Properties

 Label 2205.de Modulus $2205$ Conductor $441$ Order $42$ 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([35,0,15]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(41,2205))

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: $$42$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 441.bh sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$
$$\chi_{2205}(41,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{31}{42}\right)$$
$$\chi_{2205}(356,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$-1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{42}\right)$$
$$\chi_{2205}(461,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$-1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{42}\right)$$
$$\chi_{2205}(671,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$-1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{42}\right)$$
$$\chi_{2205}(776,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$-1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{17}{42}\right)$$
$$\chi_{2205}(986,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$-1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{25}{42}\right)$$
$$\chi_{2205}(1091,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$-1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{29}{42}\right)$$
$$\chi_{2205}(1301,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$-1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{37}{42}\right)$$
$$\chi_{2205}(1406,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$-1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{41}{42}\right)$$
$$\chi_{2205}(1721,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$-1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{42}\right)$$
$$\chi_{2205}(1931,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$-1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{19}{42}\right)$$
$$\chi_{2205}(2036,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$-1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{23}{42}\right)$$