# Properties

 Label 1224.353 Modulus $1224$ Conductor $153$ Order $12$ 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(1224)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,0,2,3]))

pari: [g,chi] = znchar(Mod(353,1224))

## Basic properties

 Modulus: $$1224$$ Conductor: $$153$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{153}(47,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1224.bw

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

$$(919,613,137,649)$$ → $$(1,1,e\left(\frac{1}{6}\right),i)$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{12})$$ Fixed field: 12.0.45943373101939347033.1