
// Make newform 4018.2.a.n 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_n();".
// 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_n();".
// 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], [0], [4], [0], [-2], [6], [6], [-4], [8], [-8], [0], [-2], [1], [-8], [0], [0], [-2], [8], [10], [-12], [-10], [16], [2], [10], [2], [10], [-4], [12], [4], [-18], [-16], [-10], [6], [-14], [0], [-8], [14], [-16], [0], [4], [-10], [2], [-20], [6], [-6], [0], [-6], [24], [20], [10], [10], [24], [10], [-6], [-6], [-16], [16], [4], [2], [-22], [6], [-26], [-26], [-24], [-26], [0], [-14], [2], [6], [16], [-18], [0], [-28], [-18], [28], [16], [-26], [34], [-34], [-14], [-34], [-28], [32], [-30], [32], [32], [-6], [6], [36], [16], [-18], [-32], [0], [-8], [-26], [0], [18], [14], [-38], [22], [2], [12], [-28], [-10], [-14], [14], [0], [-38], [16], [6], [-44], [2], [-18], [-18], [-24], [-42], [-20], [12], [-24], [26], [20], [-18], [0], [-26], [32], [46], [48], [-16], [40], [-40], [-8], [0], [48], [-32], [-6], [26], [14], [-26], [-48], [6], [30], [-22], [36], [30], [28], [-8], [-12], [26], [26], [16], [34], [-42], [-2], [40], [-8], [-40], [-36], [2], [-22], [24], [-52], [6], [-44], [20], [-42], [-24], [4], [-22], [-46], [-6], [-6], [28], [36], [-2], [8], [-10], [-58], [-10], [40], [-14], [32], [-6], [-20], [22], [24], [20], [62], [-26], [-10], [-32], [46], [18], [20], [-8], [36], [14], [-38], [-12], [-18], [-32], [66], [28], [50], [-30], [56], [42], [12], [46], [30], [-50], [62], [42], [-40], [-26], [24], [-30], [-56], [-30], [36], [-54], [-32], [44], [22], [-8], [-14], [-16], [-2], [16], [8], [-72], [-2], [16], [-40], [30], [-44], [8], [-50], [-8], [24], [64], [50], [28], [40], [-46], [62], [-64], [-12], [-16], [-12], [0], [-54], [62], [16], [-34], [42], [-52], [28], [-4], [-2], [10], [0], [50], [-6], [26], [-38], [14], [16], [54], [-40], [52], [-10], [-44], [46], [32], [-2], [68], [46], [-68], [-22], [12], [-24], [-48], [24], [-24], [20], [12], [-26], [62], [32], [62], [60], [50], [-38], [18], [28], [-14], [-48], [36], [32], [32], [-6], [-4], [-64], [-26], [32], [-2], [-36], [76], [-20], [32], [48], [-8], [-2], [-24], [-32], [-10], [42], [64], [26], [42], [38], [-74], [-40], [-4], [54], [-58], [62], [52], [88], [30], [-60], [54], [-56], [-4], [8], [-32], [0], [-74], [38], [-60], [-8], [-30], [26], [36], [74], [36], [-68], [12], [40], [-20], [40], [-14], [28], [-44], [48], [-86], [84], [38], [66], [-16], [-26], [70], [24], [96], [52], [74], [62], [-32], [-46], [18], [-16], [16], [-36], [4], [68], [-46], [0], [86], [66], [42], [38], [-94], [64], [2], [-92], [-48], [44], [-50], [4], [72], [26], [14], [-62], [50], [48], [14], [96], [50], [48], [36], [4], [-10], [32], [54], [-72], [-40], [78], [-50], [12], [-102], [38], [-22], [-78], [-10], [26], [-52], [-48], [40], [-30], [72], [62], [-18], [40], [40], [2], [-32], [18], [66], [-56], [56]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![-4, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;