Properties

 Label 161.103 Modulus $161$ Conductor $161$ Order $66$ Real no Primitive yes Minimal yes Parity even

Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(161, base_ring=CyclotomicField(66))

M = H._module

chi = DirichletCharacter(H, M([55,27]))

pari: [g,chi] = znchar(Mod(103,161))

Basic properties

 Modulus: $$161$$ Conductor: $$161$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$66$$ 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 161.o

sage: chi.galois_orbit()

order = charorder(g,chi)

[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Related number fields

 Field of values: $$\Q(\zeta_{33})$$ Fixed field: Number field defined by a degree 66 polynomial

Values on generators

$$(24,120)$$ → $$(e\left(\frac{5}{6}\right),e\left(\frac{9}{22}\right))$$

First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$\chi_{ 161 }(103, a)$$ $$1$$ $$1$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{23}{66}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 161 }(103,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 161 }(103,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 161 }(103,·),\chi_{ 161 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 161 }(103,·)) \;$$ at $$\; a,b =$$ e.g. 1,2