
// Make newform 4592.2.a.bb in Magma, downloaded from the LMFDB on 04 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4592_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_4592_2_a_bb();".
// 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_4592_2_a_bb();".
// 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 := [-3, 6, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-3, 0, 1, 0, 0], [4, -1, -6, 0, 1], [-2, -3, 5, 1, -1]];
    Rf_basisdens := [1, 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_4592_a();"
function MakeCharacter_4592_a()
    N := 4592;
    order := 1;
    char_gens := [575, 3445, 3937, 785];
    v := [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_4592_a_Hecke(Kf)
    return MakeCharacter_4592_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]];
    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, 0], [-1, 1, 0, 0, 0], [-1, 0, -1, 1, 0], [-1, 0, 0, 0, 0], [1, -1, 2, 0, -1], [1, 0, 1, 0, 1], [3, 0, 0, -1, -2], [1, -1, 1, 2, 1], [-1, -1, 0, -2, 0], [0, 1, 1, 1, -1], [-3, -2, -1, 1, 0], [-2, 1, 1, -2, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 3, 0], [0, -3, -1, 2, -2], [3, -2, 1, 3, 0], [-3, 4, -3, -1, -2], [5, -1, 0, 2, 1], [1, 2, 1, -1, -2], [5, 1, 0, 4, 3], [9, -3, 0, 0, -1], [8, -2, -2, -2, -2], [5, -3, 0, 0, -5], [1, 1, 0, -1, 0], [3, 5, 0, 2, 0], [3, 2, -1, 1, -1], [2, 1, -3, 1, 1], [4, -1, 2, 1, 1], [-6, 4, 0, -2, 4], [-5, 2, 1, -7, -3], [-5, -6, 1, 0, 3], [-7, 4, -3, -3, -2], [5, 0, 1, -1, 2], [3, 1, 2, -2, -3], [6, -2, 0, 2, -2], [3, 5, 4, -4, -1], [10, 7, -2, 1, 3], [2, -3, 3, -3, -4], [5, -2, 4, 7, 6], [-5, 3, -4, -2, -5], [1, 6, 1, 1, 4], [0, 1, 3, 1, 4], [-4, -2, -2, 0, 4], [5, -3, 2, 0, 3], [-9, 1, 3, 4, 5], [-1, 6, -1, 0, -1], [-2, 7, -1, 1, 5], [-1, 0, 3, 3, 2], [3, -1, 2, -6, -3], [6, -7, 1, -1, 0], [6, -3, -3, 1, -1], [4, -9, -5, 3, -1], [2, 2, 4, 2, 4], [3, -13, -2, 2, -5], [7, 0, 1, 7, 1], [1, 3, 0, 4, 7], [-10, -7, -1, 1, -3], [-17, 3, -6, 0, -3], [1, 6, 3, 3, 1], [11, 1, 2, 4, 3], [8, -4, 0, -4, 0], [-3, -1, -2, 2, 1], [-14, 2, -2, -4, 4], [-2, 8, -1, 0, 8], [7, 0, -2, 3, -2], [-1, 7, 2, 8, 5], [12, -4, -4, -4, -4], [4, 1, 5, 1, 6], [-20, 4, -2, -4, 0], [3, -11, 4, 4, 1], [10, -6, -6, 2, 0], [2, -11, 3, 0, -2], [6, -7, 11, -1, -1], [21, 1, 3, -2, 1], [8, 7, -5, 1, 4], [11, -8, -3, 9, 1], [0, -14, 6, 3, 3], [1, -6, 3, 3, -2], [2, 3, -1, -2, 0], [-15, 11, -6, -4, -3], [-4, 1, 7, 7, 9], [-2, 6, -2, -6, 4], [1, 3, 0, 4, 1], [1, -12, -1, 3, 4], [-9, -6, 3, -9, -5], [-3, -3, 4, 5, -2], [-8, 1, -7, 1, -2], [7, -3, -2, -6, -11], [-3, -1, -2, 2, 1], [-4, -3, -5, -9, -7], [17, -5, 4, 6, 