# Properties

 Label 4056.337 Modulus $4056$ Conductor $13$ Order $2$ Real yes Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(4056, base_ring=CyclotomicField(2))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,0,0,1]))

pari: [g,chi] = znchar(Mod(337,4056))

## Basic properties

 Modulus: $$4056$$ Conductor: $$13$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$2$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: yes Primitive: no, induced from $$\chi_{13}(12,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 4056.c

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: $$\Q$$ Fixed field: $$\Q(\sqrt{13})$$

## Values on generators

$$(1015,2029,2705,3889)$$ → $$(1,1,1,-1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$\chi_{ 4056 }(337, a)$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 4056 }(337,a) \;$$ at $$\;a =$$ e.g. 2