
// Make newform 4002.2.a.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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], [0], [2], [-6], [-8], [-2], [1], [-1], [8], [0], [-6], [-6], [4], [-6], [-4], [-8], [-4], [-8], [-10], [-2], [12], [8], [12], [6], [4], [-8], [10], [-12], [0], [-12], [-16], [4], [6], [16], [8], [16], [-8], [-18], [12], [-2], [-6], [14], [6], [12], [-8], [0], [-4], [20], [-18], [-8], [-10], [-22], [-14], [10], [10], [8], [-2], [18], [16], [0], [-8], [-12], [-10], [-6], [12], [4], [-20], [22], [30], [6], [-10], [-34], [6], [-28], [-16], [-22], [10], [14], [-12], [0], [0], [8], [-16], [24], [-6], [26], [26], [-40], [-26], [-18], [-40], [-28], [4], [-14], [-26], [-38], [36], [18], [-44], [-38], [38], [-20], [16], [22], [36], [22], [36], [22], [4], [10], [-4], [-10], [-4], [36], [-4], [32], [14], [-2], [-30], [2], [-8], [12], [-12], [30], [-34], [-8], [2], [-20], [16], [26], [22], [-4], [6], [32], [36], [12], [-4], [-50], [52], [-30], [20], [-18], [-46], [-2], [-2], [-26], [44], [-8], [18], [-40], [-12], [-48], [-6], [-54], [-24], [-18], [-22], [18], [-36], [-14], [20], [22], [54], [38], [40], [-38], [-10], [-34], [16], [2], [36], [40], [16], [-6], [-36], [30], [-2], [-26], [-32], [-6], [-38], [2], [32], [30], [6], [26], [-2], [24], [14], [-36], [-26], [38], [-10], [-26], [16], [-26], [-66], [48], [-60], [-40], [50], [-12], [22], [2], [50], [36], [10], [-8], [50], [42], [-38], [-6], [12], [-26], [-32], [22], [0], [6], [70], [64], [70], [34], [12], [-42], [-48], [-16], [22], [-20], [26], [-24], [-50], [-42], [76], [38], [56], [72], [20], [16], [-6], [-60], [8], [-18], [6], [-24], [-40], [-20], [20], [-2], [40], [10], [-58], [32], [54], [-52], [-20], [34], [58], [-62], [26], [42], [26], [-44], [-62], [-14], [44], [52], [-2], [-38], [-26], [-20], [6], [74], [-50], [-76], [0], [-54], [-40], [-36], [64], [-56], [-36], [-26], [72], [-44], [56], [-28], [-80], [58], [-58], [-80], [-42], [42], [-14], [30], [78], [-26], [-36], [16], [76], [-70], [2], [20], [-58], [-2], [-56], [14], [-40], [-26], [-8], [40], [6], [-4], [60], [10], [12], [-88], [54], [66], [-26], [14], [-38], [4], [-46], [36], [80], [-44], [-58], [-14], [-86], [62], [-16], [-48], [58], [-84], [-18], [24], [-14], [-68], [-82], [30], [62], [-80], [54], [12], [-90], [-12], [-62], [-48], [68], [22], [42], [32], [56], [-18], [42], [-44], [-50], [-72], [70], [-10], [-32], [-12], [-68], [14], [22], [-4], [88], [28], [20], [88], [44], [40], [-82], [-32], [-74], [44], [-62], [-38], [-80], [70], [-12], [-6], [-8], [-80], [-44], [38], [98], [-42], [-48], [-34], [-72], [-28], [82], [-82], [100], [-48], [-16], [22], [14], [-42], [66], [54], [6], [-64], [-86], [38], [38], [-20], [66], [54], [-4], [92], [-64], [10], [-24], [88], [-88], [70], [10], [12], [-12], [32], [-10], [-50], [-24], [2], [52], [84]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![0, 1]>,<11,R![-2, 1]>],Snew);
    return Vf;
end function;