# Properties

 Label 3520.3489 Modulus $3520$ Conductor $440$ Order $10$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(3520, base_ring=CyclotomicField(10))

M = H._module

chi = DirichletCharacter(H, M([0,5,5,1]))

pari: [g,chi] = znchar(Mod(3489,3520))

## Basic properties

 Modulus: $$3520$$ Conductor: $$440$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{440}(189,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 3520.cg

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{5})$$ Fixed field: 10.0.241453843558400000.1

## Values on generators

$$(2751,1541,2817,321)$$ → $$(1,-1,-1,e\left(\frac{1}{10}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$29$$ $$\chi_{ 3520 }(3489, a)$$ $$-1$$ $$1$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 3520 }(3489,a) \;$$ at $$\;a =$$ e.g. 2