# Properties

 Label 712.bf Modulus $712$ Conductor $712$ Order $88$ 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(712, base_ring=CyclotomicField(88))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([44,44,1]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(3,712))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$712$$ Conductor: $$712$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$88$$ 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: $\Q(\zeta_{88})$ Fixed field: Number field defined by a degree 88 polynomial

## First 31 of 40 characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{712}(3,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{37}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$
$$\chi_{712}(19,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{712}(27,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{23}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$
$$\chi_{712}(35,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{5}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$
$$\chi_{712}(43,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$
$$\chi_{712}(51,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{13}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{69}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$
$$\chi_{712}(59,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$
$$\chi_{712}(75,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{23}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{712}(83,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{61}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{23}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$
$$\chi_{712}(115,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{61}{88}\right)$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{45}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$
$$\chi_{712}(147,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$
$$\chi_{712}(155,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$
$$\chi_{712}(163,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$
$$\chi_{712}(171,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$
$$\chi_{712}(211,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{71}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$
$$\chi_{712}(219,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$
$$\chi_{712}(243,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{71}{88}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{712}(291,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{53}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{712}(315,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$
$$\chi_{712}(323,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{21}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{88}\right)$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$
$$\chi_{712}(363,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{5}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{59}{88}\right)$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{19}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$
$$\chi_{712}(371,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{71}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{5}{88}\right)$$ $$e\left(\frac{69}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{21}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$
$$\chi_{712}(379,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{59}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$
$$\chi_{712}(387,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{53}{88}\right)$$ $$e\left(\frac{45}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$
$$\chi_{712}(419,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$
$$\chi_{712}(451,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{13}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{19}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$
$$\chi_{712}(459,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{69}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{712}(475,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{21}{88}\right)$$ $$e\left(\frac{61}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{53}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$
$$\chi_{712}(483,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$
$$\chi_{712}(491,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{61}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$
$$\chi_{712}(499,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{87}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$