
// Make newform 4998.2.a.bs in Magma, downloaded from the LMFDB on 19 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_bs();".
// 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_bs();".
// 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
        poly := [-10, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, 0], [-1, 0], [0, -1], [0, 0], [0, 1], [-2, 1], [1, 0], [-6, 0], [4, -2], [0, -2], [0, 0], [-2, 1], [-6, 2], [6, -1], [8, 0], [-4, -1], [2, -2], [2, 0], [6, 1], [-4, 4], [8, 1], [10, 1], [-12, 1], [-12, 1], [-4, 1], [4, 4], [-6, 5], [-10, -2], [0, 2], [4, 3], [-12, 2], [0, 2], [-6, -2], [2, -1], [-16, -1], [8, -4], [0, -6], [-10, 0], [2, -5], [10, 0], [8, -6], [2, 2], [-2, -3], [-6, 6], [-12, 0], [8, -4], [-2, -4], [-12, -4], [-12, 4], [6, 3], [-20, -1], [18, 3], [10, 4], [-12, 7], [-8, -5], [-18, -3], [-14, 0], [6, -5], [-4, 2], [-10, 4], [-14, 3], [-8, 0], [-10, 6], [2, 3], [6, 4], [-8, -2], [10, 1], [-22, 4], [8, 3], [-6, -5], [-12, 5], [0, -8], [0, -4], [-2, -2], [2, 8], [12, -4], [10, 4], [-14, -2], [22, 0], [2, -4], [-4, 6], [-22, 0], [0, 0], [6, -6], [-16, -4], [-16, 8], [16, -1], [20, -3], [20, 2], [0, -8], [-2, 6], [-26, -3], [-30, 3], [24, 2], [2, 2], [16, 0], [-4, -4], [14, -2], [10, -10], [-6, -7], [-2, -4], [-22, 0], [-8, 9], [-26, 8], [10, -10], [-2, 8], [-28, 1], [22, -8], [0, 0], [-18, -4], [0, 0], [-34, -2], [-12, -1], [-22, 1], [16, 4], [0, 7], [14, -1], [-16, -8], [-24, -6], [16, 8], [-8, -2], [-14, 6], [-8, -9], [-22, 2], [-2, -11], [28, 5], [10, -11], [24, 0], [22, -7], [-16, -10], [16, 4], [-8, 6], [32, -4], [6, 6], [-36, 3], [2, 4], [16, -12], [-22, -7], [-36, -2], [6, -4], [-6, 3], [-16, 2], [26, 1], [-12, -7], [30, 1], [38, -3], [-22, 8], [-6, -10], [18, 0], [-2, 1], [6, -1], [-26, -4], [40, 4], [-2, 13], [-30, 6], [-4, -12], [-16, 8], [38, -2], [10, -14], [-32, -3], [-6, 6], [-6, -4], [16, 12], [-12, -11], [30, 2], [24, 0], [2, -7], [22, -10], [-22, 6], [0, -5], [-56, -2], [2, -13], [0, -4], [-12, -9], [12, 0], [18, -4], [-34, 8], [-20, 2], [-20, 4], [-44, 2], [24, -8], [4, 2], [12, -14], [14, -12], [-18, -3], [-56, 0], [-14, 1], [-22, 4], [-34, 4], [14, -3], [20, 1], [-4, 19], [26, -7], [16, -6], [36, -3], [18, 10], [46, -2], [26, -9], [14, 8], [20, 6], [44, -8], [20, 6], [-10, 17], [6, -8], [0, 13], [12, -10], [26, -3], [0, 8], [10, 12], [-2, 0], [-6, 10], [4, 10], [34, -3], [-4, -5], [38, -3], [10, 8], [-28, 2], [-12, -1], [-36, -2], [16, 7], [-10, 14], [-12, 10], [-18, 8], [10, -7], [0, -5], [-22, -2], [26, -4], [-8, 8], [-26, 5], [-38, 2], [34, 10], [20, 8], [4, -10], [6, 2], [14, 0], [24, -6], [-24, -1], [-28, -8], [-10, 10], [-8, -6], [-8, -9], [-10, 10], [-42, -1], [-2, -20], [-22, -8], [10, -1], [56, -4], [12, 4], [-22, 17], [-30, -1], [-18, -4], [14, 14], [-24, 16], [-4, -7], [32, 8], [20, 10], [34, -5], [-36, 0], [-20, 3], [54, -6], [-18, 15], [12, -10], [-50, -2], [46, 8], [-18, 8], [18, 4], [10, 0], [4, 13], [30, 6], [-56, 1], [14, -14], [-46, -5], [10, 6], [16, 4], [-34, 12], [-30, 9], [12, 4], [-10, -6], [-48, -1], [40, -1], [0, -14], [34, -5], [54, -7], [-6, 1], [-22, 9], [-6, 3], [46, -4], [-52, 8], [-28, -12], [-32, 3], [44, 6], [-20, 10], [22, -6], [32, -16], [-34, 7], [-56, 1], [58, -9], [-72, -4], [-4, -2], [-50, -2], [-2, -18], [40, -7], [6, -15], [40, -7], [-2, -10], [-14, -2], [36, -6], [38, -2], [48, 0], [42, 2], [-16, -16], [80, -2], [8, -13], [24, -12], [22, 13], [50, 2], [-6, -18], [60, 6], [-14, -4], [-30, 14], [26, 11], [54, -4], [8, -12], [4, -10], [54, 0], [-50, 10], [-34, 0], [-18, 9], [52, 6], [-36, 2], [34, -13], [44, -10], [4, 6], [-12, 5], [-38, -4], [44, -6], [10, -2], [-34, 6], [-16, 13], [42, -9], [-6, -4], [-54, 6], [-28, -3], [-36, 2], [-76, -3], [16, -13], [14, -4], [-14, -11], [-34, -7], [-20, 19], [74, -8], [30, -16], [-28, 17], [40, 0], [22, 8], [-40, -9], [32, 16], [0, 0], [22, 0], [-6, -13], [-34, 19], [38, 2], [26, 11], [20, -1], [22, -10], [40, 5], [-20, 7], [64, 0], [22, -4], [-8, 16], [44, 4], [32, 2], [-4, 3], [28, -20], [68, -10], [16, 16], [32, 10], [6, -12], [8, -7], [14, -18], [48, -11], [-34, 2], [18, -11], [-52, 15], [6, -11], [14, 5], [-20, 10], [-12, -6], [18, 18], [16, 6], [-18, -2], [-40, 9], [4, -4], [38, 12], [-10, -15], [6, 0], [-50, 5], [-14, -12], [-10, 0], [-8, -9], [34, -20], [-14, -2], [-24, 8], [46, -2], [-12, 0], [-4, 2], [-14, -18], [18, -20], [-66, 2], [-44, -8], [-36, 21], [-16, 1], [-44, 18], [10, -16], [22, 16], [60, 1], [-80, 6], [44, 6], [58, 4], [-28, 24], [-56, 11], [2, -26], [-20, 4], [-64, -1], [36, -1], [-52, 6], [36, -3], [26, 16], [-18, -10], [-64, 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_bs();".
// 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_bs(:prec:=2)
    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_bs();".
// 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_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4998_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-10, 1, 1]>,<11,R![-10, -1, 1]>,<13,R![-8, 3, 1]>,<23,R![-32, -6, 1]>],Snew);
    return Vf;
end function;