# Properties

 Label 967.l Modulus $967$ Conductor $967$ Order $138$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(967, base_ring=CyclotomicField(138))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([107]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(20,967))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$967$$ Conductor: $$967$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$138$$ 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: $\Q(\zeta_{69})$ Fixed field: Number field defined by a degree 138 polynomial (not computed)

## First 31 of 44 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{967}(20,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{68}{69}\right)$$ $$e\left(\frac{35}{46}\right)$$ $$e\left(\frac{67}{69}\right)$$ $$e\left(\frac{107}{138}\right)$$ $$e\left(\frac{103}{138}\right)$$ $$e\left(\frac{131}{138}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{35}{46}\right)$$ $$e\left(\frac{12}{23}\right)$$
$$\chi_{967}(52,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{32}{69}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{64}{69}\right)$$ $$e\left(\frac{95}{138}\right)$$ $$e\left(\frac{85}{138}\right)$$ $$e\left(\frac{17}{138}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{7}{23}\right)$$
$$\chi_{967}(54,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{69}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{10}{69}\right)$$ $$e\left(\frac{17}{138}\right)$$ $$e\left(\frac{37}{138}\right)$$ $$e\left(\frac{35}{138}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{9}{23}\right)$$
$$\chi_{967}(55,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{69}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{26}{69}\right)$$ $$e\left(\frac{127}{138}\right)$$ $$e\left(\frac{41}{138}\right)$$ $$e\left(\frac{91}{138}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{5}{23}\right)$$
$$\chi_{967}(92,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{69}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{14}{69}\right)$$ $$e\left(\frac{79}{138}\right)$$ $$e\left(\frac{107}{138}\right)$$ $$e\left(\frac{49}{138}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{8}{23}\right)$$
$$\chi_{967}(93,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{69}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{5}{69}\right)$$ $$e\left(\frac{43}{138}\right)$$ $$e\left(\frac{53}{138}\right)$$ $$e\left(\frac{121}{138}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{16}{23}\right)$$
$$\chi_{967}(96,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{52}{69}\right)$$ $$e\left(\frac{43}{46}\right)$$ $$e\left(\frac{35}{69}\right)$$ $$e\left(\frac{25}{138}\right)$$ $$e\left(\frac{95}{138}\right)$$ $$e\left(\frac{19}{138}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{43}{46}\right)$$ $$e\left(\frac{20}{23}\right)$$
$$\chi_{967}(147,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{44}{69}\right)$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{19}{69}\right)$$ $$e\left(\frac{53}{138}\right)$$ $$e\left(\frac{91}{138}\right)$$ $$e\left(\frac{101}{138}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{1}{23}\right)$$
$$\chi_{967}(197,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{64}{69}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{59}{69}\right)$$ $$e\left(\frac{121}{138}\right)$$ $$e\left(\frac{101}{138}\right)$$ $$e\left(\frac{103}{138}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{14}{23}\right)$$
$$\chi_{967}(210,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{61}{69}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{53}{69}\right)$$ $$e\left(\frac{97}{138}\right)$$ $$e\left(\frac{65}{138}\right)$$ $$e\left(\frac{13}{138}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{4}{23}\right)$$
$$\chi_{967}(211,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{56}{69}\right)$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{43}{69}\right)$$ $$e\left(\frac{11}{138}\right)$$ $$e\left(\frac{97}{138}\right)$$ $$e\left(\frac{47}{138}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{18}{23}\right)$$
$$\chi_{967}(239,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{14}{69}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{28}{69}\right)$$ $$e\left(\frac{89}{138}\right)$$ $$e\left(\frac{7}{138}\right)$$ $$e\left(\frac{29}{138}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{16}{23}\right)$$
$$\chi_{967}(249,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{69}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{4}{69}\right)$$ $$e\left(\frac{131}{138}\right)$$ $$e\left(\frac{1}{138}\right)$$ $$e\left(\frac{83}{138}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{22}{23}\right)$$
$$\chi_{967}(382,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{53}{69}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{37}{69}\right)$$ $$e\left(\frac{125}{138}\right)$$ $$e\left(\frac{61}{138}\right)$$ $$e\left(\frac{95}{138}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{8}{23}\right)$$
$$\chi_{967}(413,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{69}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{34}{69}\right)$$ $$e\left(\frac{113}{138}\right)$$ $$e\left(\frac{43}{138}\right)$$ $$e\left(\frac{119}{138}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$
