# Properties

 Label 4410.103 Modulus $4410$ Conductor $2205$ Order $84$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(4410)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([28,63,58]))

pari: [g,chi] = znchar(Mod(103,4410))

## Basic properties

 Modulus: $$4410$$ Conductor: $$2205$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$84$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{2205}(103,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 4410.ej

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

$$(3431,2647,1081)$$ → $$(e\left(\frac{1}{3}\right),-i,e\left(\frac{29}{42}\right))$$

## Values

 $$-1$$ $$1$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$-1$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{61}{84}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{84})$ Fixed field: Number field defined by a degree 84 polynomial