
// Make newform 2366.2.a.k 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_k();".
// 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_k();".
// 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], [-2], [3], [-1], [6], [0], [-3], [4], [-6], [3], [10], [1], [3], [-10], [0], [-9], [6], [-7], [-2], [0], [7], [14], [18], [6], [-2], [-3], [-4], [0], [-2], [3], [20], [0], [-3], [2], [21], [-8], [-7], [-2], [6], [6], [0], [5], [18], [-23], [-18], [-4], [-22], [28], [6], [-14], [-18], [6], [-17], [-12], [21], [-6], [18], [-2], [-1], [-27], [-22], [3], [22], [-18], [-10], [9], [10], [-13], [-6], [-2], [3], [-12], [-10], [-13], [-26], [18], [-33], [34], [-15], [-5], [6], [-35], [12], [5], [-22], [-24], [-30], [25], [-33], [22], [-18], [-42], [-32], [0], [22], [-18], [-21], [-39], [-28], [25], [-34], [9], [-18], [-18], [8], [19], [36], [15], [36], [-19], [44], [13], [45], [-26], [4], [-9], [40], [-24], [-6], [0], [-5], [35], [6], [24], [-8], [-30], [1], [-6], [-16], [-17], [16], [0], [-46], [2], [-18], [46], [-6], [10], [6], [51], [-26], [42], [-4], [-42], [-7], [-12], [19], [33], [14], [-54], [49], [45], [38], [-24], [14], [12], [-16], [-33], [-19], [-30], [42], [6], [34], [0], [-51], [24], [-10], [53], [34], [45], [-30], [31], [-48], [1], [-34], [21], [16], [-3], [-16], [-10], [-2], [12], [35], [30], [36], [-15], [2], [-56], [25], [24], [-7], [-54], [-16], [-18], [-42], [-42], [-23], [-1], [3], [-36], [-27], [44], [-41], [29], [0], [21], [22], [66], [-15], [-4], [11], [-42], [-16], [-48], [-24], [-5], [38], [-39], [36], [-30], [-49], [40], [45], [-8], [-36], [11], [-15], [48], [8], [24], [23], [-64], [22], [-57], [26], [24], [-14], [9], [-12], [-66], [-42], [50], [-64], [-47], [-9], [42], [-38], [-60], [34], [-48], [22], [30], [48], [5], [54], [0], [-10], [-8], [-42], [43], [-16], [-12], [7], [29], [3], [-10], [6], [15], [40], [15], [41], [-32], [-17], [-52], [-70], [-2], [54], [73], [-23], [12], [24], [-68], [-42], [55], [4], [48], [-61], [21], [76], [21], [-18], [0], [-63], [18], [-46], [-75], [32], [-3], [72], [58], [41], [-3], [-16], [-72], [68], [-23], [12], [-19], [18], [-49], [12], [54], [51], [-10], [48], [-67], [-66], [30], [70], [-33], [26], [22], [-75], [16], [-42], [-55], [52], [10], [18], [-81], [13], [27], [80], [-12], [-56], [-24], [13], [-18], [31], [-22], [22], [51], [-57], [-46], [-66], [-42], [77], [-80], [30], [42], [64], [-38], [33], [86], [11], [-15], [60], [-48], [-33], [84], [-71], [-27], [24], [54], [-28], [62], [78], [82], [83], [36], [-10], [72], [-21], [26], [35], [-36], [-54], [-50], [9], [17], [-66], [-39], [-38], [-54], [-14], [-36], [-62], [-82], [-74], [-18], [73], [15], [54], [50], [-18], [-10], [34], [-30], [50], [-51], [34], [30], [-20], [-45], [63], [50], [-98], [-27], [-50], [-84], [77], [-69], [60], [80], [83], [33], [60], [-82], [54], [30], [39], [-29], [0], [30], [46], [-21], [6], [-15], [-8], [-42]];
    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_k();".
// 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_k(: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_k();".
// 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_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2366_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![-3, 1]>],Snew);
    return Vf;
end function;