
// Make newform 4002.2.a.i in Magma, downloaded from the LMFDB on 17 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_i();".
// 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_i();".
// 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], [2], [-2], [-2], [4], [-4], [1], [-1], [4], [-4], [-2], [-8], [0], [2], [0], [8], [8], [0], [-10], [-14], [6], [-12], [18], [-18], [-14], [-18], [-16], [4], [8], [8], [0], [-12], [14], [-20], [-20], [4], [0], [2], [4], [0], [24], [14], [-18], [-26], [-12], [-4], [-18], [-20], [-6], [-8], [10], [30], [-6], [-12], [14], [24], [-6], [32], [12], [-14], [-20], [0], [14], [-30], [-4], [-18], [12], [-6], [14], [0], [34], [-24], [-4], [28], [-26], [38], [32], [-18], [-26], [40], [4], [6], [24], [-12], [6], [-22], [-18], [24], [6], [-24], [24], [0], [12], [28], [34], [20], [0], [-34], [-20], [14], [34], [-20], [-24], [34], [-12], [18], [-8], [10], [32], [-20], [12], [20], [42], [12], [-8], [24], [30], [14], [12], [-46], [22], [-4], [-20], [-30], [-20], [0], [22], [48], [4], [4], [2], [4], [-34], [-10], [18], [32], [-42], [-54], [44], [6], [28], [6], [-2], [-28], [-30], [2], [4], [-16], [-30], [4], [36], [-8], [-28], [36], [-2], [-46], [-38], [6], [16], [8], [12], [42], [-44], [12], [-40], [-18], [26], [-26], [34], [58], [44], [-50], [-12], [-28], [-20], [26], [62], [-56], [-36], [30], [-14], [-18], [-60], [46], [-30], [-64], [-18], [48], [-34], [-48], [40], [-62], [18], [-12], [50], [4], [36], [56], [38], [-68], [-26], [-58], [46], [34], [-26], [-64], [-22], [20], [-34], [34], [50], [-42], [8], [-42], [36], [42], [-72], [-46], [-22], [-58], [-14], [38], [40], [-46], [60], [-64], [-30], [-24], [54], [52], [58], [18], [68], [52], [66], [6], [68], [40], [-46], [-20], [32], [10], [-6], [0], [-28], [-42], [4], [-48], [-52], [-36], [12], [-26], [-46], [-48], [-52], [-24], [-34], [-74], [30], [2], [74], [-40], [-42], [68], [54], [-24], [4], [-10], [18], [32], [-18], [58], [66], [20], [-36], [-74], [-14], [-78], [16], [58], [-48], [-4], [-60], [-8], [-14], [6], [-48], [78], [-26], [-10], [-18], [-22], [22], [-42], [-2], [54], [0], [-28], [-34], [-18], [22], [-28], [-52], [22], [68], [40], [36], [-2], [-48], [10], [-28], [-36], [4], [46], [20], [-56], [-26], [-10], [24], [-2], [66], [32], [-88], [-38], [56], [-20], [-24], [54], [-58], [-34], [32], [-32], [52], [-52], [40], [-36], [22], [-62], [34], [48], [-6], [-30], [-54], [52], [50], [12], [28], [-54], [-12], [-86], [30], [-42], [80], [-22], [4], [26], [18], [-16], [44], [-58], [8], [-18], [28], [46], [18], [86], [-42], [88], [28], [-8], [26], [-22], [38], [-56], [-44], [-38], [-12], [-82], [-58], [100], [-8], [-2], [28], [-96], [40], [98], [-48], [12], [-50], [6], [4], [44], [-20], [58], [6], [0], [48], [10], [86], [-22], [-54], [-8], [10], [-4], [16], [-78], [-76], [-28], [-46], [18], [-50], [-72], [62], [-62], [56], [-64], [-32], [-24], [102], [-20], [-96], [-44], [-78], [-2], [-18], [6], [-60], [16]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : 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![2, 1]>],Snew);
    return Vf;
end function;