
// Make newform 4368.2.a.br in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4368_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_4368_2_a_br();".
// 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_4368_2_a_br();".
// 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 := [6, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [-1, -4, 0, 1], [-7, 0, 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_4368_a();"
function MakeCharacter_4368_a()
    N := 4368;
    order := 1;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    v := [1, 1, 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_4368_a_Hecke(Kf)
    return MakeCharacter_4368_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, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, -1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 0], [-1, 0, 0, -1], [-1, -1, 1, 0], [0, -1, -1, 1], [1, -1, 1, 0], [1, -1, 1, 2], [4, -2, 2, 0], [-4, 1, 2, 0], [-1, 1, 1, 0], [-2, 0, -1, -1], [1, 1, 1, 0], [5, -1, 0, -1], [4, 0, 0, 2], [4, 2, 2, -2], [-2, 3, -2, 0], [-3, -1, 3, -2], [-3, -1, 1, -2], [0, 0, -1, 1], [-3, 2, -3, 0], [-3, -1, -1, 2], [-9, 0, 2, 1], [-8, 2, 0, 0], [-3, -2, 2, 1], [0, -2, 0, -2], [-1, 1, -1, 0], [-8, -2, 0, 0], [-2, 2, 0, -2], [1, -1, -2, 3], [-6, 0, 2, 0], [-5, 1, -4, -1], [2, -2, -4, 2], [-4, 2, -6, 0], [4, 2, 0, 0], [-16, 2, -3, -1], [-3, -2, -4, 1], [2, 1, -1, 3], [0, 0, -4, 2], [-13, 4, -6, -1], [-2, 4, 0, -4], [-11, -1, 4, -3], [2, 0, 0, -2], [-1, 3, 1, 0], [-3, -3, -1, 0], [1, -1, -4, -1], [-6, 0, 4, 0], [-5, -1, -3, -2], [-8, -1, 6, -2], [1, -1, 7, -2], [2, -4, 2, 0], [-15, 0, -2, -1], [-10, -3, 1, 1], [-13, 2, 0, -1], [10, 0, 4, 2], [13, -1, 3, 2], [-9, 1, 0, 3], [-6, 4, 0, 2], [-5, 2, -5, 0], [3, -1, 3, 2], [10, 0, -2, 0], [-12, 4, 0, -2], [3, 1, -2, 5], [14, -2, -2, 0], [11, -5, 3, 0], [-7, 0, 2, -3], [5, 1, -5, 2], [-18, 1, 2, -2], [6, -1, 6, 0], [-4, 4, 6, -2], [-4, 4, -4, -2], [4, 0, -2, 2], [-13, -1, 4, 1], [6, 0, 0, 0], [9, 1, -5, 6], [11, 5, 2, -3], [1, -1, 3, 2], [-2, 4, 2, -4], [10, -4, -2, -2], [-2, -1, 4, -6], [10, -2, 2, 6], [-16, 4, -6, -2], [4, 3, -5, -3], [9, -3, 10, -1], [2, -4, -2, 2], [0, 3, -6, 0], [0, -6, 2, 6], [-2, 0, -6, 4], [2, 0, -3, 5], [4, 0, -6, -6], [-21, 2, -2, -1], [20, -4, -4, -4], [-20, -4, 2, 2], [-17, 0, 7, 0], [-19, 2, -4, -3], [-6, -4, 2, 0], [6, 0, -2, -6], [3, -5, 3, 2], [17, 3, 6, -1], [-6, 4, 0, -6], [-5, -1, -3, -2], [5, -1, -7, 2], [2, 6, 0, 0], [-6, -4, 3, 3], [-7, 0, -9, 2], [18, -5, 3, 3], [-12, 4, 0, -2], [-4, 0, 6, -2], [8, -2, -2, 8], [-9, 5, 0, 3], [10, 4, -4, -2], [-2, 0, -2, -4], [23, 3, -1, 0], [0, 4, -4, 0], [-24, -2, 6, -6], [24, -2, 4, 2], [-6, -5, -7, 1], [17, -7, 7, 2], [9, -1, -9, -2], [-23, -2, 2, -1], [-8, 7, -2, -6], [-19, 3, -5, 0], [-3, -5, -5, 2], [-4, 6, -4, -2], [12, -4, 2, 10], [2, -2, 2, -4], [-15, -1, -1, 