
// Make newform 4018.2.a.p 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_p();".
// 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_p();".
// 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], [1], [1], [0], [-6], [4], [-7], [0], [-8], [1], [-5], [-2], [-1], [-5], [6], [-3], [10], [3], [14], [3], [-8], [7], [2], [-5], [-5], [-6], [11], [-13], [2], [1], [6], [20], [-4], [18], [-15], [-15], [0], [-4], [-4], [1], [0], [-14], [-25], [-4], [12], [-4], [28], [-21], [-3], [20], [-6], [-28], [18], [20], [-19], [-16], [-6], [12], [-26], [-10], [-6], [14], [4], [34], [-22], [2], [-12], [-27], [28], [-6], [-32], [20], [-15], [-2], [5], [-30], [-34], [12], [-25], [-18], [-36], [39], [-28], [-24], [-22], [-9], [-33], [6], [37], [-16], [-6], [2], [4], [-3], [2], [-36], [20], [2], [-20], [-22], [22], [-21], [12], [39], [40], [-2], [39], [33], [44], [-39], [-5], [28], [46], [-16], [-26], [-2], [29], [27], [17], [-36], [-18], [50], [-6], [4], [17], [-20], [-31], [-12], [-8], [11], [-29], [-12], [16], [-7], [-50], [34], [42], [8], [19], [2], [-48], [-8], [-1], [22], [29], [-26], [37], [-42], [-8], [36], [-44], [4], [34], [-48], [43], [-42], [25], [6], [43], [-50], [-28], [22], [-29], [-39], [-6], [3], [24], [20], [-43], [-48], [26], [49], [36], [-21], [-8], [21], [16], [-18], [60], [-2], [-58], [42], [26], [-54], [10], [25], [-14], [-44], [-24], [2], [44], [34], [36], [-13], [-7], [44], [-48], [34], [-3], [54], [58], [44], [36], [34], [-19], [4], [-29], [60], [-66], [32], [38], [-18], [-40], [-38], [60], [28], [11], [-30], [-47], [-56], [37], [19], [36], [8], [0], [-43], [-70], [54], [48], [-4], [-24], [-19], [-2], [-9], [0], [-6], [-29], [-66], [-47], [-6], [-8], [-61], [0], [-16], [40], [21], [-56], [31], [-15], [30], [42], [22], [-8], [-52], [42], [9], [-73], [20], [-72], [25], [-4], [18], [10], [-24], [-12], [26], [-3], [-10], [56], [2], [40], [3], [-28], [32], [-19], [36], [48], [-70], [62], [-68], [52], [-51], [-81], [-38], [-45], [17], [12], [30], [46], [41], [67], [-20], [-68], [30], [-78], [-62], [86], [-86], [55], [-12], [-44], [42], [-40], [-36], [7], [13], [-52], [-14], [-55], [-47], [16], [-15], [15], [61], [-32], [38], [10], [0], [-23], [-12], [26], [-6], [-2], [-13], [7], [14], [-16], [-37], [-59], [34], [-41], [-40], [-7], [49], [-65], [84], [5], [68], [-43], [-92], [33], [-5], [-70], [11], [16], [69], [74], [-84], [10], [-66], [-13], [-21], [67], [-80], [-31], [86], [15], [-40], [-70], [-48], [62], [59], [-48], [-41], [25], [-38], [-18], [54], [31], [14], [-27], [-24], [82], [80], [3], [24], [-90], [-2], [-90], [-55], [90], [51], [-14], [-65], [-57], [-89], [33], [58], [-51], [18], [-8], [-24], [54], [64], [18], [-76], [-80], [102], [-27], [75], [77], [-78], [-52], [6], [35], [50], [-70], [-8], [17], [66], [-46], [76], [-72], [-8], [2], [-15], [-40], [68], [41], [47], [-42], [-54], [54], [64], [16], [-30], [-44], [4], [100], [60]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : 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]>,<11,R![6, 1]>],Snew);
    return Vf;
end function;