# Properties

 Label 1148.251 Modulus $1148$ Conductor $1148$ Order $20$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(1148, base_ring=CyclotomicField(20))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([10,10,11]))

pari: [g,chi] = znchar(Mod(251,1148))

## Basic properties

 Modulus: $$1148$$ Conductor: $$1148$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$20$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1148.bm

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: $$\Q(\zeta_{20})$$ Fixed field: Number field defined by a degree 20 polynomial

## Values on generators

$$(575,493,785)$$ → $$(-1,-1,e\left(\frac{11}{20}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$1$$ $$1$$ $$i$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$
 value at e.g. 2