
// Make newform 4018.2.a.b 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_b();".
// 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_b();".
// 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], [4], [6], [-3], [-4], [2], [1], [-9], [-8], [1], [-5], [6], [-3], [-14], [11], [-8], [3], [14], [7], [-16], [-5], [-1], [10], [-13], [-9], [-14], [-15], [2], [14], [-6], [22], [9], [1], [-4], [-4], [0], [1], [-4], [20], [-5], [12], [0], [0], [-12], [-9], [-19], [-20], [-20], [0], [16], [-24], [-27], [-16], [10], [16], [32], [-10], [-24], [16], [16], [-18], [-26], [6], [-2], [5], [-24], [-14], [-18], [24], [17], [36], [13], [-20], [28], [-2], [11], [-8], [-34], [-1], [20], [6], [-28], [-13], [15], [18], [21], [-32], [-24], [-14], [-18], [25], [22], [30], [-30], [26], [-26], [-20], [-4], [-21], [20], [-37], [22], [-22], [27], [1], [34], [21], [-41], [20], [-18], [0], [42], [-18], [-47], [27], [-15], [-22], [14], [36], [-30], [-8], [41], [-38], [-39], [14], [-8], [35], [7], [42], [-24], [-23], [-30], [26], [38], [28], [-9], [12], [-18], [-28], [39], [6], [13], [-44], [9], [-22], [26], [46], [16], [-48], [-32], [-50], [-17], [-40], [-27], [-58], [23], [18], [44], [-42], [-41], [17], [-42], [39], [40], [38], [-7], [30], [18], [-11], [-48], [19], [2], [-7], [32], [8], [-14], [10], [32], [-6], [34], [-20], [6], [41], [-40], [4], [-10], [-38], [-8], [6], [-28], [31], [21], [14], [16], [-54], [9], [-14], [-24], [-8], [-16], [-30], [-55], [6], [-21], [40], [-66], [-14], [-52], [-30], [-34], [22], [-30], [-48], [67], [-72], [-15], [6], [49], [35], [40], [-32], [40], [41], [-62], [28], [-10], [-36], [40], [-23], [8], [-33], [40], [8], [7], [46], [-75], [-26], [-22], [67], [22], [-34], [-16], [41], [24], [35], [-39], [24], [-18], [20], [-14], [14], [24], [53], [-17], [-52], [-50], [-47], [30], [-16], [-12], [80], [-50], [-28], [-47], [30], [-28], [-14], [68], [55], [76], [-70], [-71], [8], [22], [10], [8], [-60], [-78], [33], [-33], [42], [-13], [45], [-86], [-40], [-46], [29], [27], [-58], [-8], [-12], [82], [46], [-12], [42], [-13], [80], [0], [14], [20], [-20], [-49], [-47], [-36], [-14], [1], [29], [-36], [73], [-53], [33], [70], [-40], [30], [-32], [29], [30], [-4], [58], [2], [11], [3], [26], [26], [55], [37], [66], [-21], [-78], [61], [-19], [-49], [28], [81], [52], [-7], [60], [13], [-21], [-58], [87], [-76], [93], [-2], [-36], [46], [-2], [-29], [-77], [-5], [-56], [-63], [-26], [-21], [8], [42], [68], [-38], [91], [60], [-21], [37], [38], [44], [-50], [11], [30], [5], [34], [-6], [40], [-73], [98], [6], [-22], [-66], [-87], [20], [11], [-8], [-1], [-17], [15], [-19], [-98], [1], [-42], [-10], [74], [-14], [86], [-6], [-22], [-24], [-16], [-51], [75], [85], [44], [92], [-66], [63], [50], [66], [22], [81], [-84], [-46], [-64], [-12], [50], [20], [-91], [72], [-26], [45], [99], [68], [0], [58], [-44], [86], [-14], [24], [-78], [-44], [-46]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : 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![-4, 1]>],Snew);
    return Vf;
end function;