
// Make newform 4998.2.a.q in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4998_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_4998_2_a_q();".
// 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_4998_2_a_q();".
// 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_4998_a();"
function MakeCharacter_4998_a()
    N := 4998;
    order := 1;
    char_gens := [1667, 2551, 4117];
    v := [1, 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_4998_a_Hecke(Kf)
    return MakeCharacter_4998_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], [3], [3], [-1], [-6], [-2], [6], [-4], [-11], [-12], [3], [-12], [5], [-4], [6], [9], [12], [-15], [-11], [-7], [13], [-11], [12], [-1], [16], [6], [-15], [2], [10], [8], [1], [-15], [-16], [18], [-22], [5], [-10], [10], [-20], [-11], [-24], [0], [0], [14], [-16], [-8], [-19], [-3], [-21], [-14], [-9], [27], [-19], [-18], [9], [18], [-22], [13], [8], [8], [-3], [22], [18], [-3], [14], [9], [1], [9], [16], [-20], [-20], [22], [-20], [6], [-12], [14], [14], [6], [-10], [-4], [-32], [32], [4], [1], [-17], [30], [-16], [-32], [11], [-13], [-14], [20], [-24], [-8], [20], [4], [-31], [-38], [42], [21], [-42], [-16], [-30], [17], [34], [-24], [46], [-12], [-28], [13], [11], [32], [33], [-3], [12], [-18], [48], [-10], [48], [-33], [0], [-21], [-1], [-19], [-16], [47], [38], [20], [-46], [-20], [-12], [-41], [-2], [-4], [-53], [-6], [-30], [21], [50], [-43], [7], [-37], [35], [34], [-12], [38], [-7], [-1], [-42], [20], [-13], [36], [40], [-16], [0], [-8], [21], [28], [-6], [16], [-3], [0], [0], [13], [-60], [24], [-13], [10], [1], [-24], [-35], [-4], [-18], [10], [-14], [56], [6], [32], [-42], [-6], [-18], [29], [-4], [-15], [62], [58], [35], [-11], [45], [29], [14], [-37], [-12], [-36], [-69], [22], [-30], [20], [-2], [67], [10], [49], [-38], [23], [-24], [-26], [22], [28], [-6], [-23], [35], [3], [62], [50], [51], [-10], [41], [56], [2], [-34], [-67], [47], [-4], [-46], [16], [-37], [32], [-64], [28], [2], [-40], [34], [-30], [-1], [48], [-36], [38], [5], [20], [69], [46], [-54], [57], [68], [-4], [-11], [-9], [-74], [72], [-32], [-3], [16], [10], [47], [-52], [23], [32], [-77], [78], [-24], [-46], [-22], [6], [-30], [1], [-4], [39], [-20], [19], [16], [-32], [-26], [-69], [68], [-8], [1], [-21], [-58], [3], [47], [-17], [19], [-29], [14], [-48], [8], [-5], [62], [-2], [-84], [-84], [-17], [-7], [-7], [4], [46], [32], [4], [-73], [-23], [19], [80], [60], [-22], [28], [48], [-52], [-64], [-30], [13], [60], [-3], [-60], [-60], [-70], [-70], [0], [-19], [46], [-36], [42], [-74], [84], [-10], [7], [-42], [-30], [-73], [-82], [50], [5], [70], [14], [68], [36], [-59], [-55], [50], [8], [-15], [22], [15], [49], [-46], [1], [-1], [-9], [74], [-86], [-69], [8], [-10], [15], [-20], [4], [6], [75], [-53], [52], [-13], [-15], [-76], [-5], [59], [-4], [-74], [100], [16], [78], [-71], [80], [-34], [-20], [50], [-34], [1], [-40], [-51], [-8], [-47], [-83], [19], [-45], [98], [-26], [-56], [2], [80], [9], [0], [42], [73], [-10], [-21], [50], [-82], [33], [62], [88], [68], [92], [-48], [-50], [-44], [-46], [-20], [24], [1], [23], [34], [90], [-2], [77], [-18], [42], [2], [-104], [-59], [88], [-48], [-45], [27], [30], [99], [-70], [40], [-16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4998_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4998_2_a_q();".
// 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_4998_2_a_q(:prec:=1)
    chi := MakeCharacter_4998_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_4998_2_a_q();".
// 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_4998_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4998_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![-3, 1]>,<13,R![-3, 1]>,<23,R![2, 1]>],Snew);
    return Vf;
end function;