
// Make newform 4998.2.a.v 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_v();".
// 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_v();".
// 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], [2], [0], [2], [-4], [1], [2], [6], [8], [4], [-6], [-6], [4], [-4], [2], [-4], [10], [4], [2], [-2], [-8], [12], [-2], [10], [2], [14], [-2], [-2], [16], [8], [-12], [2], [12], [-10], [16], [-4], [12], [16], [-14], [-24], [-2], [-24], [26], [12], [-20], [20], [-2], [4], [-16], [20], [24], [10], [-16], [-2], [-4], [14], [-10], [10], [-6], [-8], [-6], [-2], [-8], [-6], [12], [20], [-22], [6], [-24], [-6], [-8], [-8], [10], [-12], [-36], [-6], [38], [-12], [12], [-4], [30], [-2], [-16], [0], [0], [36], [-10], [18], [8], [-8], [0], [4], [-24], [24], [40], [-26], [-26], [26], [34], [40], [42], [-44], [18], [-12], [-20], [-16], [-30], [-36], [-10], [36], [14], [28], [-32], [-24], [-8], [0], [24], [20], [-44], [32], [38], [-42], [-14], [-32], [-2], [22], [-24], [-10], [4], [-44], [6], [-12], [-26], [-10], [44], [18], [-32], [18], [12], [44], [-28], [-20], [-18], [40], [48], [-46], [-42], [-6], [12], [-6], [50], [-20], [0], [-8], [22], [0], [-54], [12], [30], [38], [6], [-56], [-40], [-46], [-40], [-40], [-46], [22], [-42], [0], [-48], [-30], [-18], [34], [10], [-40], [24], [-22], [4], [48], [-44], [6], [18], [-40], [-38], [2], [30], [34], [-8], [14], [-26], [-52], [-54], [-14], [-10], [-2], [-2], [14], [36], [-16], [20], [-40], [-28], [-12], [-54], [-38], [-36], [-44], [50], [-66], [-42], [4], [-12], [56], [50], [-32], [18], [42], [14], [-26], [36], [6], [-4], [-48], [-58], [50], [34], [-42], [-54], [-6], [-20], [-8], [2], [6], [-4], [-26], [64], [-26], [-16], [-26], [46], [62], [26], [-18], [-48], [-8], [12], [4], [-20], [-66], [2], [36], [-26], [18], [56], [62], [-32], [30], [-64], [-4], [-32], [-46], [54], [-66], [62], [42], [-6], [44], [34], [-78], [-20], [-16], [-8], [-64], [26], [32], [82], [-38], [0], [-68], [-40], [-8], [-40], [-44], [-36], [62], [-72], [46], [78], [-12], [-52], [-66], [44], [78], [34], [-2], [86], [84], [-50], [-20], [-6], [4], [54], [-24], [-46], [-16], [-46], [-12], [82], [24], [72], [-72], [-52], [-8], [32], [-40], [2], [-50], [70], [-52], [38], [-54], [-56], [56], [48], [0], [-84], [18], [56], [2], [72], [-24], [-8], [20], [52], [82], [-50], [86], [-66], [74], [-54], [-14], [24], [-34], [6], [-18], [-92], [24], [-54], [-54], [62], [-10], [8], [86], [-46], [-48], [20], [-32], [68], [38], [-90], [48], [-2], [-50], [-42], [30], [-40], [98], [0], [56], [-10], [26], [94], [-86], [20], [-50], [80], [50], [28], [-30], [-66], [-36], [-16], [44], [-72], [72], [48], [56], [-96], [-94], [22], [-100], [10], [-44], [82], [-54], [18], [-16], [-18], [-46], [-50], [8], [82], [78], [-40], [-14], [30], [26], [-84], [74], [18], [-16], [-104], [6], [78], [-30], [-68], [-26], [96], [38], [-26], [18], [12], [58], [-38], [-6], [84], [-38], [-64]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4998_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-2, 1]>,<13,R![4, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;