
// Make newform 4002.2.a.t 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_t();".
// 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_t();".
// 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
        poly := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, 0], [-1, 0], [0, 2], [0, 1], [4, 0], [2, -2], [0, -2], [-4, -1], [-1, 0], [1, 0], [6, 0], [2, 2], [0, 2], [4, -3], [0, 8], [4, 0], [-4, 2], [-2, 4], [6, -6], [-2, -6], [6, 6], [10, 4], [2, 11], [-8, -2], [12, -1], [-2, 2], [0, 1], [-6, -1], [-12, -3], [-4, 4], [4, -8], [2, 2], [8, 0], [10, 4], [-4, 0], [16, 0], [6, -6], [8, 6], [8, -8], [8, -6], [-8, -6], [8, -5], [-6, 1], [2, -4], [10, 0], [-4, 3], [0, 2], [16, 4], [2, -3], [-6, 4], [-2, 2], [-10, -6], [10, 6], [12, 8], [-22, -2], [-2, 11], [-14, 2], [4, 0], [6, -2], [10, -7], [10, 6], [-2, -9], [20, -2], [-8, -18], [-2, 16], [6, -10], [-8, -14], [0, 3], [0, 8], [2, 0], [-10, 0], [2, -11], [10, 0], [20, 3], [4, 13], [8, 0], [10, -11], [-14, 4], [18, -9], [18, 4], [-6, -17], [-6, -4], [8, -6], [0, -11], [-12, 4], [10, 10], [16, 14], [-6, -16], [-2, 6], [28, 0], [32, 6], [14, 5], [32, 4], [10, 6], [-10, -4], [-22, 15], [-16, -18], [22, -5], [-10, -2], [8, 24], [14, 12], [-4, 2], [16, -16], [20, -12], [-6, 8], [-10, -8], [0, -28], [-34, 0], [-12, 14], [-18, 4], [10, -12], [-4, -15], [4, -20], [-32, -11], [16, 5], [-8, -4], [-2, 2], [10, -14], [-6, -4], [0, 18], [-20, -11], [2, -8], [-26, 9], [-32, 12], [8, -8], [-20, 14], [12, 19], [24, -20], [-6, -4], [18, -2], [28, 12], [26, 1], [6, -24], [-34, -8], [-2, 0], [-8, -19], [-22, -17], [-10, -10], [-6, -23], [-32, 14], [-6, 0], [-12, 30], [12, -24], [-12, -18], [-6, 8], [14, -25], [-14, -24], [-14, 10], [0, 32], [-8, -4], [6, 10], [-8, -26], [24, 8], [-16, -26], [-20, -3], [34, 7], [32, -3], [14, 16], [-26, 8], [-40, -6], [10, -14], [18, 7], [12, -8], [0, -28], [38, -5], [-2, 9], [-48, 6], [-26, 16], [-42, 6], [2, 14], [-6, 37], [-14, -26], [32, 0], [28, 1], [4, -22], [-2, 9], [20, 28], [-38, -8], [-6, -2], [4, 29], [2, 16], [32, -12], [46, -10], [26, 10], [-12, -18], [0, 8], [8, 12], [4, -1], [30, 18], [26, -14], [-10, 16], [-26, 18], [-24, -23], [-6, -28], [0, -20], [14, 21], [-4, -23], [-28, 15], [-34, -19], [10, 18], [42, -11], [16, -22], [6, 0], [8, -43], [-28, -18], [-34, 16], [-26, -20], [12, 14], [14, -18], [-12, 6], [42, 4], [14, 8], [26, -14], [-8, 0], [-20, 8], [18, 8], [56, 0], [10, 44], [24, -12], [-2, 24], [24, -24], [-52, 9], [-14, -28], [14, -32], [-48, -6], [28, -32], [20, -6], [-16, 6], [-6, -20], [12, 26], [0, 8], [58, 10], [-22, 28], [-6, 40], [-22, -8], [-26, 9], [32, -3], [18, -27], [-8, -16], [-12, 2], [40, -14], [-56, 8], [8, 38], [-18, 36], [26, -18], [56, -12], [-40, -6], [46, 5], [-2, -20], [10, -17], [26, 18], [-6, -25], [70, 3], [-12, 41], [38, 28], [4, -16], [-6, -44], [-8, -39], [-48, 14], [-24, 0], [-30, -20], [8, -2], [2, -26], [-2, -30], [-26, -8], [44, -17], [2, -7], [4, 46], [-24, -19], [4, -2], [-30, -6], [-6, 34], [-22, 0], [2, -22], [18, 30], [8, 20], [-2, -22], [-30, 4], [-4, 9], [-18, 17], [24, -20], [20, -41], [12, -22], [-56, 3], [12, 2], [16, -30], [48, -3], [34, 9], [-44, -24], [18, -28], [0, -8], [46, -19], [10, 6], [-48, -10], [-6, 0], [-36, 12], [54, 14], [26, -8], [-20, -20], [32, -20], [-20, 11], [-20, 26], [-50, -4], [44, 28], [-4, 7], [-6, -28], [6, 14], [8, -27], [-24, 34], [-18, -26], [-30, 18], [-2, 19], [-2, -33], [-66, -12], [20, 12], [-2, 46], [-52, -16], [10, -54], [30, 28], [-30, 4], [-32, 23], [46, 4], [-18, 0], [30, -24], [-6, 11], [12, -27], [6, 0], [-52, 16], [-10, -3], [16, -34], [-34, 28], [34, -24], [-64, 0], [-12, -44], [12, 39], [-40, 30], [-60, 9], [12, 22], [38, 6], [8, 55], [-60, -8], [-14, 3], [58, 0], [8, 37], [-20, -18], [-52, 0], [-14, 12], [12, 42], [-30, -45], [-6, 21], [-54, 0], [-32, 12], [42, 8], [-64, 5], [-6, -36], [-42, -32], [50, -3], [14, -13], [-22, -24], [-10, 54], [-72, -7], [18, -2], [60, 22], [-26, -13], [-28, 12], [-38, -18], [30, -34], [4, -33], [-16, 9], [-2, 58], [-50, -8], [-18, -50], [-58, 9], [12, 13], [22, -40], [-60, 0], [2, -13], [-16, 23], [6, -9], [-8, -16], [-46, 5], [38, -33], [54, -4], [6, 8], [38, -26], [24, -44], [14, 56], [2, 54], [-24, 19], [-22, 15], [4, -15], [-22, -8], [-18, -50], [72, -2], [38, -19], [34, 34], [-8, 25], [-20, -20], [86, -12], [38, -12], [-66, 12], [4, 6], [-2, 30], [-34, 29], [-4, -18], [12, 28], [-52, 29], [-10, -36], [-40, -33], [0, -58], [-14, 28], [36, -14], [-78, -13], [-34, -30], [4, 19], [66, 6], [84, 6], [-8, -2], [0, -4], [-30, -19], [72, -6], [26, -8], [-76, -6], [-14, 10], [24, -8], [-34, 20], [62, -25], [-46, -36], [-66, 4], [16, 60]];
    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_t();".
// 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_t(:prec:=2)
    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_t();".
// 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_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 0, 1]>,<7,R![-2, 0, 1]>,<11,R![-4, 1]>],Snew);
    return Vf;
end function;