
// Make newform 2366.2.a.d in Magma, downloaded from the LMFDB on 19 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_d();".
// 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_d();".
// 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], [0], [-2], [1], [-4], [0], [-6], [0], [8], [-10], [8], [-6], [6], [4], [8], [6], [-8], [10], [-4], [8], [-2], [8], [0], [-18], [-2], [-14], [16], [-4], [2], [2], [16], [8], [6], [-8], [18], [0], [2], [4], [0], [2], [-12], [-6], [-8], [-2], [18], [-16], [20], [0], [0], [6], [10], [24], [-18], [24], [18], [0], [-30], [32], [6], [-10], [16], [14], [-32], [24], [2], [-6], [28], [2], [-12], [14], [-10], [0], [-16], [-10], [-4], [-24], [30], [-2], [-18], [38], [0], [2], [24], [-38], [8], [-36], [14], [-10], [-18], [-16], [8], [-24], [16], [28], [-4], [-8], [14], [-30], [0], [10], [28], [42], [-16], [-38], [-12], [-42], [16], [22], [16], [10], [-48], [-38], [-10], [-40], [8], [34], [-8], [-32], [-42], [12], [-2], [-14], [42], [-28], [-16], [-42], [10], [8], [16], [-26], [36], [-16], [48], [30], [6], [30], [46], [-32], [18], [-6], [-8], [-14], [48], [36], [10], [-32], [-10], [34], [8], [32], [-30], [-30], [-44], [-32], [12], [0], [48], [30], [42], [-10], [-12], [26], [-32], [48], [30], [-32], [24], [2], [46], [-22], [36], [30], [16], [38], [-24], [10], [52], [26], [-8], [26], [24], [-32], [46], [-2], [56], [58], [-50], [0], [-26], [-24], [34], [36], [4], [-2], [-44], [42], [46], [-26], [-26], [-8], [42], [56], [-2], [-14], [8], [-54], [8], [12], [6], [-24], [50], [-30], [8], [-40], [24], [-10], [-8], [26], [8], [18], [-2], [-56], [-18], [56], [24], [-34], [26], [0], [-56], [12], [-26], [16], [0], [10], [-64], [64], [-34], [50], [12], [56], [-12], [16], [24], [-38], [22], [0], [64], [32], [-68], [-24], [-38], [22], [-40], [-22], [58], [44], [30], [64], [18], [38], [64], [12], [-2], [10], [-18], [16], [-22], [-50], [36], [-66], [34], [-28], [30], [32], [2], [-72], [-20], [-6], [-58], [24], [-16], [24], [32], [46], [-72], [64], [-46], [-62], [16], [2], [30], [8], [6], [-8], [-2], [18], [48], [-70], [-8], [40], [-46], [-62], [-24], [-4], [52], [-50], [52], [-54], [-64], [6], [0], [58], [18], [76], [-56], [-70], [72], [-80], [-74], [50], [-16], [-10], [-54], [16], [22], [10], [-28], [-72], [16], [14], [26], [38], [56], [-24], [-44], [-56], [82], [70], [-66], [-88], [18], [-6], [-10], [-32], [-14], [76], [10], [28], [64], [18], [8], [38], [-54], [56], [6], [-6], [24], [-24], [-78], [-48], [-46], [42], [-16], [4], [56], [58], [30], [-16], [42], [36], [40], [64], [6], [-24], [-18], [48], [48], [-26], [-22], [82], [-18], [-74], [24], [62], [-40], [72], [48], [-94], [52], [8], [-66], [-34], [36], [80], [0], [10], [-24], [50], [60], [-6], [38], [98], [32], [38], [102], [-64], [74], [-18], [104], [72], [34], [78], [-76], [-44], [-22], [18], [-96], [16], [-66], [-24], [-106], [-82], [96], [24], [22], [38], [-4], [54], [80], [64]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2366_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![2, 1]>],Snew);
    return Vf;
end function;