Properties

 Label 529.49 Modulus $529$ Conductor $529$ Order $253$ Real no Primitive yes Minimal yes Parity even

Related objects

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

H = DirichletGroup(529, base_ring=CyclotomicField(506))

M = H._module

chi = DirichletCharacter(H, M([258]))

pari: [g,chi] = znchar(Mod(49,529))

Basic properties

 Modulus: $$529$$ Conductor: $$529$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$253$$ 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 529.g

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: $\Q(\zeta_{253})$ Fixed field: Number field defined by a degree 253 polynomial (not computed)

Values on generators

$$5$$ → $$e\left(\frac{129}{253}\right)$$

Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$\chi_{ 529 }(49, a)$$ $$1$$ $$1$$ $$e\left(\frac{247}{253}\right)$$ $$e\left(\frac{40}{253}\right)$$ $$e\left(\frac{241}{253}\right)$$ $$e\left(\frac{129}{253}\right)$$ $$e\left(\frac{34}{253}\right)$$ $$e\left(\frac{196}{253}\right)$$ $$e\left(\frac{235}{253}\right)$$ $$e\left(\frac{80}{253}\right)$$ $$e\left(\frac{123}{253}\right)$$ $$e\left(\frac{237}{253}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 529 }(49,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 529 }(49,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 529 }(49,·),\chi_{ 529 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 529 }(49,·)) \;$$ at $$\; a,b =$$ e.g. 1,2