
// Make newform 2366.2.a.n in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2366_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_2366_2_a_n();".
// 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_2366_2_a_n();".
// 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_2366_a();"
function MakeCharacter_2366_a()
    N := 2366;
    order := 1;
    char_gens := [339, 2199];
    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_2366_a_Hecke(Kf)
    return MakeCharacter_2366_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], [0], [6], [4], [3], [6], [10], [-8], [0], [8], [-6], [12], [-3], [11], [-2], [3], [-2], [-4], [0], [6], [-2], [-6], [14], [-18], [16], [-18], [11], [15], [-3], [-16], [0], [1], [2], [-20], [12], [9], [6], [5], [-24], [-5], [-12], [14], [-4], [-8], [15], [22], [-3], [-21], [28], [3], [-6], [-27], [9], [-2], [26], [-18], [-31], [6], [13], [-6], [8], [12], [10], [14], [-24], [-29], [6], [9], [-10], [32], [28], [6], [-30], [7], [-6], [22], [0], [-26], [33], [-4], [14], [18], [-9], [-29], [-21], [31], [-3], [-30], [13], [12], [22], [42], [15], [-36], [-1], [-38], [20], [-24], [-36], [-45], [8], [28], [-3], [0], [3], [26], [-10], [22], [3], [37], [-5], [-9], [-41], [-36], [30], [6], [-23], [-10], [-51], [24], [1], [-12], [10], [-48], [-52], [-35], [16], [-48], [17], [-52], [-12], [10], [30], [-53], [-15], [-42], [19], [-12], [-13], [-24], [-25], [54], [-17], [-18], [-4], [57], [22], [-6], [20], [12], [14], [21], [-7], [-18], [8], [-15], [42], [-15], [16], [3], [-39], [30], [17], [35], [7], [-3], [48], [31], [-27], [-17], [56], [0], [-38], [12], [38], [-1], [7], [12], [-28], [42], [-33], [21], [-52], [-11], [-11], [-48], [-16], [12], [2], [18], [48], [18], [-50], [44], [-60], [24], [36], [-46], [-50], [2], [-12], [-45], [22], [24], [9], [-13], [11], [-6], [-16], [39], [-6], [-32], [-61], [60], [27], [42], [32], [-14], [-42], [-8], [-33], [74], [-24], [21], [44], [-18], [-4], [44], [-68], [27], [-1], [30], [-14], [-36], [30], [18], [-66], [-4], [8], [16], [-42], [-42], [52], [15], [34], [27], [-32], [-6], [-9], [-22], [66], [-12], [-28], [37], [-33], [-20], [11], [-18], [70], [2], [12], [-1], [-18], [-30], [4], [-30], [5], [-32], [-8], [-7], [2], [70], [42], [10], [-59], [-48], [-24], [40], [-24], [37], [-41], [0], [65], [-42], [4], [-18], [-66], [0], [-18], [12], [26], [-63], [-58], [9], [27], [31], [-40], [-36], [11], [36], [14], [58], [-48], [35], [-81], [-4], [-48], [48], [39], [80], [-57], [32], [-72], [9], [34], [-39], [-73], [-59], [3], [-65], [-81], [-28], [88], [-71], [72], [-78], [22], [-60], [62], [-12], [34], [-33], [58], [0], [-14], [-58], [4], [-33], [42], [-37], [-60], [-60], [5], [-44], [-54], [18], [-44], [-65], [12], [-22], [2], [-60], [-60], [-57], [18], [-51], [10], [54], [-96], [36], [44], [-10], [-45], [-53], [65], [72], [-1], [-15], [60], [-28], [62], [-84], [24], [4], [12], [-10], [66], [-45], [-29], [33], [-5], [12], [-53], [-73], [16], [6], [-89], [-39], [42], [-85], [24], [-1], [34], [6], [86], [42], [-83], [57], [43], [-24], [18], [-4], [28], [3], [4], [-39], [-58], [-42], [-6], [-10], [-43], [21], [21], [8], [-102], [-48], [72], [97], [-48], [-33], [82], [33], [84], [39], [-35], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2366_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2366_2_a_n();".
// 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_2366_2_a_n(:prec:=1)
    chi := MakeCharacter_2366_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_2366_2_a_n();".
// 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_2366_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2366_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![3, 1]>],Snew);
    return Vf;
end function;