -17], [20, -4, 0], [30, -10, 14], [-14, -29, 19], [-8, -23, 7], [16, 25, -11], [-8, 3, 5], [-8, 24, 4], [20, 11, -7], [5, 2, 11], [-18, 6, 12], [16, -12, 12], [3, -22, 19], [-10, 15, 7], [-16, -21, 11], [-6, -2, 22], [20, 10, -4], [0, 2, 8], [66, -4, 0], [-4, -15, 15], [0, 24, -16], [16, -3, -1], [-6, -10, 14], [-4, 2, -10], [-12, 7, -15], [-28, 18, -6], [34, -15, 7], [20, 0, 16], [-13, -2, 7], [4, 1, 15], [12, 3, -9], [20, 8, -6], [-12, -8, 0], [-21, -24, 19], [-16, -15, 17], [-18, -1, 9], [2, 22, -20], [-19, 15, 16], [-3, 8, -21], [-20, 8, -6], [24, -16, 4], [3, -6, 21], [42, -5, -3], [-4, 28, -16], [-28, 14, -8], [34, 11, -9], [-12, 11, 5], [50, -2, 2], [29, -10, -1], [11, 16, -3], [0, 24, 0], [28, -5, -19], [-28, 13, -11], [-36, -2, 14], [-4, 10, -4], [7, 13, -28], [28, -13, -7], [12, -6, 24], [28, -6, 0], [18, 12, 8], [20, -14, 26], [24, 3, -11], [-21, 20, -23], [57, -9, -2], [0, 26, -26], [-37, -17, 10], [-5, -20, 3], [-8, -29, 15], [32, 11, -1], [33, -3, 6], [12, 3, 5], [-14, 9, 9], [14, 20, -2], [27, -2, -7], [4, -3, 15], [-27, 6, 1], [10, 26, 0], [-48, -5, 3], [40, -12, 0], [37, -13, -12], [0, -18, -12], [-7, -12, 13], [42, 18, -16], [-11, -2, -17], [-4, 1, -15], [8, -30, 2], [-4, 4, 0], [12, 18, -18], [7, -29, 2], [40, -7, -1], [-16, -26, 8], [-26, 18, 6], [27, 1, 10], [44, -1, 7], [-1, -19, 24], [-34, -24, 14], [-38, -13, 17], [-4, 5, -21], [-35, 8, -23], [12, -11, 3], [-14, -4, -16], [26, -4, 12], [-49, -16, 7], [-46, 6, 6], [40, -2, -16], [-34, 27, 15], [54, 4, 10], [-32, -23, 23], [-27, 14, 5], [-12, 6, -22], [-20, -5, 13], [50, -10, -12], [-3, -16, -9], [28, -10, 16], [22, 11, -11], [46, 4, 6], [4, 6, 20], [30, -14, 28], [-19, 3, 8], [-50, 8, -2], [-12, 32, -8], [17, 31, -28], [-28, 17, -1], [-10, 4, -16], [27, 6, -17], [34, 16, 2], [17, -40, 5], [-12, 30, -16], [30, 0, -2], [-3, -9, -16], [2, 8, 12], [32, -20, 0], [8, -20, -10], [8, 13, -21], [42, 10, 10], [-56, -14, 2], [2, 13, 1], [-24, -14, 10], [16, -3, 33], [6, -7, 25], [12, -2, -8], [-29, -10, 5], [-2, 21, -11], [32, -15, 7], [-29, 30, 11], [-18, 8, -8], [-45, 20, 13], [31, -11, -20], [2, 20, -22], [-20, 4, -30], [4, 0, -14], [12, 12, 6], [70, -1, -3], [42, 20, -12], [12, -23, -9], [-2, 16, 0], [-2, -19, 1], [36, -14, -2], [74, 0, 10], [8, -31, 21], [-34, 26, 12], [16, 31, -13], [32, 20, -4], [38, -5, 21], [-61, 2, -5], [65, -5, 10], [26, -24, 10], [-32, -24, 6], [-14, -24, 26], [-22, -24, 20], [-52, 38, 14], [-41, 4, -19], [-23, 11, 0], [-8, -2, 12], [76, 3, 5], [-28, 15, -7], [47, -5, 22], [32, -5, 11], [24, -20, 10], [-18, -26, 20], [30, 25, -23], [-27, 23, -14], [48, 14, 6], [18, -12, -20], [-33, -30, 19], [25, -4, -7], [-18, -8, 0], [3, -15, -2], [0, -4, -28], [0, 26, -6], [8, -8, 18], [-28, 18, -30], [24, 6, 24], [20, -1, 35], [-28, -5, 1], [18, -5, 7], [54, 2, -10], [33, 9, -12], [7, 6, 11], [-14, -6, -30], [32, -25, -23], [28, -7, 29]];
    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_be();".
// 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_be(:prec:=3)
    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_be();".
// 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_be( : 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, -8, 0, 1]>,<5,R![6, -4, -2, 1]>,<11,R![-2, -6, -2, 1]>],Snew);
    return Vf;
end function;