
// Make newform 4002.2.a.a in Magma, downloaded from the LMFDB on 22 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4002_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_4002_2_a_a();".
// 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_4002_2_a_a();".
// 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_4002_a();"
function MakeCharacter_4002_a()
    N := 4002;
    order := 1;
    char_gens := [2669, 3133, 553];
    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_4002_a_Hecke( : Kf := Kf)
    return MakeCharacter_4002_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]];
    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], [-2], [-4], [-2], [-6], [-2], [1], [-1], [-8], [2], [-6], [-2], [-8], [-6], [0], [-10], [16], [-8], [14], [8], [-14], [-6], [-8], [-10], [2], [-2], [-8], [-10], [-16], [-12], [6], [12], [18], [8], [-14], [4], [16], [6], [-16], [-12], [10], [6], [10], [22], [0], [4], [-14], [-10], [-14], [-24], [-18], [-4], [10], [30], [-6], [16], [18], [8], [-24], [-20], [20], [-12], [2], [-6], [-4], [36], [28], [-30], [6], [-10], [8], [-4], [-2], [8], [-24], [22], [-24], [30], [-14], [-2], [12], [-24], [0], [12], [18], [-14], [2], [36], [-32], [-18], [-12], [12], [36], [6], [6], [12], [24], [-30], [-28], [-26], [-4], [6], [-36], [38], [32], [-6], [40], [-2], [40], [-44], [-42], [-26], [-30], [-2], [-32], [-40], [-18], [-48], [-36], [-30], [-48], [8], [-20], [-42], [32], [0], [-40], [-30], [16], [30], [8], [-2], [14], [16], [32], [-24], [-12], [26], [-4], [-10], [-40], [48], [-10], [-30], [-10], [2], [-28], [-24], [-22], [-30], [12], [12], [-38], [-18], [-46], [10], [-22], [46], [-36], [-24], [-48], [-60], [-16], [26], [-8], [-26], [-50], [-18], [2], [18], [48], [8], [-24], [-20], [4], [6], [-4], [-52], [0], [-20], [-46], [30], [60], [-62], [18], [46], [6], [0], [-46], [-36], [-14], [-10], [4], [16], [-36], [8], [-8], [24], [36], [56], [-58], [20], [-48], [18], [-64], [16], [18], [20], [-46], [18], [-4], [-28], [68], [-30], [48], [46], [16], [6], [-14], [42], [-34], [-32], [52], [-6], [-6], [-28], [48], [-28], [18], [32], [-32], [6], [-52], [58], [-12], [24], [8], [-12], [-48], [76], [64], [14], [18], [24], [44], [58], [20], [-34], [34], [12], [-38], [-56], [22], [-24], [-50], [-66], [62], [-42], [72], [-48], [26], [26], [-50], [50], [-60], [10], [54], [-26], [42], [40], [10], [20], [-58], [-16], [36], [-58], [-52], [22], [-48], [62], [-60], [-32], [4], [28], [-4], [28], [44], [-50], [42], [-18], [-50], [32], [62], [-30], [4], [42], [-88], [-44], [-4], [46], [80], [60], [-2], [-14], [12], [4], [60], [-2], [24], [-20], [-8], [-60], [32], [86], [36], [72], [-14], [54], [-62], [-22], [-38], [-88], [16], [0], [20], [20], [-58], [54], [70], [34], [-20], [76], [-38], [24], [-20], [34], [-70], [26], [26], [12], [-58], [-82], [-66], [-36], [38], [-80], [6], [20], [36], [-58], [34], [-6], [-74], [-26], [14], [-82], [62], [72], [80], [54], [-72], [50], [32], [-34], [54], [-2], [-48], [-52], [-20], [56], [-48], [30], [-58], [52], [42], [-72], [-36], [-74], [56], [-24], [-16], [-42], [-56], [80], [16], [-22], [54], [58], [-32], [-6], [28], [-100], [6], [54], [6], [-26], [4], [30], [22], [-54], [4], [76], [-26], [40], [80], [-34], [14], [-24], [82], [86], [26], [0], [16], [42], [-92], [-44], [46], [-106], [-34], [78], [96], [-92], [6], [18], [46], [-106], [36], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4002_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_a();".
// 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_4002_2_a_a(:prec:=1)
    chi := MakeCharacter_4002_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_4002_2_a_a();".
// 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_4002_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![2, 1]>,<11,R![4, 1]>],Snew);
    return Vf;
end function;