
// Make newform 4002.2.a.r 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_r();".
// 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_r();".
// 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], [3], [1], [0], [-2], [1], [-1], [-1], [1], [10], [5], [7], [-11], [-13], [10], [-3], [10], [-4], [10], [10], [4], [8], [-6], [0], [-10], [7], [-5], [-4], [-9], [-2], [-14], [-18], [-18], [-11], [-1], [-7], [11], [-10], [23], [12], [-6], [-5], [12], [1], [4], [-23], [-8], [11], [-17], [-4], [-10], [15], [4], [10], [21], [32], [-12], [8], [-30], [0], [-6], [20], [-21], [-19], [-30], [11], [30], [23], [30], [34], [17], [-32], [36], [-20], [-24], [-4], [-20], [-10], [-6], [-3], [26], [14], [12], [-19], [-24], [15], [-7], [22], [-24], [-22], [0], [25], [0], [-24], [11], [31], [-28], [-6], [-45], [-38], [3], [30], [43], [36], [-2], [-5], [24], [24], [-20], [-14], [-38], [15], [-35], [-33], [7], [-2], [-28], [-18], [-4], [36], [27], [26], [33], [22], [51], [20], [-30], [14], [6], [-4], [11], [-26], [47], [-42], [-44], [-4], [-42], [18], [-22], [-44], [22], [-4], [-48], [11], [13], [-51], [12], [19], [-22], [-18], [50], [-42], [20], [-28], [0], [-11], [-30], [17], [-50], [26], [56], [-6], [24], [-42], [8], [-35], [21], [-13], [-10], [36], [-24], [-22], [46], [-3], [-30], [-8], [16], [32], [14], [12], [8], [-52], [48], [54], [-18], [-21], [-37], [-50], [18], [43], [12], [-56], [42], [-38], [34], [6], [44], [50], [64], [26], [-11], [1], [64], [26], [-67], [4], [-45], [32], [52], [-46], [5], [28], [62], [33], [-6], [-31], [-38], [28], [-14], [22], [0], [3], [50], [-12], [15], [21], [36], [-8], [52], [50], [4], [-10], [-9], [76], [-69], [30], [6], [-39], [41], [62], [-2], [-56], [-29], [6], [-6], [-16], [-64], [-16], [32], [14], [48], [-67], [2], [30], [-75], [35], [37], [-42], [6], [44], [22], [-2], [22], [-43], [61], [36], [29], [75], [-51], [-72], [-50], [-75], [-38], [78], [16], [12], [-32], [-4], [-41], [-34], [71], [54], [-28], [63], [45], [34], [64], [-9], [50], [50], [68], [-14], [18], [-82], [45], [-62], [42], [-19], [-61], [34], [71], [52], [-76], [68], [25], [-62], [-58], [-60], [80], [30], [54], [-42], [60], [-24], [19], [82], [-26], [-11], [19], [-27], [5], [-31], [4], [36], [-60], [70], [20], [-20], [75], [-17], [15], [-29], [-60], [52], [35], [-22], [2], [74], [-65], [11], [-66], [58], [-69], [15], [-16], [-25], [9], [-64], [-76], [-40], [-74], [-28], [24], [47], [15], [24], [-3], [38], [-56], [-26], [-38], [89], [80], [35], [68], [30], [67], [92], [-72], [4], [-66], [20], [-57], [46], [-24], [-23], [-30], [30], [49], [72], [-64], [28], [59], [62], [-64], [40], [-2], [-33], [56], [-28], [-75], [-54], [37], [-65], [39], [-25], [-37], [-60], [-44], [-16], [-34], [-12], [48], [45], [64], [-52], [13], [44], [89], [-46], [-23], [-51], [24], [82], [40], [57], [17], [-42], [31], [-25], [-78], [57], [-6], [42], [-38], [41], [-35], [0], [-31]];
    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_r();".
// 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_r(: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_r();".
// 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_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<7,R![-1, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;