$$\chi_{967}(428,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{69}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{25}{69}\right)$$ $$e\left(\frac{77}{138}\right)$$ $$e\left(\frac{127}{138}\right)$$ $$e\left(\frac{53}{138}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{11}{23}\right)$$
$$\chi_{967}(443,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{16}{69}\right)$$ $$e\left(\frac{15}{46}\right)$$ $$e\left(\frac{32}{69}\right)$$ $$e\left(\frac{13}{138}\right)$$ $$e\left(\frac{77}{138}\right)$$ $$e\left(\frac{43}{138}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{15}{46}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{967}(454,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{69}\right)$$ $$e\left(\frac{17}{46}\right)$$ $$e\left(\frac{1}{69}\right)$$ $$e\left(\frac{119}{138}\right)$$ $$e\left(\frac{121}{138}\right)$$ $$e\left(\frac{107}{138}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{17}{46}\right)$$ $$e\left(\frac{17}{23}\right)$$
$$\chi_{967}(473,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{62}{69}\right)$$ $$e\left(\frac{15}{46}\right)$$ $$e\left(\frac{55}{69}\right)$$ $$e\left(\frac{59}{138}\right)$$ $$e\left(\frac{31}{138}\right)$$ $$e\left(\frac{89}{138}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{15}{46}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{967}(522,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{65}{69}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{61}{69}\right)$$ $$e\left(\frac{83}{138}\right)$$ $$e\left(\frac{67}{138}\right)$$ $$e\left(\frac{41}{138}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{2}{23}\right)$$
$$\chi_{967}(546,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{69}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{50}{69}\right)$$ $$e\left(\frac{85}{138}\right)$$ $$e\left(\frac{47}{138}\right)$$ $$e\left(\frac{37}{138}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{22}{23}\right)$$
$$\chi_{967}(567,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{67}{69}\right)$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{65}{69}\right)$$ $$e\left(\frac{7}{138}\right)$$ $$e\left(\frac{137}{138}\right)$$ $$e\left(\frac{55}{138}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{1}{23}\right)$$
$$\chi_{967}(590,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{34}{69}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{68}{69}\right)$$ $$e\left(\frac{19}{138}\right)$$ $$e\left(\frac{17}{138}\right)$$ $$e\left(\frac{31}{138}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{6}{23}\right)$$
$$\chi_{967}(615,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{55}{69}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{41}{69}\right)$$ $$e\left(\frac{49}{138}\right)$$ $$e\left(\frac{131}{138}\right)$$ $$e\left(\frac{109}{138}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{7}{23}\right)$$
$$\chi_{967}(626,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{40}{69}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{11}{69}\right)$$ $$e\left(\frac{67}{138}\right)$$ $$e\left(\frac{89}{138}\right)$$ $$e\left(\frac{73}{138}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$
$$\chi_{967}(632,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{69}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{38}{69}\right)$$ $$e\left(\frac{37}{138}\right)$$ $$e\left(\frac{113}{138}\right)$$ $$e\left(\frac{133}{138}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{2}{23}\right)$$
$$\chi_{967}(646,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{58}{69}\right)$$ $$e\left(\frac{17}{46}\right)$$ $$e\left(\frac{47}{69}\right)$$ $$e\left(\frac{73}{138}\right)$$ $$e\left(\frac{29}{138}\right)$$ $$e\left(\frac{61}{138}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{17}{46}\right)$$ $$e\left(\frac{17}{23}\right)$$
$$\chi_{967}(687,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{50}{69}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{31}{69}\right)$$ $$e\left(\frac{101}{138}\right)$$ $$e\left(\frac{25}{138}\right)$$ $$e\left(\frac{5}{138}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{21}{23}\right)$$
$$\chi_{967}(726,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{69}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{22}{69}\right)$$ $$e\left(\frac{65}{138}\right)$$ $$e\left(\frac{109}{138}\right)$$ $$e\left(\frac{77}{138}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{6}{23}\right)$$
$$\chi_{967}(742,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{20}{69}\right)$$ $$e\left(\frac{13}{46}\right)$$ $$e\left(\frac{40}{69}\right)$$ $$e\left(\frac{137}{138}\right)$$ $$e\left(\frac{79}{138}\right)$$ $$e\left(\frac{71}{138}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{13}{46}\right)$$ $$e\left(\frac{13}{23}\right)$$
$$\chi_{967}(765,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{49}{69}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{29}{69}\right)$$ $$e\left(\frac{1}{138}\right)$$ $$e\left(\frac{59}{138}\right)$$ $$e\left(\frac{67}{138}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{10}{23}\right)$$