
// Make newform 1002.2.a.i in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1002_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_1002_2_a_i();".
// 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_1002_2_a_i();".
// 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, 1, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -2, -1, 1], [1, -4, -1, 1], [-4, -2, 2, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1002_a();"
function MakeCharacter_1002_a()
    N := 1002;
    order := 1;
    char_gens := [335, 673];
    v := [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_1002_a_Hecke(Kf)
    return MakeCharacter_1002_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, 0, 0], [-1, 0, 0, 0], [1, -1, 0, 0], [0, 0, 1, -1], [0, 0, 0, 0], [2, 0, 0, 1], [3, 1, -1, 0], [0, 0, -2, 0], [0, 0, 2, 0], [4, 2, 0, 0], [0, 0, 1, 1], [2, 0, -3, 0], [3, 1, 3, 0], [1, 1, -1, 1], [2, 2, 1, -1], [1, -1, -2, 0], [-2, -2, 1, -2], [0, -2, 0, 0], [-3, -3, 2, 2], [2, 2, -4, 0], [0, -2, 0, -2], [-3, -3, -1, 0], [4, 4, 1, 0], [4, 2, -1, -1], [2, 4, 1, -1], [1, -1, -2, 4], [-3, 5, -3, 2], [-4, -4, -2, 4], [-2, 4, 4, -3], [-3, -5, 3, 0], [-6, 2, -5, 1], [-2, -2, -2, 3], [6, 4, -7, -1], [-5, 3, 4, -2], [3, 1, 0, 0], [-5, 3, 1, 2], [-2, 0, -4, 0], [-7, 1, 6, 0], [-1, 0, 0, 0], [0, 2, 0, 0], [-4, 0, 4, 0], [-12, 2, 6, -2], [-4, 0, -5, -3], [-6, 0, 0, -2], [-7, -1, 1, -1], [0, 0, 0, 2], [-16, 0, -2, 0], [0, 0, -1, 1], [-10, -2, -5, -2], [-4, 2, 4, -2], [8, 6, 1, -3], [0, 0, 0, -2], [4, 2, 2, -2], [0, -4, -6, 4], [7, -3, -3, 0], [10, 2, 1, 5], [11, 9, 2, -4], [-11, -3, -5, -2], [0, -10, 3, 0], [4, -6, 1, 1], [8, -8, 4, -2], [16, -2, 0, 0], [-1, -1, -8, 4], [8, 4, -3, 5], [18, 8, -2, -4], [-8, 2, 0, 4], [-3, -3, -1, 1], [-4, 2, 9, 1], [2, -6, 5, -2], [-2, -4, 3, 0], [12, -6, 1, 1], [-6, -2, -2, -2], [6, -2, 2, -2], [-2, 4, -5, 8], [3, 3, 4, 0], [10, -2, -6, 4], [3, 1, -9, -1], [12, 2, -4, 2], [5, 3, -5, 2], [-10, -8, 0, 2], [0, 0, 4, -2], [-14, 4, 6, -6], [-8, -4, 5, -3], [10, 0, -9, 7], [11, -5, -11, 8], [-14, 2, 3, -6], [6, -4, 5, -1], [-4, 2, 10, -2], [-6, 4, 2, 2], [15, -9, -1, 0], [0, 4, 6, 2], [-10, 6, 10, -8], [-5, 3, 3, -2], [-4, -12, -8, 0], [-9, -1, -7, 3], [14, -6, -6, 4], [-14, -4, 2, -2], [13, -5, -1, 2], [12, 4, 8, 2], [-4, 2, -3, -8], [5, -3, 8, -6], [-10, -4, 6, -2], [8, 4, -2, 4], [-9, -11, 7, 4], [-15, 1, 2, -4], [0, -6, -9, 1], [-2, 6, 5, -4], [1, -1, 7, 0], [2, -6, 1, 1], [14, 0, -15, 7], [5, 5, 5, -8], [16, -2, -10, 8], [12, -6, 10, -4], [5, 5, 7, -9], [10, 2, 1, -1], [-7, 7, -3, 8], [5, -3, 3, -5], [-2, -2, -2, -6], [-12, 6, 4, 0], [-2, 6, -5, 4], [-4, -14, -6, -3], [26, 8, 0, -6], [-20, 2, 4, -4], [6, -2, -14, 3], [-1, -1, 9, -3], [22, 0, 10, 2], [-4, 2, 3, -12], [2, -6, 2, 2], [13, -3, -7, -2], [16, -2, -8, -2], [9, 1, 4, 8], [10, 2, 3, -9], [7, 