# Properties

 Label 363.m Modulus $363$ Conductor $121$ Order $55$ Real no Primitive no Minimal yes Parity even

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(110))

M = H._module

chi = DirichletCharacter(H, M([0,2]))

chi.galois_orbit()

[g,chi] = znchar(Mod(4,363))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$363$$ Conductor: $$121$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$55$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 121.g 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: $\Q(\zeta_{55})$ Fixed field: Number field defined by a degree 55 polynomial

## First 31 of 40 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{363}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$
$$\chi_{363}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$
$$\chi_{363}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$
$$\chi_{363}(31,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$
$$\chi_{363}(37,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$
$$\chi_{363}(49,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$
$$\chi_{363}(58,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$
$$\chi_{363}(64,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$
$$\chi_{363}(70,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$
$$\chi_{363}(82,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$
$$\chi_{363}(91,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$
$$\chi_{363}(97,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$
$$\chi_{363}(103,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$
$$\chi_{363}(115,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$
$$\chi_{363}(136,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$
$$\chi_{363}(157,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$
$$\chi_{363}(163,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$
$$\chi_{363}(169,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$
$$\chi_{363}(181,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$
$$\chi_{363}(190,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$
$$\chi_{363}(196,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$
$$\chi_{363}(214,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$
$$\chi_{363}(223,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$
$$\chi_{363}(229,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$
$$\chi_{363}(235,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$
$$\chi_{363}(247,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{53}{55}\right)$$
$$\chi_{363}(256,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$
$$\chi_{363}(262,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$
$$\chi_{363}(268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$
$$\chi_{363}(280,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$
$$\chi_{363}(289,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$