
// Make newform 4018.2.a.a 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_a();".
// 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_a();".
// 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], [-3], [-1], [0], [-5], [6], [0], [1], [-4], [-8], [6], [-2], [-1], [4], [0], [6], [2], [13], [16], [-3], [-2], [-11], [-14], [14], [-8], [8], [-14], [-6], [16], [21], [2], [4], [-18], [2], [18], [-8], [4], [-10], [0], [5], [-1], [-2], [16], [6], [-3], [0], [3], [-24], [13], [-16], [-14], [-21], [-22], [18], [0], [-16], [-1], [8], [-31], [20], [-6], [-28], [-28], [-24], [-16], [12], [-20], [-13], [-27], [2], [-27], [-30], [-2], [-9], [22], [35], [13], [2], [-1], [29], [4], [2], [-16], [27], [-11], [-28], [9], [-36], [-3], [1], [12], [35], [6], [-26], [-8], [27], [36], [10], [-16], [10], [-28], [-6], [13], [-10], [37], [-38], [15], [44], [16], [-6], [-10], [-25], [-3], [-30], [-42], [-42], [17], [18], [-48], [20], [43], [-18], [27], [-8], [-41], [13], [0], [25], [32], [46], [-2], [0], [39], [28], [-30], [-5], [-2], [-28], [-30], [-24], [-18], [5], [-9], [24], [-22], [-1], [-39], [13], [-2], [-50], [-53], [-15], [4], [-19], [10], [-4], [-51], [-14], [-44], [-39], [-16], [-6], [37], [4], [-6], [-36], [-17], [-14], [38], [-6], [27], [2], [3], [37], [31], [-44], [-43], [5], [11], [-34], [62], [0], [-50], [-10], [12], [-47], [2], [26], [26], [29], [26], [-45], [50], [-37], [-6], [-26], [-20], [36], [-42], [41], [-51], [26], [-20], [-42], [16], [-18], [-54], [-11], [-9], [-4], [50], [30], [14], [-52], [-9], [-15], [-47], [54], [-48], [30], [40], [4], [-56], [43], [-16], [-4], [-1], [-44], [55], [12], [73], [-28], [2], [63], [-7], [43], [14], [-2], [24], [8], [-4], [41], [29], [38], [-26], [-32], [30], [-35], [36], [18], [9], [73], [-74], [-35], [18], [-10], [4], [-44], [-22], [-28], [-42], [17], [-42], [76], [-25], [-26], [-50], [-33], [44], [40], [28], [-44], [-55], [44], [-16], [-68], [52], [46], [20], [51], [-12], [9], [-36], [33], [-68], [-3], [-26], [-1], [64], [-50], [0], [34], [-20], [24], [-62], [-46], [27], [-63], [55], [37], [-63], [-82], [-52], [-44], [-64], [58], [-12], [23], [67], [26], [24], [58], [-29], [-12], [-38], [31], [24], [-8], [64], [18], [40], [49], [-35], [-19], [66], [-92], [11], [71], [1], [-15], [66], [30], [-4], [61], [-46], [-52], [66], [-34], [34], [0], [16], [27], [34], [-15], [-46], [30], [-7], [-18], [32], [2], [-28], [-14], [28], [-76], [-54], [-76], [-66], [58], [-78], [-22], [-29], [20], [3], [-30], [20], [-34], [-44], [-26], [65], [9], [37], [16], [6], [-22], [-88], [64], [-48], [-101], [60], [72], [-62], [-67], [52], [23], [-55], [24], [71], [2], [76], [63], [-64], [70], [25], [-74], [-60], [-58], [45], [1], [-60], [0], [-85], [32], [-28], [30], [-6], [88], [51], [-20], [75], [54], [7], [35], [-63], [-25], [50], [-74], [6], [-61], [-96], [-60], [-58], [18], [10], [-55], [61], [-7], [-27], [60], [5], [55]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4018_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![1, 1]>,<11,R![5, 1]>],Snew);
    return Vf;
end function;