
// Make newform 4018.2.a.bh in Magma, downloaded from the LMFDB on 25 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_bh();".
// 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_bh();".
// 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, -4, -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], [1, 0, 0], [-1, 0, 1], [0, 0, 0], [-2, 1, 0], [-1, 0, -2], [2, -2, -1], [-2, 2, -1], [-2, -3, 0], [-3, -2, 1], [3, 2, 0], [-6, 1, 1], [-1, 0, 0], [-5, 4, 2], [1, -5, 0], [-2, 1, 3], [-1, -3, 1], [-5, 6, 1], [-7, -1, 1], [-1, 1, -2], [3, 2, 3], [-2, 2, -5], [0, -3, 1], [5, -2, 3], [1, 3, 3], [6, 4, -1], [1, -3, 2], [-9, 1, 3], [-5, 10, 1], [-9, -3, -7], [0, -5, -4], [1, 5, 6], [1, -4, 6], [-6, -1, -2], [-1, -1, -3], [8, -4, -3], [2, -5, -5], [16, 0, -1], [0, 4, 3], [3, -6, 2], [5, -12, -1], [-4, 2, -1], [-1, 3, -3], [8, -6, 0], [-18, 1, -4], [6, 3, 10], [-6, 3, 5], [-6, -8, -4], [-1, 7, -2], [-2, 2, -3], [4, 7, 6], [0, 1, 6], [26, 0, 2], [0, 0, -5], [2, -12, 0], [7, 4, 7], [6, 5, 1], [-9, 1, -10], [14, 0, 10], [-8, 7, -5], [-1, 5, 3], [1, -7, 3], [-17, -4, -8], [9, -4, 6], [-7, -3, -12], [-11, 6, -7], [-6, -9, 6], [-12, -2, -4], [-14, 2, -11], [3, 5, 1], [9, 0, 5], [-5, 4, 1], [-1, 1, -8], [-4, 11, 2], [-7, 14, 1], [-10, 4, 1], [2, 10, 2], [12, -5, -1], [-14, -6, 2], [1, 3, -2], [-13, 16, 4], [-12, 3, 0], [-9, 5, 2], [-2, -12, -2], [3, -5, 3], [13, 8, 8], [4, -2, -9], [5, 0, 8], [3, 2, -12], [12, -16, -4], [-2, -1, 8], [-23, 0, -4], [15, -5, -5], [33, -1, -2], [3, 3, 1], [5, 2, 6], [2, -3, -3], [-7, 4, -7], [7, -10, -2], [-7, 6, 0], [14, -9, 4], [-20, -7, 2], [8, -12, -4], [2, -5, -5], [2, 7, 15], [8, 1, 9], [8, -2, 1], [-18, 11, -5], [2, -5, 8], [-26, 6, 1], [11, 8, -1], [13, -7, -5], [-4, -11, 6], [-23, 4, -2], [5, 1, -9], [-10, 4, -3], [-13, 10, 8], [-26, -2, -6], [-10, -5, -1], [19, -6, 7], [10, -16, -4], [-9, -2, 6], [16, -1, -7], [3, -10, 6], [-15, -3, -11], [17, -4, -4], [-31, 0, 0], [1, -15, 13], [20, -1, 8], [4, 11, -9], [-12, 6, -10], [22, 8, 5], [-10, 3, 4], [5, 11, 4], [-20, 13, 5], [-17, 9, -3], [27, -4, 7], [-13, 3, -15], [14, 9, -2], [-13, -1, 1], [-11, -1, -7], [24, -10, 9], [8, 5, 1], [10, 14, 4], [-13, 7, 2], [23, -21, -11], [-22, 3, 7], [-24, 5, 1], [5, -13, 5], [-22, 4, -5], [14, -21, -8], [-17, -6, 0], [-13, 11, -13], [-6, 11, 1], [-3, 17, -12], [23, -10, 9], [10, 10, 17], [-17, 4, 9], [9, -12, -1], [-17, -7, -5], [-12, -9, -2], [-10, -2, 9], [22, 6, 9], [26, 5, 0], [-51, -2, 2], [13, -12, -11], [-1, 0, -13], [11, -15, 15], [-13, 5, -17], [32, -14, 4], [16, -13, -12], [-3, 8, 15], [-23, -4, -1], [35, -14, -2], [-41, -2, -16], [23, 3, -10], [-4, -5, -9], [17, 3, -7], [30, 11, 4], [-3, -3, 2], [9, 14, 8], [29, -12, -10], [2, -2, -13], [31, 5, -1], [26, -20, 3], [-13, 9, 13], [-25, -9, 4], [19, 9, -9], [-11, 19, 3], [12, 0, -14], [-41, 15, 8], [16, -24, -6], [3, -14, -10], [29, -17, -7], [0, 12, 19], [4, 13, -5], [0, 19, -10], [7, -16, -5], [19, 11, 11], [16, -7, -7], [-20, 8, -17], [8, -6, -12], [-10, 9, 11], [9, 0, -16], [30, 12, 10], [12, 7, -14], [-9, -12, -25], [-44, -8, -9], [8, 22, -11], [-15, 11, 11], [-26, -10, -18], [26, 16, 1], [-13, 2, 10], [-13, -17, 9], [-17, -4, 5], [-20, -5, -22], [24, 16, 0], [-8, -13, -7], [6, 23, 3], [24, 18, -7], [-13, -7, -7], [-2, -19, -11], [-25, 4, 1], [-9, -11, -9], [4, -24, -17], [-11, 