
// Make newform 4018.2.a.i in Magma, downloaded from the LMFDB on 18 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_i();".
// 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_i();".
// 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], [-4], [0], [4], [-4], [2], [6], [-8], [6], [-4], [2], [1], [0], [-4], [2], [-14], [-4], [-8], [8], [14], [-8], [-6], [-10], [-6], [0], [12], [-4], [6], [-10], [-8], [-6], [-6], [22], [-6], [16], [-4], [-24], [-20], [-24], [-24], [-20], [0], [-18], [-10], [-20], [8], [-24], [6], [20], [10], [0], [26], [6], [-2], [-16], [0], [16], [-18], [10], [-14], [16], [26], [-8], [34], [-14], [-12], [30], [-4], [16], [22], [24], [-8], [-34], [-32], [-20], [18], [28], [-34], [2], [6], [-6], [0], [-14], [-8], [12], [-30], [38], [16], [-32], [-14], [36], [32], [-20], [32], [0], [-20], [-14], [34], [-10], [36], [-6], [10], [-42], [12], [38], [-38], [6], [24], [-34], [-16], [10], [22], [-30], [-8], [-18], [-22], [-28], [30], [-12], [4], [-14], [0], [-8], [-34], [2], [46], [4], [-28], [-20], [-8], [-8], [16], [-18], [10], [26], [48], [18], [16], [22], [42], [-18], [-16], [16], [-12], [36], [-16], [18], [6], [56], [-14], [-18], [8], [20], [-12], [40], [-32], [-38], [-2], [28], [24], [-22], [-16], [-18], [18], [-36], [-40], [8], [-2], [20], [-32], [-16], [-8], [-6], [52], [-42], [32], [18], [16], [40], [-8], [54], [14], [-50], [16], [-24], [30], [-6], [10], [12], [-58], [16], [52], [-24], [36], [-66], [6], [46], [14], [-24], [6], [-28], [-56], [-10], [30], [56], [4], [-60], [-66], [6], [-2], [0], [-24], [2], [40], [42], [32], [18], [0], [-54], [54], [40], [10], [-32], [-30], [6], [-2], [8], [0], [24], [-30], [-38], [8], [42], [-10], [48], [2], [26], [60], [-16], [-12], [22], [-48], [6], [-66], [-44], [-16], [-50], [-24], [24], [62], [30], [56], [-26], [-56], [-12], [-2], [18], [40], [-62], [-40], [44], [-32], [40], [50], [-58], [18], [30], [-28], [46], [-12], [-40], [-74], [-40], [-26], [-32], [12], [10], [38], [30], [-8], [8], [12], [12], [22], [0], [74], [30], [4], [54], [2], [42], [-18], [18], [2], [-4], [48], [-8], [-78], [50], [-70], [-26], [-40], [-60], [16], [58], [24], [76], [-64], [-46], [24], [78], [-38], [8], [0], [-90], [-10], [88], [54], [-30], [6], [18], [-68], [-64], [-42], [26], [-84], [-10], [-88], [26], [-6], [-78], [-44], [66], [-44], [78], [46], [-18], [78], [20], [58], [54], [12], [-8], [-6], [-72], [-72], [64], [-24], [-12], [-94], [-22], [50], [-76], [42], [-66], [-36], [18], [-78], [48], [62], [6], [40], [4], [22], [-82], [84], [-40], [6], [-80], [50], [24], [-46], [40], [-38], [18], [-24], [-2], [-6], [-62], [-60], [6], [-48], [-46], [58], [40], [-64], [72], [36], [-52], [70], [-56], [56], [-34], [24], [58], [-44], [-6], [4], [70], [-60], [74], [-88], [-6], [8], [20], [-34], [62], [66], [6], [-26], [66], [28], [-80], [-70], [64], [72], [4], [10], [-96], [-62], [-80], [-72], [26], [26], [-44], [84], [-38], [26], [-36]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : 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![4, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;