2], [32, 2, 6, -2], [-22, 3, 2, 0], [-9, -7, -3, 8], [-29, 3, -1, 0], [21, 2, -1, -2], [-17, 3, 5, -6], [-2, 1, -2, 6], [-1, 1, 5, -4], [1, -6, 2, -1], [-13, 1, -13, 0], [16, -2, -12, 0], [3, 1, 0, 3], [4, -4, -8, -4], [-10, -1, -2, 4], [6, -4, -12, 4], [9, -5, 4, 11], [3, 3, 9, -2], [-19, 0, -2, -5], [4, -6, -6, 6], [-14, 1, 6, 4], [20, -6, 10, 8], [3, 4, 6, -3], [6, -2, -16, 6], [-6, 0, 4, 2], [3, -9, 5, 0], [-10, -1, -3, 5], [-2, 2, 0, -6], [-3, -6, -5, 2], [-14, 6, 6, 2], [11, -4, -11, -2], [10, -1, 4, -6], [15, 5, -5, -4], [-22, 4, 2, -4], [-2, 6, 2, -4], [-19, -1, 4, 1], [16, 0, 5, -1], [14, 0, 0, 10], [28, 2, 6, -4], [4, -2, -8, -2], [-27, -2, 4, -7], [34, 3, 0, -2], [-6, -10, 0, 4], [12, 1, -3, -9], [-2, -4, -10, 6], [14, 2, 4, 6], [3, 0, 0, 3], [10, -4, 0, -6], [1, 1, 8, 5], [-26, 0, -6, 0], [4, -8, 4, -2], [2, 4, -4, 2], [22, 6, 2, -4], [-12, 0, -12, -2], [5, 8, 11, -6], [14, -3, -12, 2], [-31, 6, -6, -1], [-1, 5, 1, 2], [-7, -5, 5, 2], [-4, -2, 0, 6], [14, -10, 13, 1], [38, -6, 10, 2], [-8, 5, -4, -4], [11, -3, 17, -4], [-1, -2, -5, 4], [22, -1, -3, 1], [25, -2, 4, 9], [-18, 4, -20, 0], [15, 3, -5, 0], [9, 6, 13, -4], [-12, 6, -14, 2], [-5, 13, -2, -3], [-14, -2, 8, -2], [-27, 7, -5, -6], [10, -14, 10, -2], [-10, 2, -1, 3], [-17, -4, -4, -1], [23, -1, 9, 4], [14, -9, 11, 3], [-19, 5, -18, -1], [-8, 2, 10, 2], [-15, -3, -11, 0], [5, 0, 0, 5], [-6, -6, 8, 6], [-11, 5, 0, 7], [8, -2, -9, 5], [-5, 7, -7, -6], [-21, 11, -3, -4], [27, 0, 4, -1], [24, -5, 2, 10], [-29, -5, -2, -3], [-15, -5, 7, -6], [14, 4, 12, -2], [37, -7, 6, 7], [-4, 8, 6, -2], [-12, 0, 6, -6], [-5, 9, -7, 6], [13, 4, 4, -3], [6, 9, -3, -3], [-18, -6, 2, 0], [-12, -1, -16, 4], [-6, -12, 4, 0], [18, -2, -4, 6], [-12, 4, 0, -4], [-2, 4, -12, -8], [-16, 0, 6, -2], [-60, 0, 3, -3], [-22, 0, 0, 0], [5, -1, -6, -1], [-4, -1, 13, -5], [-6, 6, 2, 4], [-24, -5, 2, -2], [-22, 6, -2, -12], [10, -4, -4, 10], [18, 8, 0, 4], [13, 2, -7, -4], [20, 6, -4, 0], [5, 7, 3, 4], [14, -4, -3, 5], [8, -10, 4, 0], [-26, -7, 5, 5], [-10, 12, 2, -2], [10, -7, -6, 10], [42, -1, -8, 2], [-42, 0, 8, -4], [-9, 4, -10, -11], [14, 3, -6, 8], [1, 9, 5, -12], [-38, -2, 0, -6], [35, 0, 4, -5], [-11, 5, -1, -6], [-7, 3, 17, 2], [0, 3, 1, 11], [13, -5, 7, -2], [-2, -4, -16, 0], [16, -9, -6, 4], [-30, 2, 2, 0], [1, 3, 16, -3], [17, -12, 5, 12], [0, 2, -6, 2], [0, -14, 4, 2], [-16, -4, 8, -2], [12, 0, -12, -4], [22, -4, -12, 8], [5, -1, -17, 0], [-8, 6, 6, -4], [-22, -12, -4, 2], [-16, 1, 2, 6], [28, -2, 2, 0], [4, -2, -2, -8], [8, 10, 0, -4], [-30, 10, 2, 4], [-28, 10, -14, -6], [-4, -8, 4, -8], [13, -11, 11, -6], [-46, 4, -12, -2], [20, -7, -1, -3], [-7, 13, -3, 0], [1, -9, 2, 11], [24, -6, -12, -4], [7, 10, -4, -13], [41, -7, 5, 12], [-8, 12, 4, 0], [13, -1, 16, -3], [22, -12, -7, -7], [17, -5, 7, 2], [-21, 8, 8, 7], [-24, -8, -2, 