# Properties

 Label 177.f Modulus $177$ Conductor $177$ Order $58$ 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(177, base_ring=CyclotomicField(58))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([29,1]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(2,177))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$177$$ Conductor: $$177$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$58$$ 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: $\Q(\zeta_{29})$ Fixed field: Number field defined by a degree 58 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{177}(2,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$
$$\chi_{177}(8,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$
$$\chi_{177}(11,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$
$$\chi_{177}(14,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{27}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$
$$\chi_{177}(23,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$
$$\chi_{177}(32,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$
$$\chi_{177}(38,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$
$$\chi_{177}(44,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$
$$\chi_{177}(47,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$
$$\chi_{177}(50,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$
$$\chi_{177}(56,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{39}{58}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{17}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$
$$\chi_{177}(65,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{45}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{15}{29}\right)$$
$$\chi_{177}(77,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{11}{58}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$
$$\chi_{177}(83,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{19}{29}\right)$$
$$\chi_{177}(89,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{23}{58}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{51}{58}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{13}{58}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$
$$\chi_{177}(92,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{15}{58}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{11}{58}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{5}{29}\right)$$
$$\chi_{177}(98,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{14}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{27}{58}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$
$$\chi_{177}(101,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{19}{58}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{31}{58}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{22}{29}\right)$$
$$\chi_{177}(113,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{7}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{13}{58}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$
$$\chi_{177}(128,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{13}{58}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{49}{58}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{14}{29}\right)$$
$$\chi_{177}(131,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{29}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{9}{58}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$
$$\chi_{177}(149,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{20}{29}\right)$$ $$e\left(\frac{33}{58}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{1}{58}\right)$$ $$e\left(\frac{16}{29}\right)$$ $$e\left(\frac{11}{29}\right)$$
$$\chi_{177}(152,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{12}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{55}{58}\right)$$ $$e\left(\frac{5}{29}\right)$$ $$e\left(\frac{47}{58}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{24}{29}\right)$$
$$\chi_{177}(155,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{11}{58}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{10}{29}\right)$$ $$e\left(\frac{37}{58}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{39}{58}\right)$$ $$e\left(\frac{15}{29}\right)$$ $$e\left(\frac{23}{29}\right)$$
$$\chi_{177}(158,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{25}{58}\right)$$ $$e\left(\frac{23}{29}\right)$$ $$e\left(\frac{28}{29}\right)$$ $$e\left(\frac{5}{58}\right)$$ $$e\left(\frac{11}{29}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{13}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$
$$\chi_{177}(161,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$ $$e\left(\frac{53}{58}\right)$$ $$e\left(\frac{7}{29}\right)$$ $$e\left(\frac{6}{29}\right)$$ $$e\left(\frac{57}{58}\right)$$ $$e\left(\frac{21}{29}\right)$$ $$e\left(\frac{35}{58}\right)$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{8}{29}\right)$$
$$\chi_{177}(170,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{29}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{21}{58}\right)$$ $$e\left(\frac{17}{29}\right)$$ $$e\left(\frac{27}{29}\right)$$ $$e\left(\frac{39}{58}\right)$$ $$e\left(\frac{22}{29}\right)$$ $$e\left(\frac{27}{58}\right)$$ $$e\left(\frac{26}{29}\right)$$ $$e\left(\frac{7}{29}\right)$$
$$\chi_{177}(173,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{29}\right)$$ $$e\left(\frac{2}{29}\right)$$ $$e\left(\frac{41}{58}\right)$$ $$e\left(\frac{18}{29}\right)$$ $$e\left(\frac{3}{29}\right)$$ $$e\left(\frac{43}{58}\right)$$ $$e\left(\frac{25}{29}\right)$$ $$e\left(\frac{3}{58}\right)$$ $$e\left(\frac{19}{29}\right)$$ $$e\left(\frac{4}{29}\right)$$