# Properties

 Label 3724.123 Modulus $3724$ Conductor $3724$ Order $126$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(3724, base_ring=CyclotomicField(126))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([63,48,56]))

pari: [g,chi] = znchar(Mod(123,3724))

## Basic properties

 Modulus: $$3724$$ Conductor: $$3724$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$126$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 3724.ev

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: $\Q(\zeta_{63})$ Fixed field: Number field defined by a degree 126 polynomial (not computed)

## Values on generators

$$(1863,3041,3137)$$ → $$(-1,e\left(\frac{8}{21}\right),e\left(\frac{4}{9}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$23$$ $$25$$ $$27$$ $$-1$$ $$1$$ $$e\left(\frac{83}{126}\right)$$ $$e\left(\frac{10}{63}\right)$$ $$e\left(\frac{20}{63}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{50}{63}\right)$$ $$e\left(\frac{103}{126}\right)$$ $$e\left(\frac{61}{63}\right)$$ $$e\left(\frac{109}{126}\right)$$ $$e\left(\frac{20}{63}\right)$$ $$e\left(\frac{41}{42}\right)$$
 value at e.g. 2