# Properties

 Label 3724.75 Modulus $3724$ Conductor $3724$ Order $42$ 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(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([21,17,21]))

pari: [g,chi] = znchar(Mod(75,3724))

## Basic properties

 Modulus: $$3724$$ Conductor: $$3724$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$42$$ 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.de

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: $$\Q(\zeta_{21})$$ Fixed field: Number field defined by a degree 42 polynomial

## Values on generators

$$(1863,3041,3137)$$ → $$(-1,e\left(\frac{17}{42}\right),-1)$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$23$$ $$25$$ $$27$$ $$-1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$
 value at e.g. 2