
// Make newform 4998.2.a.x in Magma, downloaded from the LMFDB on 25 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_x();".
// 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_x();".
// 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], [4], [0], [0], [6], [1], [-4], [6], [-4], [6], [-4], [10], [-4], [-4], [-2], [-12], [4], [-12], [-6], [-2], [10], [12], [2], [-6], [-14], [-4], [0], [16], [2], [8], [16], [-6], [-8], [-6], [-24], [-6], [12], [2], [4], [-12], [20], [-4], [6], [8], [-14], [8], [4], [-4], [2], [-6], [-20], [18], [-12], [-18], [-12], [12], [16], [-8], [-18], [-32], [-2], [12], [-30], [26], [-16], [20], [-6], [-4], [30], [-14], [0], [-10], [-14], [-4], [-28], [-14], [-20], [30], [26], [12], [34], [14], [18], [10], [12], [-26], [22], [10], [-4], [36], [-10], [-38], [-20], [32], [26], [-26], [10], [20], [-12], [-8], [14], [4], [-6], [44], [30], [-36], [30], [-16], [38], [38], [30], [-6], [20], [20], [-2], [-4], [-28], [20], [12], [14], [-26], [-8], [12], [-16], [-18], [-16], [-42], [8], [-18], [-12], [38], [-34], [14], [-10], [-26], [-42], [0], [-46], [-26], [12], [0], [18], [16], [-6], [30], [16], [10], [-20], [-48], [-24], [-50], [-52], [-18], [24], [26], [16], [34], [-22], [-48], [-24], [22], [-44], [-20], [2], [38], [-34], [20], [-18], [48], [4], [14], [-46], [-26], [-56], [-2], [52], [0], [-44], [-62], [-12], [8], [20], [6], [36], [-14], [44], [44], [6], [42], [-50], [-4], [20], [54], [12], [-18], [-22], [-28], [-18], [0], [48], [-10], [-22], [-18], [-52], [62], [-20], [-52], [26], [-4], [22], [30], [14], [36], [-66], [-62], [12], [-42], [-6], [-66], [-10], [6], [-50], [58], [28], [-62], [62], [38], [-8], [4], [-14], [68], [-60], [-58], [76], [8], [-70], [-32], [-48], [48], [-44], [-68], [-32], [-10], [30], [-14], [-34], [-12], [44], [0], [70], [-2], [12], [-18], [14], [36], [-44], [76], [72], [-58], [-46], [-76], [-12], [8], [26], [-44], [70], [-70], [28], [42], [72], [4], [-26], [-76], [50], [52], [-44], [10], [6], [-20], [-36], [-14], [-42], [62], [16], [56], [-58], [-72], [-72], [-14], [-12], [84], [74], [56], [36], [12], [-76], [-70], [-60], [-68], [74], [-78], [-10], [-20], [52], [22], [44], [28], [-24], [30], [26], [-52], [22], [-4], [16], [74], [-84], [-58], [6], [-34], [28], [70], [-42], [20], [-18], [54], [-24], [24], [34], [-16], [-8], [-88], [26], [28], [-44], [56], [-30], [30], [22], [12], [10], [10], [8], [-60], [62], [20], [20], [4], [26], [-48], [-76], [-18], [-30], [70], [22], [54], [80], [-28], [-18], [68], [52], [-98], [36], [-92], [44], [54], [84], [32], [-98], [68], [-16], [38], [78], [-32], [-32], [-60], [18], [-18], [-14], [-26], [60], [-58], [38], [42], [-4], [-10], [-32], [2], [-76], [-24], [66], [-24], [60], [60], [12], [-66], [-16], [42], [-32], [30], [100], [2], [4], [-102], [70], [-18], [94], [-46], [-28], [28], [-26], [-54], [-68], [-16], [62], [40], [-18], [62], [-94], [-60], [30], [0], [-78], [100], [-70], [86], [72], [66], [-4], [78]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4998_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 1]>,<11,R![0, 1]>,<13,R![-6, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;