# Properties

 Label 2500.899 Modulus $2500$ Conductor $500$ Order $50$ Real no Primitive no Minimal no Parity odd

# Learn more

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

sage: H = DirichletGroup(2500, base_ring=CyclotomicField(50))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([25,21]))

pari: [g,chi] = znchar(Mod(899,2500))

## Basic properties

 Modulus: $$2500$$ Conductor: $$500$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$50$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{500}(479,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 2500.n

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: $$\Q(\zeta_{25})$$ Fixed field: Number field defined by a degree 50 polynomial

## Values on generators

$$(1251,1877)$$ → $$(-1,e\left(\frac{21}{50}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 2500 }(899, a)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{25}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{22}{25}\right)$$ $$e\left(\frac{21}{50}\right)$$ $$e\left(\frac{19}{50}\right)$$ $$e\left(\frac{33}{50}\right)$$ $$e\left(\frac{3}{50}\right)$$ $$e\left(\frac{16}{25}\right)$$ $$e\left(\frac{13}{25}\right)$$ $$e\left(\frac{8}{25}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 2500 }(899,a) \;$$ at $$\;a =$$ e.g. 2