
// Make newform 4018.2.a.bg in Magma, downloaded from the LMFDB on 24 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_bg();".
// 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_bg();".
// 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 := [-2, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, -2, 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, 0, 1], [0, -1, 0], [0, 0, 0], [2, 1, 1], [0, 0, 0], [2, 0, -1], [-2, 0, -2], [2, -2, 0], [2, -1, 0], [-2, 0, 1], [2, 0, 0], [1, 0, 0], [-2, 0, 3], [-2, 0, 0], [-2, -3, -2], [0, 3, 1], [4, -1, -2], [6, 1, -3], [2, 4, 3], [10, 0, 2], [6, 2, 1], [4, 1, -1], [-8, 2, 1], [8, 2, 1], [-8, 6, 0], [-2, 4, -3], [-2, 6, 1], [0, -7, -5], [-6, -2, -5], [-6, -2, -2], [-4, 5, -3], [2, 6, 4], [8, -5, 1], [14, -5, 0], [6, 0, -3], [8, 0, 0], [8, 2, -2], [-4, 4, 4], [0, -7, -6], [6, -1, -3], [-4, -2, -8], [14, 0, -1], [6, 4, 10], [2, 2, 0], [10, -8, -2], [6, -5, 1], [-2, 6, -1], [-4, 2, 3], [-4, -2, 2], [-2, 0, -8], [-4, 6, -2], [6, 0, 4], [-4, -7, -7], [4, -10, -5], [16, -4, -4], [10, -3, 5], [-20, 0, 0], [6, 2, -6], [10, -4, 0], [0, -3, 3], [-4, -4, -10], [-4, 9, 5], [2, -8, 4], [-4, -10, -4], [8, -7, -5], [-10, 5, -1], [0, -2, -11], [-2, -7, -1], [26, -3, -3], [-10, 10, 0], [-8, 8, 0], [-10, -2, -3], [-6, 12, 4], [-14, 6, 3], [0, -4, 2], [6, 4, 12], [0, -6, 2], [2, 2, -7], [6, 8, 8], [0, 9, 5], [-10, 9, 8], [-10, 2, 4], [22, 4, -6], [-10, 8, 0], [10, -8, -5], [24, 2, 7], [-18, -4, 0], [4, 11, 4], [-16, -4, -4], [16, 3, 9], [2, 4, 12], [-16, 8, 0], [26, 4, 1], [-6, -1, -5], [-8, -8, -4], [12, -14, 2], [-24, -2, 0], [8, -7, 1], [-18, 4, 0], [-2, -9, 3], [-10, 7, -6], [34, 4, 2], [-14, 2, 1], [-2, 5, -1], [20, -2, 0], [-8, -8, -1], [32, -2, -5], [-16, 8, 0], [22, -8, 3], [6, -18, -9], [10, -10, 8], [8, 12, 6], [4, 3, 7], [-10, -6, 6], [14, -4, 8], [0, 10, 7], [-10, 16, 1], [2, 7, 0], [14, -3, 11], [-22, 7, 3], [2, 2, 14], [6, 11, -1], [-18, 3, -7], [8, 8, 1], [18, 4, -6], [26, 9, 10], [4, 8, -12], [-4, 0, 8], [-24, 13, 8], [-2, -16, 3], [0, 0, 4], [0, -10, -10], [-22, 15, 0], [-14, -6, -2], [-26, -6, -14], [12, 8, -6], [20, -5, -5], [0, 3, -2], [10, -4, -8], [12, -3, -7], [-10, 10, 4], [14, 0, -1], [18, -5, -5], [8, -13, 0], [-10, -8, -4], [0, -7, 6], [2, -8, -8], [-28, 3, -9], [16, -8, 8], [-14, 2, -8], [6, -10, 4], [-2, 1, -15], [22, -4, -6], [26, -12, 3], [26, 10, -2], [14, -8, -11], [22, 0, 4], [12, 10, 5], [-10, -3, 5], [-16, 16, 4], [-4, -8, 14], [6, -10, 9], [8, -14, -5], [-14, 10, 2], [2, -8, -3], [-8, 10, -6], [-8, 10, -2], [0, -14, 9], [0, 6, -10], [-14, -13, -13], [-4, 7, -4], [-40, -6, -2], [-2, -10, 13], [-20, -8, 0], [30, 8, -9], [-18, 7, 1], [6, -12, -16], [-26, -12, -14], [-4, 2, 16], [4, -12, 6], [16, -1, 1], [8, 1, 11], [14, 14, 10], [-12, -4, 10], [-44, -1, 4], [14, -8, 4], [4, 13, 5], [30, -6, -8], [8, 0, -18], [-10, -2, -16], [14, -3, 5], [12, -8, -8], [-8, -13, -10], [-46, -6, 3], [38, 0, 6], [-14, 14, 4], [32, -19, -1], [22, 4, 3], [22, -20, -12], [6, 6, 14], [-12, 6, 6], [-24, 6, 6], [6, 12, 20], [-12, 20, 13], [28, -22, -6], [6, 0, 17], [10, 7, 5], [0, 0, -6], [-8, -13, -5], [-2, -8, 16], [16, 10, 4], [0, 16, -8], [8, -7, 3], [-16, 12, 12], [2, -14, -8], [26, 10, 17], [-10, -16, 20], [-10, -16, -19], [-42, -6, -12], [2, -11, 2], [-42, -6, -5], [22, 14, 0], [-4, 