
// Make newform 4002.2.a.l in Magma, downloaded from the LMFDB on 25 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_l();".
// 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_l();".
// 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], [8], [10], [10], [4], [-8], [-6], [12], [-14], [16], [-8], [2], [-16], [-12], [10], [6], [-10], [4], [-12], [-2], [10], [8], [12], [10], [4], [10], [0], [2], [12], [16], [-18], [-20], [-2], [12], [26], [22], [4], [-20], [-16], [-4], [-6], [-6], [-24], [2], [-8], [-14], [4], [-18], [-8], [-10], [6], [16], [6], [28], [-32], [-30], [30], [-20], [22], [-4], [-2], [10], [-4], [-32], [-10], [-4], [-32], [6], [-18], [38], [2], [-12], [2], [-16], [-10], [40], [-20], [2], [2], [14], [8], [-24], [12], [-32], [-36], [4], [12], [14], [-10], [24], [30], [28], [2], [-8], [26], [16], [-22], [-4], [18], [-24], [-30], [-8], [30], [34], [-28], [-28], [-46], [40], [-48], [-18], [24], [-18], [-14], [30], [-36], [20], [-30], [-18], [-32], [8], [50], [36], [-28], [-40], [26], [-30], [-10], [6], [-24], [-10], [-22], [-12], [30], [40], [0], [46], [-36], [-34], [-54], [36], [48], [-50], [-30], [36], [8], [-4], [-28], [-20], [50], [-54], [10], [-28], [14], [56], [48], [6], [-60], [-8], [-10], [34], [-10], [-52], [-2], [-56], [22], [-32], [14], [-28], [-10], [8], [30], [-8], [0], [-58], [-62], [0], [-30], [38], [28], [-46], [16], [18], [20], [-60], [-2], [32], [6], [22], [54], [-18], [-32], [-58], [32], [54], [-26], [8], [-50], [0], [-12], [-6], [4], [-62], [62], [8], [0], [-72], [-30], [8], [18], [-24], [62], [-34], [-44], [-14], [-56], [36], [62], [10], [16], [-16], [12], [-2], [-8], [56], [-6], [-56], [44], [14], [6], [44], [8], [-16], [-28], [32], [38], [34], [-56], [-56], [36], [72], [60], [-22], [70], [28], [6], [-66], [60], [-54], [-68], [-74], [-38], [-56], [-56], [6], [74], [-30], [52], [-26], [-38], [52], [54], [-50], [56], [26], [32], [-6], [-16], [-60], [62], [46], [-12], [48], [60], [-24], [-42], [84], [-24], [-66], [-18], [16], [26], [34], [-20], [-14], [24], [46], [-78], [-56], [14], [36], [-28], [-26], [50], [16], [36], [-52], [-30], [-20], [-58], [-80], [6], [24], [54], [22], [-20], [-16], [-46], [28], [80], [-14], [-30], [-28], [-14], [-10], [56], [-18], [-42], [-32], [12], [-4], [-6], [46], [38], [-8], [-12], [-36], [52], [-50], [-14], [42], [12], [94], [54], [54], [68], [-30], [76], [14], [60], [-60], [6], [4], [90], [-18], [68], [-6], [-6], [12], [84], [66], [24], [6], [50], [8], [-60], [12], [34], [-18], [-44], [62], [52], [-52], [96], [22], [-52], [94], [-12], [60], [46], [30], [-22], [-34], [62], [88], [50], [-32], [72], [-96], [22], [92], [12], [-50], [-10], [84], [36], [12], [-22], [52], [58], [-40], [-58], [14], [34], [40], [86], [78], [-64], [-2], [66], [-68], [60], [2], [30], [-68], [16], [-58], [78], [0], [-56], [66], [60], [66], [18], [96], [36], [26], [54], [-12], [90], [-108], [32]];
    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_l();".
// 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_l(: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_l();".
// 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_l( : 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;