# Properties

 Label 1380.1273 Modulus $1380$ Conductor $115$ Order $44$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(1380, base_ring=CyclotomicField(44))

M = H._module

chi = DirichletCharacter(H, M([0,0,33,12]))

pari: [g,chi] = znchar(Mod(1273,1380))

## Basic properties

 Modulus: $$1380$$ Conductor: $$115$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{115}(8,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1380.bq

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Values on generators

$$(691,461,277,1201)$$ → $$(1,1,-i,e\left(\frac{3}{11}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$\chi_{ 1380 }(1273, a)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{27}{44}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1380 }(1273,a) \;$$ at $$\;a =$$ e.g. 2