6, 2], [64, 3, -1], [2, -19, 2], [-26, -14, 2], [-4, 4, 6], [-22, 12, -6], [12, 6, -10], [-10, 4, -22], [24, 12, 17], [-2, 10, -6], [22, -6, 13], [-10, 0, 6], [-20, 8, -10], [10, -20, -19], [-40, -3, -5], [14, 12, -15], [12, 8, 2], [10, 11, 13], [-32, 2, 13], [-14, 0, 10], [-2, 22, 8], [6, 22, -4], [26, 0, 7], [4, 0, -12], [-16, -8, 7], [-2, -10, -5], [-10, -2, 2], [10, 2, -18], [22, -4, 16], [6, -26, -16], [26, -24, 2], [-20, -10, -12], [6, 18, -17], [-6, 3, -2], [-62, 0, -2], [8, -4, 8], [16, 22, 27], [8, -12, -12], [26, -3, -3], [12, 6, 0], [28, 18, 22], [6, 2, -8], [-44, 9, 3], [22, -3, -2], [-14, -2, 18], [-4, -10, 10], [32, 3, -11], [24, 20, 24], [-6, 14, -19], [2, -2, 8], [-42, 6, 0], [6, 0, 15], [20, 0, 0], [-38, -9, 5], [36, 1, -1], [-22, -10, -2], [-18, 0, 10], [22, 12, 18], [6, 22, 15], [6, -8, -11], [26, 15, 23], [-8, 22, 9], [-26, -10, 19], [-6, 14, -4], [14, -8, -16], [0, 20, 14], [14, -12, 15], [6, 23, 10], [8, 21, 21], [-6, 22, 8], [-4, 11, -15], [20, 11, 9], [-8, -6, 20], [-10, -12, -12], [26, -11, -3], [-20, 4, 7], [22, -22, -12], [-6, 14, 20], [0, 27, 21], [10, -18, -4], [14, 9, 19], [42, -16, -5], [-2, -14, -9], [12, 4, 12], [14, 5, 25], [-22, 22, -9], [22, -29, 0], [-48, 28, 8], [10, -9, -8], [-10, -22, 5], [-6, -4, 9], [8, 0, 0], [14, -4, 16], [-16, -17, -11], [66, -2, 8], [44, 14, 7], [34, 0, 2], [12, 11, -19], [40, 8, 6], [-54, -1, 3], [-50, 12, 7], [26, 6, 21], [32, -18, -4], [-6, -25, -7], [-28, -6, -7], [-54, 0, -3], [-26, 4, 20], [-58, 1, 8], [6, 24, -2], [-34, 8, -1], [-4, -22, 7], [-2, -4, 19], [6, -2, -24], [10, -27, -14], [-14, -20, -2], [14, 8, -9], [-28, -14, -22], [14, 25, 14], [16, 22, 7], [24, 2, 4], [10, -16, 7], [50, -14, -8], [-18, -10, -1], [34, 17, 1], [24, -4, -8], [36, 16, 22], [14, 9, 25], [-48, 12, -9], [20, -30, -9], [6, -1, 2], [-8, -26, -26], [6, 7, 20], [26, 0, 8], [10, -20, -3], [-4, 17, -23], [-14, 8, -16], [8, 8, -20], [-46, 24, 0], [-18, -8, 21], [-20, 14, 2], [30, -8, -1], [72, -6, 3], [-14, -2, 10], [0, 12, 10], [-10, -2, 22], [0, -10, -5], [46, 13, -7], [12, -16, 1], [18, -6, 8], [60, 9, 15], [8, -16, 0], [14, -1, 10], [-48, 19, -1], [-60, 0, -6], [30, 18, -2], [14, 4, 16], [-16, 10, -9], [52, 12, 6], [28, -10, -21], [30, 6, -10], [-28, 22, 15], [-4, 6, -15], [-10, -22, -9], [30, 8, 5], [-48, 32, 6], [-18, 22, 7], [-10, 4, -12], [10, 22, -16], [-56, -8, 8], [-46, 11, -1], [12, 17, 13], [20, 0, 22], [-10, -6, -4], [-4, -12, 8], [-38, -12, -8], [-14, 20, 1], [-6, -9, 14], [40, -1, 12], [14, -2, 26], [22, 14, 8], [-4, 2, -4], [-20, -23, -16], [-18, 16, -12], [-22, 20, 8], [-2, 20, -6], [-8, 2, 3], [8, 25, -13], [42, 4, 0], [42, -14, 4], [2, 11, 13], [-18, -5, -23], [-30, -12, 12], [-32, 45, 14], [8, 0, -28], [-28, -16, -28], [-24, -2, -13], [-38, 6, 7], [-10, -28, -20], [28, -8, -14], [50, -10, 2], [-6, 10, -4], [-22, 4, -14], [-26, -15, -3], [26, 27, 21], [30, -16, 14], [34, -18, 12], [46, -32, -2]];
    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_bg();".
// 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_bg(: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_bg();".
// 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_bg( : 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, 1, 1]>,<5,R![2, -6, 1, 1]>,<11,R![20, 2, -6, 1]>],Snew);
    return Vf;
end function;