Properties

 Label 1045.31 Modulus $1045$ Conductor $209$ Order $30$ Real no Primitive no Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(1045, base_ring=CyclotomicField(30))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,18,25]))

pari: [g,chi] = znchar(Mod(31,1045))

Basic properties

 Modulus: $$1045$$ Conductor: $$209$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$30$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{209}(31,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 1045.bt

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: $$\Q(\zeta_{15})$$ Fixed field: 30.0.916778600930282229451203736288894059046202165847586688659.1

Values on generators

$$(837,761,496)$$ → $$(1,e\left(\frac{3}{5}\right),e\left(\frac{5}{6}\right))$$

Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$-1$$ $$1$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$-1$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{19}{30}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1045 }(31,a) \;$$ at $$\;a =$$ e.g. 2