7, -9, 4], [6, 0, 10, 6], [10, 0, -1, -7], [6, 12, -2, 0], [-1, -3, 14, 0], [9, -7, 7, -7], [17, -1, 5, -3], [-6, 0, 5, -1], [-7, 9, 0, -2], [-19, -5, 5, -5], [7, 7, -7, 2], [4, 4, -8, 5], [12, -14, -7, 4], [-4, 0, -13, -3], [4, -6, 0, -2], [-6, -8, -13, 5], [-6, -6, -2, -4], [-14, -10, -6, -6], [-14, -4, -6, 4], [9, -9, -5, 0], [-16, 8, -8, 2], [-22, -6, -4, 10], [6, -2, -2, 0], [16, -4, 15, -1], [4, 12, -1, -3], [6, 4, 7, -3], [-10, -4, 10, -2], [-21, -15, 3, 7], [-40, -8, -2, 2], [11, -15, -7, 2], [4, -12, 4, -10], [-20, 4, 13, -4], [1, 7, 9, -8], [-10, 6, 14, 0], [9, 17, 5, 0], [-4, 2, 10, -6], [-28, -10, -3, 3], [10, -4, -8, -4], [-10, -10, 15, 4], [-18, -8, 16, 2], [10, 10, -5, 3], [2, -8, 5, -3], [-9, -1, 11, 6], [-8, -10, -2, 4], [-2, -6, -10, -2], [-1, -3, 12, -12], [46, 6, 2, -4], [-6, 0, -1, 8], [16, 0, -9, -5], [-4, -8, 6, 2], [4, 2, -11, 12], [21, 3, -3, -6], [-18, 6, 6, 2], [-6, 0, -6, -2], [-4, 10, 0, -6], [0, 4, -6, 10], [-34, 12, -5, 3], [18, -14, -4, 4], [-24, -2, 0, -10], [28, 4, -3, -8], [-8, 4, -10, -2], [-2, 0, 4, -4], [4, 4, -8, 16], [30, -12, 11, -7], [-20, 14, -11, 3], [-40, -10, 6, 6], [-20, 2, 3, -1], [16, -4, -7, 7], [-7, 7, 9, 5], [-4, -12, 15, -1], [0, 14, 3, -4], [-12, 10, -12, 10], [30, 6, 10, 7], [12, 2, 12, -8], [7, -9, 19, 2], [-12, -4, -10, -10], [29, 3, -3, 2], [4, 0, -12, 20], [10, 0, 3, 3], [26, -4, 0, 8], [7, -1, -7, -2], [20, -4, -8, -3], [8, 0, 0, -6], [18, 0, -5, 3], [15, -9, -19, 10], [-16, 6, 10, -8], [12, -8, 19, -1], [21, -5, 4, -8], [-18, 4, -12, -5], [14, -18, -15, 7], [17, 15, 11, 6], [-34, -10, 10, 0], [20, -4, 12, -3], [0, -2, 18, 6], [-6, 0, 23, 1], [10, 18, -2, -8], [9, 1, 7, -6], [-12, 0, 2, -10], [28, 2, 3, 4], [-13, -5, 5, -13], [-41, -1, 9, -2], [-1, 5, -11, -4], [18, -10, 4, -4], [6, -18, -8, 0], [-24, 6, 16, -12], [12, 2, 10, 6], [18, 10, 5, -4], [8, 8, 16, -10], [26, -6, -18, 5], [-4, 8, 12, 0], [-21, -5, 13, -4], [18, 8, -13, -4], [20, 18, -6, -4], [2, -10, -10, 12], [48, -8, 7, -7], [-10, 6, -9, -10], [6, -14, -8, 4], [34, 10, -8, -4], [20, -14, 2, -2], [28, -14, -14, 4], [-24, 16, 2, 2], [-40, -2, -12, -6], [11, 9, -5, -11], [-8, -4, 18, -8], [-20, -6, 11, 0], [20, 8, -12, 0], [-21, 1, 2, -4], [-8, -26, -7, 5], [39, 7, 3, -6], [4, 20, 11, 0], [-16, -18, -7, -4], [22, 12, -3, -4], [-44, 10, -6, 4], [14, 18, -4, -8], [-25, 5, 19, -5], [35, 1, -11, 12], [-47, 9, 14, -6], [-8, -10, -14, 6], [-20, -6, 2, 5], [12, 4, 6, -12], [-20, 2, 20, 2], [30, 14, -9, -5], [-24, 14, -6, 2], [35, 3, -17, 6], [12, 4, 21, 4], [38, 4, -3, 4], [22, 12, 16, 0], [-8, 0, 0, -14], [18, 2, -2, 2], [35, -5, 1, -2], [52, 4, -4, -6], [28, -6, -2, 2], [-5, -21, -4, 0], [20, -4, -18, 6], [18, 8, 13, -11], [-3, 3, 9, 9], [14, -2, 3, 5], [-37, -7, -9, -2], [22, 8, 0, 0], [-52, -12, -1, 2], [4, 2, 11, -25], [12, 20, 6, -10], [-20, 2, -2, 