# Properties

 Label 2500.11 Modulus $2500$ Conductor $2500$ Order $250$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(2500, base_ring=CyclotomicField(250))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([125,238]))

pari: [g,chi] = znchar(Mod(11,2500))

## Basic properties

 Modulus: $$2500$$ Conductor: $$2500$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$250$$ 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 2500.t

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: $\Q(\zeta_{125})$ Fixed field: Number field defined by a degree 250 polynomial (not computed)

## Values on generators

$$(1251,1877)$$ → $$(-1,e\left(\frac{119}{125}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$\chi_{ 2500 }(11, a)$$ $$-1$$ $$1$$ $$e\left(\frac{91}{250}\right)$$ $$e\left(\frac{11}{50}\right)$$ $$e\left(\frac{91}{125}\right)$$ $$e\left(\frac{163}{250}\right)$$ $$e\left(\frac{41}{125}\right)$$ $$e\left(\frac{87}{125}\right)$$ $$e\left(\frac{109}{250}\right)$$ $$e\left(\frac{73}{125}\right)$$ $$e\left(\frac{203}{250}\right)$$ $$e\left(\frac{23}{250}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 2500 }(11,a) \;$$ at $$\;a =$$ e.g. 2