7], [-9, -5, -6, 2, -2], [2, -3, -2, -3, -1], [2, 8, -8, 2, -1], [4, -2, 8, -2, -4], [-22, 0, -2, -6, -5], [-6, -2, 2, -1, 1], [7, -9, -4, -1, -6], [19, -1, 6, 8, 7], [-11, 1, -1, -2, -3], [-1, 1, 6, -8, 1], [-2, -1, 1, -5, 1], [-5, -2, -5, -4, -9], [1, 7, -5, -4, -9], [-2, 10, -10, 4, 8], [11, -12, 5, 3, -3], [-1, -2, -10, 1, -6], [2, -1, -7, 6, 4], [12, -7, -11, 0, 0], [10, -2, -3, -2, -12], [-3, -5, 0, 4, -3], [23, -5, 8, 4, -4], [-19, 4, 0, -5, 6], [-2, -13, 7, -1, 1], [25, 2, -1, 8, 1], [15, 3, -2, -4, 1], [8, 2, -2, 11, 3], [-1, 11, -10, -6, -5], [-21, 14, 7, -3, 0], [1, 1, 8, -8, 1], [0, -8, 14, -8, -8], [25, -5, -2, -2, -9], [9, -8, 1, 5, 4], [-3, 1, 4, 2, 7], [-20, 2, 2, -1, 7], [7, -17, 14, 7, 10], [-8, -2, -4, -8, -12], [-17, -1, -6, -2, 1], [8, 16, -3, -2, -8], [15, -5, 4, -8, -11], [2, 12, 8, -9, -7], [0, 14, 7, 2, 10], [27, -11, 4, 10, -3], [0, -2, -8, -2, -8], [0, -12, -2, 4, 2], [8, 0, -6, 0, -12], [1, 2, -4, -5, -8], [-4, -14, 2, -2, -2], [1, 5, -2, 2, -3], [9, -8, 3, -3, -6], [3, -1, 2, -10, -7], [-5, 14, -6, -9, -4], [4, -13, -1, -7, -5], [-1, 9, 2, 0, 1], [23, 7, 6, -8, -1], [-12, 15, 9, -3, 6], [6, -12, 0, 0, -4], [-20, 5, -7, -11, 1], [19, -10, 1, 5, -8], [3, 15, -6, 2, 13], [-14, 4, 0, -8, 3], [-22, 12, -4, -12, -8], [-13, 0, -1, -3, 8], [2, -3, -7, -4, 2], [20, -8, -2, -5, -5], [-10, 19, -4, -5, -3], [12, -9, -3, 3, -5], [0, -12, 4, -2, -7], [27, -8, -1, 10, 1], [-31, 0, -5, 5, -4], [6, 0, 0, 2, -11], [-28, 14, -2, -7, 5], [-20, 10, -12, -8, -10], [-3, -28, 2, 1, -4], [22, 5, -13, 1, 1], [-15, 7, 2, 8, 19], [-5, 13, 4, -8, 5], [4, -4, -4, 2, 5], [6, -4, -5, 14, 6], [13, -5, -3, -10, 3], [27, -11, 0, 2, -11], [-10, 8, -2, -4, -8], [3, 14, -1, 7, 2], [14, 1, 1, -2, 8], [-20, -9, 6, -9, -3], [-4, 3, 3, -9, -11], [-8, 11, -5, -13, 1], [-19, 4, -3, 9, 13], [17, 16, -1, -2, 7], [15, 3, 0, 0, -7], [6, -4, -4, 11, 5], [-15, 1, -8, 4, 5], [-4, 10, 4, 16, 14], [0, 5, 5, 9, 3], [15, -9, 4, 1, -4], [-28, 8, -6, -8, 6], [21, -4, -1, 8, 9], [4, 13, 9, -5, -1], [-9, 17, -16, 2, 3], [-19, 3, -4, 1, -6], [-19, 2, -15, 5, 10], [-11, -1, 6, 2, 3], [23, 2, 14, -1, 6], [-3, 5, -2, 6, -3], [-33, 1, -4, 8, -5], [11, -20, 1, 3, -14], [-7, 7, 0, 16, -1], [11, -16, 9, 5, -14], [-3, -21, 12, 6, 6], [6, -3, -7, -7, 7], [-11, -15, 0, 8, 3], [5, 7, 6, -8, -5], [-14, -11, 6, 7, 9], [18, -10, -5, 14, 0], [-2, -18, 0, 3, 5], [0, 6, 12, 8, 10], [-27, 9, 6, -12, -5], [-1, -11, -2, -12, -17], [8, 9, 1, -15, -8], [23, -10, -9, 17, 6], [-10, -2, -18, 4, 8], [18, -7, -13, -4, -2], [-13, 6, 5, -6, -7], [7, 5, 6, -8, -5], [-4, -29, 5, 5, -4], [0, -11, 11, -5, 1], [-9, 7, -2, -2, 