Properties

Label 547.99
Modulus $547$
Conductor $547$
Order $273$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(547, base_ring=CyclotomicField(546))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([382]))
 
pari: [g,chi] = znchar(Mod(99,547))
 

Basic properties

Modulus: \(547\)
Conductor: \(547\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(273\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 547.o

\(\chi_{547}(4,\cdot)\) \(\chi_{547}(6,\cdot)\) \(\chi_{547}(15,\cdot)\) \(\chi_{547}(16,\cdot)\) \(\chi_{547}(19,\cdot)\) \(\chi_{547}(25,\cdot)\) \(\chi_{547}(34,\cdot)\) \(\chi_{547}(36,\cdot)\) \(\chi_{547}(49,\cdot)\) \(\chi_{547}(51,\cdot)\) \(\chi_{547}(53,\cdot)\) \(\chi_{547}(56,\cdot)\) \(\chi_{547}(60,\cdot)\) \(\chi_{547}(62,\cdot)\) \(\chi_{547}(66,\cdot)\) \(\chi_{547}(67,\cdot)\) \(\chi_{547}(69,\cdot)\) \(\chi_{547}(73,\cdot)\) \(\chi_{547}(74,\cdot)\) \(\chi_{547}(76,\cdot)\) \(\chi_{547}(78,\cdot)\) \(\chi_{547}(82,\cdot)\) \(\chi_{547}(86,\cdot)\) \(\chi_{547}(97,\cdot)\) \(\chi_{547}(99,\cdot)\) \(\chi_{547}(110,\cdot)\) \(\chi_{547}(111,\cdot)\) \(\chi_{547}(113,\cdot)\) \(\chi_{547}(115,\cdot)\) \(\chi_{547}(116,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(2\) → \(e\left(\frac{191}{273}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{191}{273}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{109}{273}\right)\)\(e\left(\frac{64}{273}\right)\)\(e\left(\frac{230}{273}\right)\)\(e\left(\frac{17}{273}\right)\)\(e\left(\frac{9}{91}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{85}{91}\right)\)\(e\left(\frac{38}{39}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{273})$
Fixed field: Number field defined by a degree 273 polynomial (not computed)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 547 }(99,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{547}(99,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(99,r) e\left(\frac{2r}{547}\right) = -19.2361168594+13.3030751396i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 547 }(99,·),\chi_{ 547 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{547}(99,\cdot),\chi_{547}(1,\cdot)) = \sum_{r\in \Z/547\Z} \chi_{547}(99,r) \chi_{547}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 547 }(99,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{547}(99,·)) = \sum_{r \in \Z/547\Z} \chi_{547}(99,r) e\left(\frac{1 r + 2 r^{-1}}{547}\right) = 14.0216407321+-19.3459103309i \)