Properties

 Label 3744.235 Modulus $3744$ Conductor $32$ Order $8$ Real no Primitive no Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(3744, base_ring=CyclotomicField(8))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([4,5,0,0]))

pari: [g,chi] = znchar(Mod(235,3744))

Basic properties

 Modulus: $$3744$$ Conductor: $$32$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{32}(11,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 3744.ef

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: $$\Q(\zeta_{8})$$ Fixed field: 8.0.2147483648.1

Values on generators

$$(703,2341,2081,2017)$$ → $$(-1,e\left(\frac{5}{8}\right),1,1)$$

Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 3744 }(235,a) \;$$ at $$\;a =$$ e.g. 2