# Properties

 Label 363.p Modulus $363$ Conductor $363$ Order $110$ Real no Primitive yes 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([55,1]))

chi.galois_orbit()

[g,chi] = znchar(Mod(2,363))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$363$$ Conductor: $$363$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$110$$ 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: $\Q(\zeta_{55})$ Fixed field: Number field defined by a degree 110 polynomial (not computed)

## First 31 of 40 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$13$$ $$14$$ $$16$$ $$17$$
$$\chi_{363}(2,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{7}{110}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{2}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$
$$\chi_{363}(8,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{83}{110}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$
$$\chi_{363}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{7}{110}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{18}{55}\right)$$
$$\chi_{363}(29,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{67}{110}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{4}{55}\right)$$
$$\chi_{363}(35,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{43}{110}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$
$$\chi_{363}(41,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{107}{110}\right)$$ $$e\left(\frac{51}{110}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$
$$\chi_{363}(50,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{37}{110}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$
$$\chi_{363}(62,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{59}{110}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{97}{110}\right)$$ $$e\left(\frac{91}{110}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{14}{55}\right)$$
$$\chi_{363}(68,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{27}{110}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{91}{110}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$
$$\chi_{363}(74,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{69}{110}\right)$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$
$$\chi_{363}(83,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{29}{55}\right)$$ $$e\left(\frac{1}{110}\right)$$ $$e\left(\frac{93}{110}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{69}{110}\right)$$ $$e\left(\frac{67}{110}\right)$$ $$e\left(\frac{3}{55}\right)$$ $$e\left(\frac{23}{55}\right)$$
$$\chi_{363}(95,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{43}{55}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{101}{110}\right)$$ $$e\left(\frac{39}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$
$$\chi_{363}(101,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{69}{110}\right)$$ $$e\left(\frac{37}{110}\right)$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{47}{55}\right)$$
$$\chi_{363}(107,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{97}{110}\right)$$ $$e\left(\frac{1}{110}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{93}{110}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{31}{55}\right)$$
$$\chi_{363}(116,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{97}{110}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{53}{55}\right)$$
$$\chi_{363}(128,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{38}{55}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{1}{110}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$
$$\chi_{363}(134,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{81}{110}\right)$$ $$e\left(\frac{93}{110}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{27}{55}\right)$$
$$\chi_{363}(140,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{37}{110}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{42}{55}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{23}{110}\right)$$ $$e\left(\frac{59}{110}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{26}{55}\right)$$
$$\chi_{363}(149,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$ $$e\left(\frac{61}{110}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{28}{55}\right)$$
$$\chi_{363}(167,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{73}{110}\right)$$ $$e\left(\frac{32}{55}\right)$$ $$e\left(\frac{7}{55}\right)$$
$$\chi_{363}(173,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{87}{110}\right)$$ $$e\left(\frac{61}{110}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{109}{110}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{21}{55}\right)$$
$$\chi_{363}(182,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{91}{110}\right)$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{47}{110}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{3}{55}\right)$$
$$\chi_{363}(194,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{46}{55}\right)$$ $$e\left(\frac{37}{55}\right)$$ $$e\left(\frac{43}{110}\right)$$ $$e\left(\frac{39}{110}\right)$$ $$e\left(\frac{28}{55}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{107}{110}\right)$$ $$e\left(\frac{21}{110}\right)$$ $$e\left(\frac{19}{55}\right)$$ $$e\left(\frac{54}{55}\right)$$
$$\chi_{363}(200,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{41}{55}\right)$$ $$e\left(\frac{9}{110}\right)$$ $$e\left(\frac{67}{110}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{71}{110}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{27}{55}\right)$$ $$e\left(\frac{42}{55}\right)$$
$$\chi_{363}(206,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{27}{110}\right)$$ $$e\left(\frac{91}{110}\right)$$ $$e\left(\frac{47}{55}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{103}{110}\right)$$ $$e\left(\frac{49}{110}\right)$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{16}{55}\right)$$
$$\chi_{363}(227,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{55}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{89}{110}\right)$$ $$e\left(\frac{23}{55}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{27}{110}\right)$$ $$e\left(\frac{31}{110}\right)$$ $$e\left(\frac{49}{55}\right)$$ $$e\left(\frac{9}{55}\right)$$
$$\chi_{363}(248,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{34}{55}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{73}{110}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{107}{110}\right)$$ $$e\left(\frac{13}{55}\right)$$ $$e\left(\frac{8}{55}\right)$$
$$\chi_{363}(260,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{55}\right)$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{63}{110}\right)$$ $$e\left(\frac{29}{110}\right)$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{57}{110}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{19}{55}\right)$$
$$\chi_{363}(266,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{18}{55}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{79}{110}\right)$$ $$e\left(\frac{87}{110}\right)$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{61}{110}\right)$$ $$e\left(\frac{13}{110}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{2}{55}\right)$$
$$\chi_{363}(272,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{54}{55}\right)$$ $$e\left(\frac{53}{55}\right)$$ $$e\left(\frac{17}{110}\right)$$ $$e\left(\frac{41}{110}\right)$$ $$e\left(\frac{52}{55}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{73}{110}\right)$$ $$e\left(\frac{39}{110}\right)$$ $$e\left(\frac{51}{55}\right)$$ $$e\left(\frac{6}{55}\right)$$
$$\chi_{363}(281,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{12}{55}\right)$$ $$e\left(\frac{24}{55}\right)$$ $$e\left(\frac{71}{110}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{36}{55}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{59}{110}\right)$$ $$e\left(\frac{27}{110}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{38}{55}\right)$$