8], [-14, 8, -2, 2], [4, -4, -9, 17], [-17, 13, -14, -4], [20, 4, -23, 8], [28, -4, 8, 2], [14, -12, -4, 2], [-25, 5, -16, 8], [2, 2, -21, 7], [-12, 12, -8, 9], [-28, -16, 4, 4], [4, 10, -18, -6], [-10, -2, -15, 0], [-6, 12, 16, -6], [18, -22, -4, 2], [6, -8, 24, 0], [48, 8, 2, 4], [-8, -14, -14, -6], [2, 24, 4, 4], [3, 3, 5, -13], [-22, -22, 0, 2], [-12, -18, 14, 8], [40, 16, 1, -14], [-18, -18, 1, 13], [0, 14, 10, 0], [21, -21, -21, 8], [-50, 6, 2, 4], [-2, -8, 3, 13], [2, 8, -12, 8], [17, 17, -9, -6], [-5, -15, 5, -6], [-42, 0, 15, -7], [-32, -24, 10, 8], [-14, 14, 0, 4], [6, 14, -21, 5], [38, 4, 12, 8], [-4, 18, -28, 2], [-26, -4, 12, 4], [-17, -1, -1, 12], [-4, 4, -12, 18], [-29, 3, 1, -3], [-20, 8, -2, 8], [22, -8, -2, 0], [19, 1, -19, 16], [0, -2, 4, -14], [28, 20, -1, -9], [10, 12, -4, -8], [26, 0, 8, 12], [-11, -13, -6, 8], [21, -3, -3, 16], [-12, -10, -16, -4], [32, 12, -2, -16], [-18, -16, 0, 16], [-18, 6, 2, -12], [58, 2, -22, 4], [-42, -12, 4, 4], [-18, -14, -8, 0], [4, -14, 16, -4], [11, -23, 2, -8], [-15, 9, 1, 0], [-14, -8, 7, -8], [27, 9, 15, -6], [-8, -20, 7, -5], [-4, -4, -13, 18], [51, 9, -7, 2], [8, -4, -9, 17], [-10, 16, -26, 2], [-7, -17, 19, 0], [-2, 14, -16, 8], [-40, -8, 22, 4], [17, 9, -12, 16], [-54, -16, -4, 4], [-17, -3, 10, 4], [23, -1, -23, -2], [-44, 2, 7, -1], [12, 12, -15, 8], [35, 11, 9, 9], [-30, -18, -2, 22], [-34, -8, -22, -10], [31, -1, -17, 12], [0, 6, 17, -24], [-18, -2, -2, -1], [18, 26, 14, -2], [-26, -12, -19, -11], [-9, 21, -9, -4], [0, -14, 22, -8], [-18, 4, -8, 4], [48, -2, -17, 7], [17, -7, 15, 2], [-4, -6, 21, -3], [18, -2, -22, -6], [6, 22, 14, 6], [1, 25, 7, -6], [36, 10, -19, -4], [-2, -14, -6, -6], [-56, 0, -14, 2], [0, -10, 4, 2], [-12, -6, -6, -6], [-52, 8, 6, -4], [35, -21, 4, -4], [10, 2, -8, 10], [8, -2, 14, 18], [-34, -18, -11, 5], [6, -4, -7, 11], [23, 7, -18, 12], [27, 9, 8, -8], [42, 4, 6, 2], [-48, -2, 2, 12], [49, -7, -5, -4], [-37, -7, -15, -10], [15, 13, 2, 0], [41, -15, -3, 4], [48, 6, 15, 4], [-9, -19, 1, 2], [-19, 13, -14, 12], [-16, -4, -26, -12], [14, -20, 10, -6], [-9, 5, 15, -9], [0, 0, 12, 4], [-48, -12, 6, 10], [34, 24, 20, 4], [10, 24, -22, -2], [-4, -12, -18, 12], [-26, -2, 11, -3], [-46, 0, 15, 1], [-10, -18, 3, -7], [7, -3, -9, 19], [38, 12, -9, -4], [30, -10, -3, 3], [-8, 12, 12, -14], [-30, 0, -17, 7], [-21, -7, 3, -3], [-16, -8, 8, -4], [12, 2, 19, 15], [8, 8, 8, -2], [-16, 0, 26, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1002_2_a_i();".
// 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_1002_2_a_i(:prec:=4)
    chi := MakeCharacter_1002_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_1002_2_a_i();".
// 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_1002_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 20, 0, -5, 1]>],Snew);
    return Vf;
end function;