9], [-20, -6, -14, 8, 6], [15, 0, 13, 5, 4], [7, -6, 5, -3, 2], [-7, 14, -1, 5, 10], [-22, -4, 4, 14, 12], [7, -15, 10, 4, 13], [10, 6, -10, 12, 0], [-6, 5, -13, -1, -11], [10, -17, 1, -5, -15], [-26, 6, 2, 4, 4], [15, -11, 15, -4, -5], [17, -17, 13, -2, 13], [-20, 10, -4, -15, 5], [38, -3, -5, -9, -12], [-6, -3, 1, 6, -8], [8, -15, -11, 3, 2], [17, -9, 18, 14, 2], [41, -23, 2, 10, -5], [-24, 3, 1, -2, 2], [4, 15, 7, -12, 4], [-11, 4, -11, -1, 0], [-14, -15, 3, -12, 2], [9, -11, 1, -8, -13], [7, 0, 9, -11, 10], [35, 2, -1, -13, -15], [-10, -16, 10, -10, -3], [-9, -15, 6, 6, 14], [-24, -8, -4, 4, 14], [26, -11, 11, -9, 7], [0, 5, -1, -7, 11], [-17, 18, -13, -5, 13], [-21, 4, 2, -23, -12], [-26, -20, 10, -2, -8], [-20, 9, -10, 3, 1], [-4, -4, -2, -24, -8], [-12, -4, 7, 12, 12], [-17, -16, -1, 11, 8], [12, -6, 12, -7, -11], [-3, 11, 7, 16, 5], [-1, -5, 12, 4, 17], [-20, 9, -13, -9, -6], [4, -21, 1, 3, -16], [11, 24, 0, 3, 18], [25, -12, -7, 3, -3], [24, -7, 1, 11, 7], [-11, 3, 17, -6, -5], [-12, -15, 8, -3, 5], [-20, 14, -10, -14, -2], [7, 21, 6, 0, 15], [14, 13, -1, -5, -3], [-11, 13, 14, -8, 1], [19, -14, 14, 7, 14], [25, -20, 1, 9, 8], [-14, 17, -5, 11, -1], [-8, -1, -1, 8, 16], [22, 9, -3, 15, 1], [6, -13, -11, -4, -10], [24, 8, -6, -1, 3], [-1, 0, -13, -3, -4], [28, 17, 5, -11, -17], [-34, 9, -7, -3, 9], [11, 13, -2, -8, -1], [22, 6, -12, 8, 3], [10, 3, -8, 13, 1], [32, 6, 10, -12, -16], [-13, 26, 13, -13, -4], [27, -6, 11, -15, 8], [-9, 1, 0, 1, 0], [33, -4, -11, -11, -10], [-21, 4, -17, 7, -2], [42, -8, 4, 4, -4], [-33, 4, -7, 1, 21], [37, 2, 7, -4, 9], [-6, 0, 8, 2, -2], [21, -11, -10, -16, -23], [48, 0, 0, 10, 2], [-14, 9, 3, -9, 11], [7, 6, -11, -9, -2], [-19, 5, 7, 4, 9], [-31, 24, 15, 0, 5], [26, -4, 12, -16, -10], [3, 23, -6, -5, 14], [-36, 0, -8, 0, -5], [18, -7, -7, 5, -3], [25, -11, 12, 8, -3], [23, -2, 8, 19, 10], [-29, 5, 4, -4, -3], [39, -4, 3, 2, 5], [-21, 6, -15, 9, 11], [22, 3, 14, -17, 1], [-16, 20, -16, -4, 16], [21, -11, 4, 18, 9], [26, -6, 4, -12, -18], [17, 15, -4, -4, -10], [-7, 9, 14, -4, 11], [-11, -7, -3, -18, -19], [13, 7, -6, -11, -10], [-44, -2, -1, -14, -2], [-9, -27, -2, 0, -13], [-50, 3, -7, -3, -7], [30, -8, 20, -10, -5], [0, -8, 21, 10, -2], [-25, 9, 0, -12, -5], [18, -8, 10, -20, -12], [-14, -7, -6, 17, -7], [25, 33, -4, 0, 5], [-8, 13, 3, 1, -7], [-1, -8, -7, -24, -17], [12, -8, 2, -14, -20], [-20, 9, -15, -15, -3], [26, -9, -8, -3, -7], [-13, 27, -10, -4, -13], [-33, -3, -3, 12, 21], [-55, 18, -13, -15, 0], [53, -3, 12, -4, -1], [-1, 11, -10, -10, -3], [-9, -8, -8, -1, 4], [-31, 2, 3, -7, -3], [2, -17, 11, 7, -6], [7, 9, 20, 0, 