
// Make newform 4002.2.a.c 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_c();".
// 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_c();".
// 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], [1], [4], [-3], [1], [6], [6], [-1], [1], [-1], [3], [1], [6], [4], [-10], [-3], [7], [3], [9], [2], [12], [-10], [12], [-6], [-15], [11], [-10], [-10], [-2], [-21], [10], [-6], [12], [3], [-12], [10], [-3], [11], [12], [-4], [2], [-9], [14], [-18], [-13], [-13], [18], [-10], [3], [-8], [29], [6], [9], [12], [28], [15], [25], [5], [10], [27], [-24], [19], [4], [-8], [1], [-8], [5], [4], [23], [4], [0], [10], [4], [0], [-20], [-6], [30], [-11], [20], [10], [13], [12], [-2], [-10], [36], [-19], [-28], [-27], [14], [21], [-15], [-22], [-8], [-26], [24], [-4], [-26], [-25], [-22], [-8], [-45], [-13], [-14], [-23], [14], [28], [36], [-26], [-44], [32], [-22], [22], [36], [1], [4], [20], [17], [-1], [-12], [20], [26], [24], [9], [-10], [-13], [-38], [-21], [6], [3], [-29], [-5], [22], [-17], [-30], [49], [36], [-43], [0], [31], [22], [-18], [-21], [-21], [-30], [33], [-54], [-58], [44], [11], [-38], [28], [42], [20], [10], [-29], [-29], [36], [20], [-15], [38], [18], [-20], [35], [-3], [-29], [32], [-32], [-4], [-26], [-54], [-62], [4], [-10], [28], [-15], [-22], [-42], [14], [-20], [45], [-9], [27], [-4], [56], [-59], [-28], [-14], [-52], [12], [-37], [-24], [-60], [50], [-3], [14], [37], [-52], [-26], [33], [-44], [-44], [-64], [-38], [12], [44], [20], [-23], [26], [5], [58], [-66], [-44], [-36], [-30], [-34], [-70], [53], [-52], [3], [54], [-23], [-21], [-58], [40], [-72], [-32], [-6], [-44], [39], [46], [0], [20], [42], [65], [52], [-1], [-34], [-9], [-32], [12], [4], [6], [42], [24], [64], [-22], [10], [-33], [-9], [-42], [-15], [40], [-65], [-74], [3], [-14], [18], [8], [39], [-26], [-28], [3], [-31], [82], [4], [-54], [20], [40], [10], [51], [-52], [-22], [40], [-2], [4], [78], [42], [-59], [-6], [63], [-32], [50], [-72], [-8], [36], [-34], [42], [84], [-66], [10], [-44], [48], [-60], [22], [19], [-16], [-6], [-51], [36], [3], [63], [-24], [-54], [38], [22], [20], [54], [0], [-10], [-63], [-80], [-15], [26], [-72], [28], [-60], [11], [-68], [35], [-52], [-8], [-18], [-80], [66], [-19], [-71], [-44], [69], [55], [-34], [50], [-50], [78], [28], [-60], [60], [30], [20], [25], [-64], [-9], [65], [65], [90], [-66], [-8], [19], [-49], [64], [-26], [23], [-17], [19], [18], [-51], [-39], [60], [-41], [-67], [82], [24], [-90], [-50], [-1], [16], [-50], [-51], [-22], [90], [58], [48], [20], [-83], [79], [-4], [41], [-1], [-37], [-2], [-96], [-3], [59], [87], [-52], [23], [56], [-6], [-32], [-81], [91], [30], [-64], [-97], [-73], [-12], [36], [-44], [-67], [-66], [-11], [-63], [-64], [47], [-34], [-44], [36], [-35], [-46], [13], [21], [-90], [24], [-28], [31], [34], [-46], [-16], [-100], [69], [83], [22], [96], [-44], [-71], [18], [4], [6], [-36], [102]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<7,R![-4, 1]>,<11,R![3, 1]>],Snew);
    return Vf;
end function;