Properties

 Label 1441.1118 Modulus $1441$ Conductor $1441$ Order $10$ Real no Primitive yes Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(1441, base_ring=CyclotomicField(10))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([7,1]))

pari: [g,chi] = znchar(Mod(1118,1441))

Basic properties

 Modulus: $$1441$$ Conductor: $$1441$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ 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 1441.m

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

Related number fields

 Field of values: $$\Q(\zeta_{5})$$ Fixed field: Number field defined by a degree 10 polynomial

Values on generators

$$(1311,133)$$ → $$(e\left(\frac{7}{10}\right),e\left(\frac{1}{10}\right))$$

Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$\chi_{ 1441 }(1118, a)$$ $$1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1441 }(1118,a) \;$$ at $$\;a =$$ e.g. 2