
// Make newform 4018.2.a.d 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_d();".
// 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_d();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [-2], [2], [0], [-6], [4], [2], [-2], [0], [0], [0], [10], [1], [4], [-8], [-4], [8], [-10], [-14], [-12], [6], [0], [-4], [14], [10], [-12], [8], [12], [4], [-14], [0], [0], [6], [-8], [-20], [4], [-12], [12], [12], [-18], [6], [24], [-12], [-26], [-6], [-16], [2], [-24], [-10], [16], [-6], [-12], [2], [-16], [6], [24], [26], [8], [-22], [2], [0], [0], [-8], [-12], [-30], [-28], [-34], [-26], [-6], [10], [2], [-24], [-8], [14], [0], [-28], [2], [-16], [34], [10], [20], [-16], [-40], [18], [-20], [-24], [-6], [-18], [-18], [0], [-8], [32], [32], [32], [-26], [-32], [-24], [-38], [12], [-18], [30], [40], [10], [-6], [22], [30], [-42], [-42], [-24], [46], [0], [-2], [-38], [16], [32], [-34], [2], [24], [24], [30], [2], [-2], [-30], [-18], [34], [-6], [-16], [12], [-4], [-34], [52], [8], [52], [24], [-34], [30], [32], [-32], [2], [-10], [0], [10], [-4], [30], [50], [16], [22], [-6], [32], [8], [-42], [-46], [-18], [-8], [-40], [-24], [-36], [2], [-18], [38], [48], [22], [-48], [30], [-18], [-24], [-52], [-4], [46], [8], [-30], [22], [20], [-10], [60], [6], [14], [26], [44], [28], [8], [0], [28], [-14], [16], [-50], [30], [-16], [38], [-36], [18], [66], [20], [10], [-44], [38], [50], [4], [-42], [-44], [26], [-64], [-44], [14], [-54], [4], [-40], [-14], [-22], [36], [-18], [-52], [8], [-8], [-32], [-50], [-40], [2], [-28], [54], [-32], [16], [-42], [20], [-20], [-64], [66], [-40], [-44], [60], [34], [-74], [16], [46], [14], [-24], [-22], [-60], [-64], [8], [30], [70], [-4], [18], [42], [48], [8], [-6], [0], [-56], [66], [62], [-8], [-10], [36], [-4], [20], [-14], [24], [58], [-16], [30], [44], [-52], [46], [-4], [44], [-14], [-8], [-60], [52], [56], [10], [-40], [70], [64], [58], [-56], [2], [54], [36], [72], [80], [22], [-34], [8], [-26], [6], [36], [30], [32], [28], [-86], [20], [24], [24], [-4], [38], [-42], [22], [46], [44], [80], [-14], [12], [62], [-56], [-70], [24], [36], [-72], [-24], [78], [-44], [-48], [-26], [-16], [16], [14], [-70], [44], [-10], [22], [68], [-30], [-30], [-38], [38], [-52], [18], [16], [10], [-64], [34], [88], [-30], [-8], [-50], [14], [-16], [-52], [6], [32], [76], [-42], [-56], [-82], [0], [-48], [58], [-2], [82], [64], [-16], [4], [-74], [8], [0], [34], [-80], [-26], [-90], [-44], [0], [-66], [-78], [44], [-40], [54], [-62], [-34], [40], [-48], [-8], [-56], [60], [80], [-54], [58], [2], [-12], [6], [0], [-30], [74], [64], [-12], [48], [-4], [96], [10], [-16], [-86], [24], [80], [-26], [68], [-22], [96], [52], [-54], [46], [-88], [2], [66], [16], [22], [-42], [-38], [56], [-34], [-66], [38], [38], [-94], [30], [-24], [-60], [-46], [48], [90], [48], [64], [38], [-86], [-58], [30], [-62], [-58], [-24]];
    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_d();".
// 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_d(:prec:=1)
    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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-2, 1]>,<11,R![6, 1]>],Snew);
    return Vf;
end function;