Properties

 Label 2205.cr Modulus $2205$ Conductor $441$ Order $21$ Real no Primitive no Minimal yes Parity even

Related objects

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

sage: H = DirichletGroup(2205, base_ring=CyclotomicField(42))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([28,0,20]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(16,2205))

pari: order = charorder(g,chi)

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

Basic properties

 Modulus: $$2205$$ Conductor: $$441$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$21$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 441.z sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Related number fields

 Field of values: $$\Q(\zeta_{21})$$ Fixed field: 21.21.2972491714150324080426899160865869074720055489.2

Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$
$$\chi_{2205}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$
$$\chi_{2205}(256,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{2205}(331,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{2205}(571,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$
$$\chi_{2205}(646,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{2205}(886,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{2205}(1201,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$
$$\chi_{2205}(1276,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$
$$\chi_{2205}(1516,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{2205}(1591,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$
$$\chi_{2205}(1906,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$
$$\chi_{2205}(2146,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$