# Properties

 Label 1375.312 Modulus $1375$ Conductor $1375$ Order $100$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(1375, base_ring=CyclotomicField(100))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([49,20]))

pari: [g,chi] = znchar(Mod(312,1375))

## Basic properties

 Modulus: $$1375$$ Conductor: $$1375$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$100$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1375.cs

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: $\Q(\zeta_{100})$ Fixed field: Number field defined by a degree 100 polynomial

## Values on generators

$$(1002,376)$$ → $$(e\left(\frac{49}{100}\right),e\left(\frac{1}{5}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$\chi_{ 1375 }(312, a)$$ $$-1$$ $$1$$ $$e\left(\frac{69}{100}\right)$$ $$e\left(\frac{3}{100}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{18}{25}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{100}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{41}{100}\right)$$ $$e\left(\frac{31}{100}\right)$$ $$e\left(\frac{37}{50}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1375 }(312,a) \;$$ at $$\;a =$$ e.g. 2