# Properties

 Label 2205.37 Modulus $2205$ Conductor $245$ Order $84$ Real no Primitive no Minimal yes Parity odd

# Learn more about

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

sage: H = DirichletGroup(2205, base_ring=CyclotomicField(84))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,21,64]))

pari: [g,chi] = znchar(Mod(37,2205))

## Basic properties

 Modulus: $$2205$$ Conductor: $$245$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$84$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{245}(37,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 2205.ei

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

## Values on generators

$$(1226,442,1081)$$ → $$(1,i,e\left(\frac{16}{21}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$ $$-1$$ $$1$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{59}{84}\right)$$
 value at e.g. 2