
// Make newform 4002.2.a.b 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_b();".
// 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_b();".
// 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], [0], [-4], [0], [6], [0], [0], [-1], [-1], [-8], [2], [10], [0], [0], [0], [0], [6], [-10], [16], [6], [10], [6], [-12], [-4], [2], [-4], [-14], [12], [0], [-12], [-4], [8], [0], [-20], [0], [-22], [-20], [-8], [2], [0], [-24], [6], [-2], [18], [-24], [20], [0], [-6], [-14], [-18], [-8], [-6], [12], [-2], [18], [-2], [-28], [-14], [18], [-26], [-6], [-12], [12], [26], [-10], [-12], [-24], [-12], [-6], [30], [10], [-22], [4], [-8], [-20], [30], [-34], [-10], [30], [-18], [-14], [20], [16], [0], [28], [-6], [-18], [-2], [8], [8], [-18], [-40], [-12], [-4], [-18], [-14], [-42], [-26], [-18], [-4], [-20], [12], [12], [18], [-10], [12], [2], [20], [46], [36], [-20], [-24], [12], [20], [-12], [-14], [32], [-18], [24], [28], [38], [-46], [36], [32], [-8], [-4], [40], [22], [14], [-20], [-14], [22], [34], [6], [-24], [-30], [10], [-14], [-18], [12], [46], [-20], [-56], [2], [-42], [22], [-18], [44], [-56], [-6], [36], [-16], [-36], [8], [22], [52], [-14], [-38], [0], [-20], [-42], [-4], [60], [-46], [-46], [-8], [-6], [-22], [2], [-14], [-6], [-60], [44], [-16], [26], [-20], [-10], [6], [44], [-48], [-36], [-38], [34], [-36], [-16], [10], [-32], [38], [-24], [-46], [52], [28], [30], [0], [54], [4], [-4], [30], [-8], [-14], [0], [14], [-24], [60], [66], [-10], [-40], [-34], [28], [-22], [-18], [18], [72], [-40], [-46], [56], [30], [-32], [-34], [-10], [-32], [38], [-38], [-16], [22], [-36], [60], [-10], [72], [30], [-18], [-18], [-50], [-14], [-6], [32], [46], [76], [-28], [-56], [-56], [-32], [-46], [6], [8], [-40], [-42], [50], [-34], [-10], [-38], [70], [76], [78], [-12], [10], [4], [34], [26], [74], [20], [66], [34], [50], [-56], [-46], [40], [-20], [18], [-54], [-46], [-26], [-22], [62], [64], [76], [-66], [-28], [2], [16], [48], [16], [-24], [-28], [36], [-16], [-22], [-16], [14], [28], [-38], [66], [64], [-14], [-36], [-54], [-66], [-60], [-28], [44], [36], [22], [52], [64], [6], [-20], [8], [-48], [-42], [-8], [-38], [-62], [64], [-16], [42], [-28], [-32], [-74], [34], [4], [-6], [70], [-84], [-66], [60], [70], [-44], [6], [20], [-34], [-66], [40], [-84], [-40], [24], [76], [-36], [-6], [68], [54], [18], [6], [60], [-48], [-12], [-14], [-28], [-18], [18], [48], [82], [66], [-12], [6], [62], [-22], [-18], [-66], [-80], [16], [2], [-12], [-6], [-4], [46], [18], [40], [-80], [-36], [-28], [-32], [-78], [-80], [78], [84], [-22], [-32], [-10], [42], [-18], [-6], [-32], [74], [-14], [64], [8], [50], [28], [2], [-60], [-70], [60], [20], [78], [-62], [12], [64], [22], [-10], [-58], [54], [-58], [18], [30], [-88], [76], [-74], [76], [0], [78], [-18], [-74], [-74], [-44], [-82], [-92], [88], [0], [10], [-84], [22], [36], [44], [-58], [22], [-82], [-70], [-60], [24]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;