3, -13], [-44, -1, -3], [-24, -2, -25], [-3, 5, -2], [21, -24, -11], [-24, 16, -1], [19, 16, 22], [-8, 5, 19], [4, -16, 14], [16, 16, -3], [11, 6, 2], [-22, 9, 10], [-14, 19, -6], [-14, -3, -18], [41, 9, 11], [-49, 16, 6], [-2, -29, -2], [-24, 8, 7], [-22, -15, 12], [-21, -14, -14], [-12, 29, -8], [24, -8, 7], [20, -10, -1], [-1, -14, 4], [15, 0, 9], [-32, -14, 6], [-35, 15, 3], [0, -29, 4], [-4, -23, 8], [-21, 21, 15], [4, -29, 3], [-7, -14, -13], [10, -2, 19], [18, 3, 12], [-9, -7, -6], [-25, 20, -2], [-14, 2, -16], [10, -26, -1], [-28, -4, 0], [0, 2, 22], [-55, 10, 8], [-6, -18, -18], [-5, -6, -13], [-1, -16, 19], [14, 13, 2], [-38, -15, 9], [1, 1, -12], [4, -16, 9], [35, -16, -5], [-28, 0, -15], [-51, 19, 5], [-60, 3, 0], [-7, -3, 6], [-29, 21, 5], [21, 0, -16], [-4, -12, -8], [65, -7, -6], [5, 19, 4], [-21, -14, -20], [24, -4, -8], [-6, -4, -7], [39, 1, 8], [-16, 7, -5], [-48, 15, -1], [-33, 10, -6], [27, -5, 4], [-2, -4, 19], [48, -24, -10], [51, 2, 4], [-1, 15, 20], [-46, -7, -7], [-31, 33, 9], [33, 21, -13], [6, -6, -10], [-5, 21, 7], [4, 8, 1], [-28, 19, 4], [17, 24, 0], [-3, 0, 12], [-20, -11, 2], [-48, -6, -13], [-24, -9, 10], [-65, -3, -12], [-15, 15, -5], [36, 17, 6], [-9, -24, 5], [-2, -16, 14], [-22, -8, 10], [5, 16, 27], [-35, -5, 14], [-26, -14, -32], [8, -17, 9], [-9, 9, -3], [-16, 6, 22], [-11, 5, 11], [-7, -20, 20], [-76, 13, 4], [33, -14, -2], [-5, -19, -27], [-4, -10, -1], [-17, 14, 12], [-4, 11, -8], [67, -7, 9], [-20, 23, 19], [-13, -1, -13], [8, -3, 19], [-26, -22, 9], [23, 20, 4], [37, 21, -12], [24, -21, 2], [18, -5, 17], [-10, -11, -10], [-25, -20, 5], [-20, 37, 13], [-17, -20, -16], [-3, 5, -23], [11, -1, -16], [-44, -10, 6], [-1, 20, -11], [1, 6, 6], [0, 11, 28], [-55, -3, -9], [52, -4, 23], [-48, -5, 10], [-6, -12, -12], [-14, 3, -13], [-49, 27, 11], [-42, 15, -16], [23, -1, -4], [24, 13, 0], [13, 20, 0], [1, 23, -7], [13, -6, 2], [-30, -14, -18], [-25, 15, -28], [-12, 7, -12], [-44, -7, -15], [2, -5, -2], [19, -13, 10], [-18, -21, -16], [12, -14, -23], [47, -14, 13], [-12, -13, 22], [-35, -14, -4], [-5, -11, -17], [14, 29, 0], [48, -16, -2], [22, -28, 7], [21, 25, 3], [-40, 9, -1], [24, -10, 16], [-6, 30, 21], [2, 9, -23], [-1, 11, 3], [2, 6, -6], [-13, 31, 4], [-21, -9, -15], [-34, 29, 17], [-36, -21, -4], [17, 9, 2], [-2, -7, 4], [17, -15, 2], [-2, 21, 12], [-1, -17, -23], [29, -17, -5], [-5, -40, 16], [59, -11, 1], [-59, -4, -4], [3, 3, 11], [12, -10, 13], [-28, 16, 14], [-41, 5, -11], [16, 26, -13], [-5, -9, 1], [-17, 2, 5], [12, -9, 7], [16, -37, 2], [-51, -10, -22], [40, 17, 23], [21, 14, 11], [12, -7, -7], [-43, 1, 5], [-33, 11, 3], [-11, -32, 22], [-13, 4, 1], [36, -37, -22], [-10, 10, 21], [-38, 0, -14], [8, -19, -16], [-33, 9, -20], [10, 8, -4], [14, -11, 8], [56, -13, 13], [-42, -23, 12], [-25, 17, 4], [-24, 30, 6], [-23, 48, 4], [5, 22, -5], [-37, -2, 0], [-3, 28, 13], [-29, 25, -6], [-39, 9, -22], [58, -3, 24], [37, -20, -13], [-68, 14, 5]];
    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_bh();".
// 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_bh(: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_bh();".
// 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_bh( : 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![-1, -1, 4, 1]>,<11,R![-3, 4, 5, 1]>],Snew);
    return Vf;
end function;