Properties

 Label 2001.5 Modulus $2001$ Conductor $2001$ Order $154$ Real no Primitive yes Minimal yes Parity even

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

sage: H = DirichletGroup(2001, base_ring=CyclotomicField(154))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([77,7,121]))

pari: [g,chi] = znchar(Mod(5,2001))

Basic properties

 Modulus: $$2001$$ Conductor: $$2001$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$154$$ 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 2001.br

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

$$(668,1132,553)$$ → $$(-1,e\left(\frac{1}{22}\right),e\left(\frac{11}{14}\right))$$

Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{29}{77}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{64}{77}\right)$$ $$e\left(\frac{45}{154}\right)$$ $$e\left(\frac{10}{77}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{103}{154}\right)$$ $$e\left(\frac{39}{77}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $\Q(\zeta_{77})$ Fixed field: Number field defined by a degree 154 polynomial (not computed)