# Properties

 Label 3381.bz Modulus $3381$ Conductor $3381$ Order $154$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(3381, base_ring=CyclotomicField(154))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([77,55,84]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(41,3381))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$3381$$ Conductor: $$3381$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$154$$ 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: $\Q(\zeta_{77})$ Fixed field: Number field defined by a degree 154 polynomial (not computed)

## First 31 of 60 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{3381}(41,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{31}{77}\right)$$ $$e\left(\frac{97}{154}\right)$$ $$e\left(\frac{43}{154}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{65}{154}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{19}{77}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{3381}(62,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{101}{154}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{50}{77}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{47}{154}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{3}{154}\right)$$ $$e\left(\frac{48}{77}\right)$$ $$e\left(\frac{53}{77}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{3381}(104,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{15}{154}\right)$$ $$e\left(\frac{13}{154}\right)$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{145}{154}\right)$$ $$e\left(\frac{10}{77}\right)$$ $$e\left(\frac{72}{77}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{3381}(167,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{154}\right)$$ $$e\left(\frac{43}{77}\right)$$ $$e\left(\frac{19}{77}\right)$$ $$e\left(\frac{129}{154}\right)$$ $$e\left(\frac{81}{154}\right)$$ $$e\left(\frac{1}{154}\right)$$ $$e\left(\frac{15}{154}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{34}{77}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{3381}(188,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{154}\right)$$ $$e\left(\frac{23}{77}\right)$$ $$e\left(\frac{3}{77}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{65}{154}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{46}{77}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{3381}(209,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{154}\right)$$ $$e\left(\frac{17}{77}\right)$$ $$e\left(\frac{29}{77}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{115}{154}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{34}{77}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{3381}(335,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{38}{77}\right)$$ $$e\left(\frac{27}{154}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{79}{154}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{18}{77}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{3381}(377,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{151}{154}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{13}{77}\right)$$ $$e\left(\frac{145}{154}\right)$$ $$e\left(\frac{23}{154}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{71}{77}\right)$$ $$e\left(\frac{3}{77}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{3381}(524,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{25}{77}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{65}{154}\right)$$ $$e\left(\frac{151}{154}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{50}{77}\right)$$ $$e\left(\frac{52}{77}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{3381}(545,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{145}{154}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{127}{154}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{47}{154}\right)$$ $$e\left(\frac{59}{77}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{3381}(629,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{51}{77}\right)$$ $$e\left(\frac{10}{77}\right)$$ $$e\left(\frac{153}{154}\right)$$ $$e\left(\frac{71}{154}\right)$$ $$e\left(\frac{37}{154}\right)$$ $$e\left(\frac{93}{154}\right)$$ $$e\left(\frac{25}{77}\right)$$ $$e\left(\frac{26}{77}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{3381}(650,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{87}{154}\right)$$ $$e\left(\frac{10}{77}\right)$$ $$e\left(\frac{8}{77}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{103}{154}\right)$$ $$e\left(\frac{45}{154}\right)$$ $$e\left(\frac{59}{154}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{3381}(671,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{69}{77}\right)$$ $$e\left(\frac{47}{154}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{95}{154}\right)$$ $$e\left(\frac{57}{77}\right)$$ $$e\left(\frac{10}{77}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{3381}(692,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{61}{154}\right)$$ $$e\left(\frac{61}{77}\right)$$ $$e\left(\frac{18}{77}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{97}{154}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{75}{154}\right)$$ $$e\left(\frac{45}{77}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{3381}(818,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{154}\right)$$ $$e\left(\frac{53}{77}\right)$$ $$e\left(\frac{27}{77}\right)$$ $$e\left(\frac{5}{154}\right)$$ $$e\left(\frac{107}{154}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{151}{154}\right)$$ $$e\left(\frac{29}{77}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{3381}(860,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{2}{77}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{45}{154}\right)$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{111}{154}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{3381}(923,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{154}\right)$$ $$e\left(\frac{37}{77}\right)$$ $$e\left(\frac{45}{77}\right)$$ $$e\left(\frac{111}{154}\right)$$ $$e\left(\frac{127}{154}\right)$$ $$e\left(\frac{51}{154}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{40}{77}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{3381}(1007,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{69}{154}\right)$$ $$e\left(\frac{69}{77}\right)$$ $$e\left(\frac{9}{77}\right)$$ $$e\left(\frac{53}{154}\right)$$ $$e\left(\frac{87}{154}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{153}{154}\right)$$ $$e\left(\frac{61}{77}\right)$$ $$e\left(\frac{8}{77}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{3381}(1070,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{93}{154}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{59}{77}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{57}{154}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{79}{154}\right)$$ $$e\left(\frac{32}{77}\right)$$ $$e\left(\frac{61}{77}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{3381}(1112,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{95}{154}\right)$$ $$e\left(\frac{18}{77}\right)$$ $$e\left(\frac{76}{77}\right)$$ $$e\left(\frac{131}{154}\right)$$ $$e\left(\frac{93}{154}\right)$$ $$e\left(\frac{81}{154}\right)$$ $$e\left(\frac{137}{154}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{59}{77}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{3381}(1133,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{131}{154}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{89}{154}\right)$$ $$e\left(\frac{103}{154}\right)$$ $$e\left(\frac{31}{77}\right)$$ $$e\left(\frac{23}{77}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{3381}(1154,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{111}{154}\right)$$ $$e\left(\frac{34}{77}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{25}{154}\right)$$ $$e\left(\frac{73}{154}\right)$$ $$e\left(\frac{153}{154}\right)$$ $$e\left(\frac{139}{154}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{43}{77}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{3381}(1301,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{97}{154}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{137}{154}\right)$$ $$e\left(\frac{129}{154}\right)$$ $$e\left(\frac{13}{154}\right)$$ $$e\left(\frac{41}{154}\right)$$ $$e\left(\frac{40}{77}\right)$$ $$e\left(\frac{57}{77}\right)$$ $$e\left(\frac{1}{22}\right)$$
$$\chi_{3381}(1343,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{8}{77}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{101}{154}\right)$$ $$e\left(\frac{67}{154}\right)$$ $$e\left(\frac{113}{154}\right)$$ $$e\left(\frac{1}{154}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{69}{77}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{3381}(1406,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{81}{154}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{34}{77}\right)$$ $$e\left(\frac{89}{154}\right)$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{95}{154}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{8}{77}\right)$$ $$e\left(\frac{73}{77}\right)$$ $$e\left(\frac{9}{22}\right)$$
$$\chi_{3381}(1490,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{113}{154}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{75}{77}\right)$$ $$e\left(\frac{31}{154}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{85}{154}\right)$$ $$e\left(\frac{43}{154}\right)$$ $$e\left(\frac{72}{77}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{15}{22}\right)$$
$$\chi_{3381}(1511,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{79}{154}\right)$$ $$e\left(\frac{2}{77}\right)$$ $$e\left(\frac{17}{77}\right)$$ $$e\left(\frac{83}{154}\right)$$ $$e\left(\frac{113}{154}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{135}{154}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{75}{77}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{3381}(1553,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{137}{154}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{48}{77}\right)$$ $$e\left(\frac{103}{154}\right)$$ $$e\left(\frac{79}{154}\right)$$ $$e\left(\frac{39}{154}\right)$$ $$e\left(\frac{123}{154}\right)$$ $$e\left(\frac{43}{77}\right)$$ $$e\left(\frac{17}{77}\right)$$ $$e\left(\frac{3}{22}\right)$$
$$\chi_{3381}(1595,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{139}{154}\right)$$ $$e\left(\frac{62}{77}\right)$$ $$e\left(\frac{65}{77}\right)$$ $$e\left(\frac{109}{154}\right)$$ $$e\left(\frac{115}{154}\right)$$ $$e\left(\frac{125}{154}\right)$$ $$e\left(\frac{27}{154}\right)$$ $$e\left(\frac{47}{77}\right)$$ $$e\left(\frac{15}{77}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{3381}(1637,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{154}\right)$$ $$e\left(\frac{1}{77}\right)$$ $$e\left(\frac{47}{77}\right)$$ $$e\left(\frac{3}{154}\right)$$ $$e\left(\frac{95}{154}\right)$$ $$e\left(\frac{43}{154}\right)$$ $$e\left(\frac{29}{154}\right)$$ $$e\left(\frac{2}{77}\right)$$ $$e\left(\frac{76}{77}\right)$$ $$e\left(\frac{5}{22}\right)$$
$$\chi_{3381}(1658,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{149}{154}\right)$$ $$e\left(\frac{72}{77}\right)$$ $$e\left(\frac{73}{77}\right)$$ $$e\left(\frac{139}{154}\right)$$ $$e\left(\frac{141}{154}\right)$$ $$e\left(\frac{93}{154}\right)$$ $$e\left(\frac{9}{154}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{19}{22}\right)$$