19], [-1, -4, 1, -3, -8], [-10, -13, 13, 2, -8], [18, -9, 15, 15, 13], [-1, -16, -7, -7, 4], [-21, -21, 12, 0, 12], [24, -17, 7, 19, 5], [45, -8, 13, -5, 12], [-11, -9, -6, -2, 4], [-26, -2, 2, 1, 7], [5, 25, 10, -14, 11], [-20, -6, 4, 18, 9], [21, -16, 3, -4, 13], [-9, -5, 6, 8, 1], [31, -6, -8, 21, 6], [27, -2, -2, -5, -8], [5, 23, 14, -10, 5], [27, -12, 7, -3, -20], [-43, 7, 6, 4, 11], [4, 8, 8, 14, 15], [-23, -9, -6, 2, -15], [20, 11, -1, -15, -17], [-24, -3, -4, 7, -7], [4, -3, -16, 3, -1], [-9, -15, 4, 9, 8], [29, 3, -5, -2, 3], [8, -10, 2, 6, -2], [10, 3, -7, 11, 0], [12, -6, -10, 30, 7], [-3, 1, 18, 22, 25], [5, -3, -14, 0, 10], [27, 25, -19, 4, 3], [-23, -10, -3, 8, -19], [24, 14, 14, -6, 24], [9, 4, 15, -5, -15], [19, -8, -21, -3, -7], [-5, -37, 18, 2, 3], [32, -17, -11, 7, 1], [-19, 27, -8, -18, -9], [-2, -11, 1, -1, 9], [-48, 12, 0, -12, 12], [41, 5, 14, 8, 13], [-31, -11, 7, -4, -7], [0, 5, -17, -19, -6], [15, -32, -8, 3, -18], [-5, 6, -14, 3, 8], [13, -12, -17, -9, 0], [-27, 22, -2, 9, 12], [-6, 31, -17, -5, -3], [-12, -14, -8, 14, 4], [9, -30, 23, -3, -2], [-16, 37, -1, -11, 2], [-5, 4, 1, 13, -4], [-5, 9, 24, 0, -1], [8, -35, 21, 13, 17], [19, 2, -7, 0, -13], [22, -26, 14, 14, 0], [1, 5, 2, -10, 1], [-2, 5, -29, 1, -8], [18, -23, -3, 1, 1], [-19, -4, -5, 2, -21], [-19, 5, -18, -16, -13], [22, -2, 2, 28, 17], [39, -22, 1, 1, -4], [22, 23, -9, 5, 5], [17, 7, -24, -4, -1], [13, 30, -1, -3, 9], [9, 6, -3, -5, 11], [-62, 9, -5, -25, -3], [4, 4, -4, 16, -13], [25, 6, -18, -15, -20], [26, 4, 14, -2, 24], [22, -7, 9, -4, 10], [-20, -18, -10, 8, -6], [57, -5, 8, 16, -5], [1, 0, 9, -11, 7], [1, 11, 6, 20, 21], [-1, 0, 11, -9, -4], [21, -3, 14, 0, -7], [5, 18, -13, 7, 18], [-19, 0, 5, 14, 11], [16, -16, -15, -22, -32], [17, -6, 8, 27, 16], [-7, -16, -3, 1, 0], [-10, 30, 0, -7, -1], [26, 10, 8, 10, 5], [32, -6, 4, -15, 13], [-30, 29, -19, -7, -8], [18, -8, -22, -20, -20], [-21, 5, 12, 4, 5], [-2, -2, -20, 6, -4], [29, -7, -18, -4, -7], [4, -17, 1, -26, -10], [-35, -16, 6, -3, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4592_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4592_2_a_bb();".
// 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_4592_2_a_bb(:prec:=5)
    chi := MakeCharacter_4592_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_4592_2_a_bb();".
// 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_4592_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4592_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -3, -10, 0, 4, 1]>,<5,R![24, -96, -86, -11, 5, 1]>,<11,R![2472, 972, -140, -63, 2, 1]>],Snew);
    return Vf;
end function;