# Properties

 Label 1764.449 Modulus $1764$ Conductor $147$ Order $14$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(1764)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,7,12]))

pari: [g,chi] = znchar(Mod(449,1764))

## Basic properties

 Modulus: $$1764$$ Conductor: $$147$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$14$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{147}(8,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1764.bu

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

$$(883,785,1081)$$ → $$(1,-1,e\left(\frac{6}{7}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$37$$ $$-1$$ $$1$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$1$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$1$$ $$e\left(\frac{3}{7}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{7})$$ Fixed field: 14.0.418988153029298748294987.1