Properties

 Label 2205.cw Modulus $2205$ Conductor $735$ Order $28$ 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(28))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([14,21,24]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(8,2205))

pari: order = charorder(g,chi)

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

Basic properties

 Modulus: $$2205$$ Conductor: $$735$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$28$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 735.bj 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: $$\Q(\zeta_{28})$$ Fixed field: 28.28.83709274472667248662717543234670163936409630856037139892578125.1

Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$
$$\chi_{2205}(8,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$-1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{28}\right)$$
$$\chi_{2205}(323,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$-1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{1}{28}\right)$$
$$\chi_{2205}(512,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$-1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{27}{28}\right)$$
$$\chi_{2205}(827,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$-1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{19}{28}\right)$$
$$\chi_{2205}(953,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$-1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{28}\right)$$
$$\chi_{2205}(1142,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$-1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{28}\right)$$
$$\chi_{2205}(1268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$-1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{28}\right)$$
$$\chi_{2205}(1457,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$-1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{28}\right)$$
$$\chi_{2205}(1583,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$-1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{25}{28}\right)$$
$$\chi_{2205}(1772,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$-1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{23}{28}\right)$$
$$\chi_{2205}(1898,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$-1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{17}{28}\right)$$
$$\chi_{2205}(2087,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$-1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{15}{28}\right)$$