
// Make newform 4002.2.a.n in Magma, downloaded from the LMFDB on 19 April 2024.
// 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_n();".
// 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_n();".
// 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)
    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+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], [-4], [0], [-2], [6], [4], [-1], [1], [0], [10], [2], [4], [-8], [-10], [12], [10], [-4], [0], [10], [4], [8], [14], [10], [-10], [12], [0], [-14], [6], [8], [-4], [22], [12], [14], [-16], [18], [-4], [0], [-2], [12], [-6], [0], [2], [6], [4], [-28], [-8], [-24], [-22], [26], [0], [-30], [-16], [-30], [16], [-18], [8], [-2], [30], [-20], [14], [-20], [24], [-14], [30], [-4], [-30], [-12], [-10], [-6], [-8], [-12], [26], [-20], [-24], [6], [30], [-10], [-6], [-8], [-14], [24], [2], [-24], [-4], [10], [-22], [-18], [16], [8], [0], [40], [20], [36], [16], [-34], [22], [4], [30], [12], [38], [0], [-2], [36], [18], [20], [-46], [-16], [10], [-24], [2], [30], [-20], [12], [22], [28], [32], [-18], [-24], [-14], [2], [6], [-12], [12], [46], [-30], [-40], [4], [26], [36], [16], [4], [42], [18], [-14], [46], [28], [-2], [-22], [-44], [22], [-24], [-48], [-34], [-32], [14], [42], [44], [8], [22], [-10], [28], [0], [12], [40], [4], [50], [-38], [-10], [-4], [-34], [-56], [-56], [-2], [48], [0], [6], [2], [-10], [-24], [6], [48], [26], [-8], [-10], [-28], [6], [-28], [34], [-48], [8], [-2], [18], [-56], [-18], [-26], [-52], [10], [48], [18], [52], [44], [62], [-48], [54], [26], [-6], [30], [24], [-34], [-56], [6], [-46], [16], [-2], [-36], [-60], [-38], [-28], [-46], [-10], [-52], [-48], [8], [-46], [-56], [2], [-72], [-34], [-18], [-60], [18], [-20], [-12], [-50], [-34], [-24], [12], [12], [30], [4], [20], [26], [-4], [-24], [-70], [6], [36], [16], [32], [-52], [-56], [6], [66], [-56], [24], [-24], [-36], [-48], [34], [-2], [48], [42], [30], [20], [-70], [-28], [38], [26], [44], [-48], [-2], [-38], [-38], [36], [46], [54], [-20], [54], [-42], [-20], [50], [52], [-62], [-24], [12], [-82], [66], [-16], [-24], [76], [24], [82], [68], [0], [74], [54], [56], [10], [-10], [-72], [66], [-72], [-2], [70], [-52], [22], [-4], [-36], [-6], [6], [12], [-76], [28], [50], [28], [82], [-80], [-10], [40], [14], [78], [20], [16], [34], [-28], [-16], [-6], [-86], [28], [50], [-66], [-56], [46], [-70], [60], [-20], [0], [30], [78], [30], [-64], [-20], [-68], [-60], [18], [22], [-86], [20], [46], [-66], [-42], [-84], [30], [84], [-50], [4], [-24], [14], [60], [-54], [-18], [44], [2], [90], [-56], [72], [18], [24], [-46], [2], [-96], [-80], [-20], [-22], [30], [52], [-38], [-52], [44], [24], [-50], [28], [46], [76], [72], [10], [-50], [-86], [62], [6], [-32], [-6], [-28], [-24], [40], [-2], [-28], [-24], [82], [-10], [-4], [-28], [-80], [-86], [-20], [-42], [100], [86], [-66], [-14], [-92], [78], [-90], [-16], [18], [-62], [4], [-36], [-46], [22], [-96], [44], [-38], [-50], [-8], [-8], [-82], [-24], [-90], [-38], [72], [-36], [-38], [-18], [-24], [-30], [20], [64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_n();".
// 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_n(: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_n();".
// 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_n( : 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![4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;