# Properties

 Label 2205.ej Modulus $2205$ Conductor $2205$ Order $84$ Real no Primitive yes Minimal yes Parity even

# Learn more about

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([28,63,66]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(13,2205))

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: even 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 $$-1$$ $$1$$ $$2$$ $$4$$ $$8$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$ $$22$$ $$23$$
$$\chi_{2205}(13,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$1$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{65}{84}\right)$$
$$\chi_{2205}(202,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$1$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{71}{84}\right)$$
$$\chi_{2205}(223,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$1$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{25}{84}\right)$$
$$\chi_{2205}(328,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$1$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{5}{84}\right)$$
$$\chi_{2205}(412,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$1$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{31}{84}\right)$$
$$\chi_{2205}(517,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$1$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{11}{84}\right)$$
$$\chi_{2205}(643,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$1$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{29}{84}\right)$$
$$\chi_{2205}(727,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$1$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{55}{84}\right)$$
$$\chi_{2205}(853,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$1$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{73}{84}\right)$$
$$\chi_{2205}(958,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{23}{84}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$1$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{53}{84}\right)$$
$$\chi_{2205}(1042,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$1$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{79}{84}\right)$$
$$\chi_{2205}(1147,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$1$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{59}{84}\right)$$
$$\chi_{2205}(1168,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$1$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{13}{84}\right)$$
$$\chi_{2205}(1357,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$1$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{19}{84}\right)$$
$$\chi_{2205}(1462,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{17}{84}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{28}\right)$$ $$1$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{83}{84}\right)$$
$$\chi_{2205}(1483,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{71}{84}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{15}{28}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$1$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{37}{84}\right)$$
$$\chi_{2205}(1588,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{47}{84}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$1$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{17}{84}\right)$$
$$\chi_{2205}(1672,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{84}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$1$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{43}{84}\right)$$
$$\chi_{2205}(1777,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{84}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{29}{84}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{25}{28}\right)$$ $$1$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{23}{84}\right)$$
$$\chi_{2205}(1798,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{27}{28}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{55}{84}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$1$$ $$e\left(\frac{19}{84}\right)$$ $$e\left(\frac{61}{84}\right)$$
$$\chi_{2205}(1903,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{84}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{3}{28}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{59}{84}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{19}{28}\right)$$ $$1$$ $$e\left(\frac{83}{84}\right)$$ $$e\left(\frac{41}{84}\right)$$
$$\chi_{2205}(1987,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{65}{84}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{9}{28}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{37}{84}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{28}\right)$$ $$1$$ $$e\left(\frac{25}{84}\right)$$ $$e\left(\frac{67}{84}\right)$$
$$\chi_{2205}(2092,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{84}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{17}{28}\right)$$ $$1$$ $$e\left(\frac{5}{84}\right)$$ $$e\left(\frac{47}{84}\right)$$
$$\chi_{2205}(2113,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{84}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{28}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{67}{84}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{23}{28}\right)$$ $$1$$ $$e\left(\frac{43}{84}\right)$$ $$e\left(\frac{1}{84}\right)$$