10], [5, 4, -4, 9], [-22, 4, 12, -10], [-13, 1, 9, 4], [-14, -4, 0, 8], [5, -3, -10, 3], [21, -7, 1, -2], [12, 9, 3, -3], [19, -5, -5, 2], [-24, 2, 4, 6], [-27, 8, -2, 5], [6, 6, 2, -10], [30, 3, 4, 2], [-3, -3, 5, 8], [14, -6, -4, 6], [-45, 15, -16, -5], [6, -10, -16, 4], [-3, 1, 11, 0], [-12, -5, -2, 2], [4, -2, 6, 8], [-12, 4, -7, -5], [16, -13, 0, 4], [-41, 13, -3, 2], [3, -1, 3, 0], [-50, 6, 4, 2], [-52, 6, -14, -4], [9, -6, -2, 11], [-2, 2, -12, 6], [-15, 9, 10, -1], [0, 2, 2, 8], [-16, -2, 0, 4], [-17, 3, -17, 2], [-33, 8, -2, -1], [-16, 10, 0, -10], [36, -2, 12, -2], [9, -17, 13, -4], [-2, -6, -10, 4], [22, 8, -5, -9], [10, 12, -6, 0], [8, 2, -11, -5], [-16, -6, 2, 4], [28, -7, 10, 0], [-34, -4, 4, 8], [-14, 0, 18, -12], [-24, 10, -22, -4], [-22, -2, 8, 0], [-20, -3, -2, 0], [30, 12, 4, -2], [5, -3, -2, -5], [15, 4, 14, -5], [-10, 4, -4, 4], [-6, -14, -4, 6], [10, -2, -11, -3], [-23, 2, 6, -5], [25, -3, -19, 10], [-40, -2, -8, 2], [1, 3, -10, -1], [8, 2, 2, -10], [-44, 4, 4, -2], [-5, 8, 18, 1], [16, 0, 11, -7], [10, 6, -1, -1], [60, 2, 0, 6], [-10, 3, 8, 6], [6, -4, 4, 12], [-5, -6, -2, -3], [-70, 3, -9, -1], [16, -3, -2, -6], [-38, 0, -2, -4], [39, 5, 5, 2], [27, -8, 27, 0], [-34, 4, 2, 8], [-1, -11, 13, 2], [-9, -14, -2, 11], [-40, 4, -18, -2], [38, -5, 2, 12], [-19, 1, 5, 0], [-4, 8, -12, -8], [-37, 5, -3, -6], [-3, -3, -8, -1], [-3, -4, -10, 1], [38, 2, 12, 0], [-21, 0, 8, -5], [4, -8, 8, 6], [39, -2, -1, -8], [1, 1, 0, 1], [20, 4, 0, 4], [1, 1, 14, -13], [11, -3, 7, 6], [18, 14, 3, -9], [-12, 12, 4, -8], [28, 2, 2, 0], [-12, 6, 4, 4], [16, 10, -8, 0], [40, 2, 8, -6], [-21, 4, -15, -6], [-38, -1, 6, -8], [-18, 20, -12, 2], [34, 3, 14, -4], [-69, 3, -5, 0], [-24, -4, 16, -8], [21, -14, 10, -1], [-33, 5, -5, -2], [-7, 1, -18, -1], [35, -5, 17, -2], [-5, -7, 5, -10], [-36, -10, -6, 2], [25, -14, 5, -4], [23, -4, 11, 12], [-6, -4, -20, 10], [-14, -4, 6, -10], [9, -7, 10, -1], [-11, -5, 7, -4], [2, -16, -9, -1], [4, -6, -10, 0], [-7, -1, -1, -6], [-32, 3, 13, -9], [-2, 8, -4, 10], [15, 9, -19, 2], [49, 4, -8, 9], [0, -1, 18, -6], [16, -4, 8, 16], [24, -5, 6, 0], [-8, -6, 0, -8], [13, -5, -19, 8], [-34, 2, 8, -8], [-38, 3, 6, 4], [2, 14, 4, -2], [-18, -10, 0, 4], [39, 6, -9, 0], [48, -9, -5, -7], [3, -11, 14, 1], [31, 1, -9, -6], [28, 2, -6, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4368_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4368_2_a_br();".
// 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_4368_2_a_br(:prec:=4)
    chi := MakeCharacter_4368_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_4368_2_a_br();".
// 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_4368_2_a_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![24, -20, -10, 3, 1]>,<11,R![96, 32, -24, -2, 1]>,<17,R![96, -40, -28, 2, 1]>],Snew);
    return Vf;
end function;