# Properties

 Label 115.k Modulus $115$ Conductor $115$ Order $44$ 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(115, base_ring=CyclotomicField(44))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([11,4]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(2,115))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$115$$ Conductor: $$115$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ 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)

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$
$$\chi_{115}(2,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$
$$\chi_{115}(3,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$
$$\chi_{115}(8,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$
$$\chi_{115}(12,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$
$$\chi_{115}(13,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{115}(18,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{115}(27,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{13}{44}\right)$$
$$\chi_{115}(32,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{115}(48,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$
$$\chi_{115}(52,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{9}{44}\right)$$
$$\chi_{115}(58,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$
$$\chi_{115}(62,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$
$$\chi_{115}(72,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$
$$\chi_{115}(73,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{35}{44}\right)$$
$$\chi_{115}(77,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$
$$\chi_{115}(78,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$
$$\chi_{115}(82,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$
$$\chi_{115}(87,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$
$$\chi_{115}(98,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$
$$\chi_{115}(108,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{15}{44}\right)$$