
// Make newform 2233.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2233_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_2233_2_a_b();".
// 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_2233_2_a_b();".
// 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_2233_a();"
function MakeCharacter_2233_a()
    N := 2233;
    order := 1;
    char_gens := [1277, 1828, 2003];
    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_2233_a_Hecke(Kf)
    return MakeCharacter_2233_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 := [[0], [1], [0], [1], [-1], [2], [6], [-7], [-3], [1], [-4], [2], [9], [-4], [-9], [9], [-6], [2], [-13], [-9], [-1], [-10], [-6], [-15], [-7], [-15], [-4], [-3], [5], [12], [8], [0], [-6], [-16], [9], [17], [23], [8], [6], [-6], [-12], [-16], [-24], [14], [-3], [-10], [-28], [20], [12], [-1], [3], [12], [8], [21], [-24], [6], [-3], [-25], [-1], [-18], [14], [15], [-4], [-3], [14], [24], [32], [32], [36], [2], [-18], [-6], [17], [-10], [-16], [12], [-36], [-16], [-3], [-25], [0], [26], [-3], [-1], [2], [12], [-24], [-31], [18], [29], [-9], [27], [11], [-12], [-7], [12], [42], [24], [-28], [2], [35], [15], [-12], [-24], [11], [2], [30], [-30], [-36], [35], [-7], [5], [36], [44], [11], [12], [26], [-18], [-48], [-36], [-10], [-43], [-15], [12], [8], [-21], [-10], [36], [-13], [41], [-34], [-24], [-10], [2], [-18], [-34], [-51], [32], [18], [-30], [2], [-42], [-22], [-30], [47], [21], [50], [6], [20], [3], [5], [-21], [-37], [-9], [38], [-6], [-43], [-12], [38], [-54], [-42], [9], [-4], [-48], [-18], [24], [-25], [-19], [35], [-15], [-45], [-16], [24], [-34], [8], [-42], [44], [-54], [-52], [-46], [26], [-21], [-19], [-18], [24], [42], [-40], [35], [-10], [-18], [-4], [0], [50], [30], [-12], [54], [38], [-31], [-30], [18], [-24], [14], [-10], [59], [-63], [36], [-13], [3], [45], [-1], [62], [6], [50], [36], [-51], [-52], [17], [-27], [-51], [6], [-4], [56], [-54], [56], [36], [-46], [33], [0], [-16], [15], [38], [8], [-34], [-30], [8], [72], [50], [6], [12], [-3], [-42], [-28], [-4], [14], [24], [-30], [14], [42], [-13], [-12], [-4], [36], [18], [-37], [69], [-15], [-28], [-37], [-78], [20], [8], [18], [50], [23], [51], [-1], [-30], [-18], [56], [-6], [-64], [17], [-46], [20], [-43], [11], [-24], [-79], [74], [0], [-54], [68], [-48], [8], [-61], [-78], [-52], [-6], [-46], [54], [-54], [42], [-3], [12], [-4], [-54], [-37], [-54], [54], [44], [-49], [33], [50], [66], [44], [47], [78], [-76], [24], [-70], [0], [30], [-63], [68], [-45], [-70], [72], [-27], [-64], [24], [-28], [83], [60], [-46], [-63], [2], [59], [26], [-66], [9], [-10], [-15], [32], [-51], [-40], [-78], [-55], [54], [-13], [-70], [14], [42], [3], [-16], [-18], [0], [38], [-34], [63], [-6], [50], [-58], [-48], [-64], [68], [42], [-15], [-6], [12], [-84], [-7], [-48], [0], [93], [-28], [44], [-75], [-16], [74], [12], [-22], [45], [60], [56], [65], [9], [12], [-1], [90], [44], [18], [9], [-88], [48], [14], [-18], [-70], [50], [56], [-48], [92], [54], [-12], [65], [18], [41], [-10], [42], [-1], [-36], [-40], [36], [68], [-72], [-12], [104], [-82], [-66], [101], [12], [8], [78], [-21], [44], [2], [45], [102], [-10], [-9], [24], [-9], [74], [78], [9], [-76], [-42], [-24], [36], [-88], [-21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2233_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2233_2_a_b();".
// 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_2233_2_a_b(:prec:=1)
    chi := MakeCharacter_2233_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_2233_2_a_b();".
// 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_2233_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2233_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;