# Properties

 Label 2205.em Modulus $2205$ Conductor $2205$ Order $84$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(2205, base_ring=CyclotomicField(84))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([14,63,38]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(38,2205))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$2205$$ Conductor: $$2205$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$84$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{84})$ Fixed field: Number field defined by a degree 84 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$
$$\chi_{2205}(38,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{23}{84}\right)$$
$$\chi_{2205}(257,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{73}{84}\right)$$
$$\chi_{2205}(353,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{71}{84}\right)$$
$$\chi_{2205}(383,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{19}{84}\right)$$
$$\chi_{2205}(542,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{41}{84}\right)$$
$$\chi_{2205}(572,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{1}{84}\right)$$
$$\chi_{2205}(698,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{31}{84}\right)$$
$$\chi_{2205}(857,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{5}{84}\right)$$
$$\chi_{2205}(887,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{13}{84}\right)$$
$$\chi_{2205}(983,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{83}{84}\right)$$
$$\chi_{2205}(1013,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{43}{84}\right)$$
$$\chi_{2205}(1172,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{53}{84}\right)$$
$$\chi_{2205}(1202,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{25}{84}\right)$$
$$\chi_{2205}(1298,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{47}{84}\right)$$
$$\chi_{2205}(1328,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{55}{84}\right)$$
$$\chi_{2205}(1487,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{17}{84}\right)$$
$$\chi_{2205}(1517,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{37}{84}\right)$$
$$\chi_{2205}(1613,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{11}{84}\right)$$
$$\chi_{2205}(1643,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{67}{84}\right)$$
$$\chi_{2205}(1802,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{65}{84}\right)$$
$$\chi_{2205}(1928,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{59}{84}\right)$$
$$\chi_{2205}(1958,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{79}{84}\right)$$
$$\chi_{2205}(2117,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{29}{84}\right)$$
$$\chi_{2205}(2147,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{61}{84}\right)$$