Properties

 Label 4008.101 Modulus $4008$ Conductor $4008$ Order $166$ Real no Primitive yes Minimal yes Parity even

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

sage: H = DirichletGroup(4008)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,83,83,79]))

pari: [g,chi] = znchar(Mod(101,4008))

Basic properties

 Modulus: $$4008$$ Conductor: $$4008$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$166$$ 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 4008.y

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

Values on generators

$$(3007,2005,1337,673)$$ → $$(1,-1,-1,e\left(\frac{79}{166}\right))$$

Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$1$$ $$1$$ $$e\left(\frac{79}{166}\right)$$ $$e\left(\frac{13}{83}\right)$$ $$e\left(\frac{27}{83}\right)$$ $$e\left(\frac{43}{83}\right)$$ $$e\left(\frac{60}{83}\right)$$ $$e\left(\frac{17}{166}\right)$$ $$e\left(\frac{51}{83}\right)$$ $$e\left(\frac{79}{83}\right)$$ $$e\left(\frac{32}{83}\right)$$ $$e\left(\frac{69}{83}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $\Q(\zeta_{83})$ Fixed field: Number field defined by a degree 166 polynomial