Properties

 Label 4000.3871 Modulus $4000$ Conductor $500$ Order $50$ Real no Primitive no Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(4000)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([25,0,26]))

pari: [g,chi] = znchar(Mod(3871,4000))

Basic properties

 Modulus: $$4000$$ Conductor: $$500$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$50$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{500}(371,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 4000.cl

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

$$(2751,2501,1377)$$ → $$(-1,1,e\left(\frac{13}{25}\right))$$

Values

 $$-1$$ $$1$$ $$3$$ $$7$$ $$9$$ $$11$$ $$13$$ $$17$$ $$19$$ $$21$$ $$23$$ $$27$$ $$-1$$ $$1$$ $$e\left(\frac{7}{50}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{1}{50}\right)$$ $$e\left(\frac{7}{25}\right)$$ $$e\left(\frac{24}{25}\right)$$ $$e\left(\frac{43}{50}\right)$$ $$e\left(\frac{21}{25}\right)$$ $$e\left(\frac{31}{50}\right)$$ $$e\left(\frac{21}{50}\right)$$
 value at e.g. 2

Related number fields

 Field of values: $$\Q(\zeta_{25})$$ Fixed field: Number field defined by a degree 50 polynomial