# Properties

 Modulus $712$ Structure $$C_{88}\times C_{2}\times C_{2}$$ Order $352$

# Learn more

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(712)

pari: g = idealstar(,712,2)

## Character group

 sage: G.order()  pari: g.no Order = 352 sage: H.invariants()  pari: g.cyc Structure = $$C_{88}\times C_{2}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{712}(535,\cdot)$, $\chi_{712}(357,\cdot)$, $\chi_{712}(537,\cdot)$

## First 32 of 352 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

Character Orbit Order Primitive $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{712}(1,\cdot)$$ 712.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{712}(3,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{1}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{37}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$
$$\chi_{712}(5,\cdot)$$ 712.z 44 yes $$1$$ $$1$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{8}{11}\right)$$
$$\chi_{712}(7,\cdot)$$ 712.bd 88 no $$1$$ $$1$$ $$e\left(\frac{37}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{5}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$
$$\chi_{712}(9,\cdot)$$ 712.ba 44 no $$1$$ $$1$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$
$$\chi_{712}(11,\cdot)$$ 712.w 22 yes $$-1$$ $$1$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$
$$\chi_{712}(13,\cdot)$$ 712.bc 88 yes $$-1$$ $$1$$ $$e\left(\frac{67}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{45}{88}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$
$$\chi_{712}(15,\cdot)$$ 712.bd 88 no $$1$$ $$1$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{49}{88}\right)$$ $$e\left(\frac{69}{88}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$
$$\chi_{712}(17,\cdot)$$ 712.ba 44 no $$1$$ $$1$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{712}(19,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{35}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{57}{88}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{712}(21,\cdot)$$ 712.z 44 yes $$1$$ $$1$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{10}{11}\right)$$
$$\chi_{712}(23,\cdot)$$ 712.bd 88 no $$1$$ $$1$$ $$e\left(\frac{13}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$
$$\chi_{712}(25,\cdot)$$ 712.u 22 no $$1$$ $$1$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$
$$\chi_{712}(27,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{23}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$
$$\chi_{712}(29,\cdot)$$ 712.bc 88 yes $$-1$$ $$1$$ $$e\left(\frac{15}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{27}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{81}{88}\right)$$ $$e\left(\frac{53}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$
$$\chi_{712}(31,\cdot)$$ 712.bd 88 no $$1$$ $$1$$ $$e\left(\frac{75}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{3}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{45}{88}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{73}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$
$$\chi_{712}(33,\cdot)$$ 712.be 88 no $$-1$$ $$1$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{21}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{19}{88}\right)$$ $$e\left(\frac{51}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{71}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$
$$\chi_{712}(35,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{63}{88}\right)$$ $$e\left(\frac{27}{44}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{85}{88}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{13}{44}\right)$$ $$e\left(\frac{5}{88}\right)$$ $$e\left(\frac{9}{44}\right)$$
$$\chi_{712}(37,\cdot)$$ 712.p 8 yes $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$
$$\chi_{712}(39,\cdot)$$ 712.v 22 no $$-1$$ $$1$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$
$$\chi_{712}(41,\cdot)$$ 712.be 88 no $$-1$$ $$1$$ $$e\left(\frac{21}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{31}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$
$$\chi_{712}(43,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{23}{44}\right)$$
$$\chi_{712}(45,\cdot)$$ 712.x 22 yes $$1$$ $$1$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{13}{22}\right)$$
$$\chi_{712}(47,\cdot)$$ 712.bb 44 no $$-1$$ $$1$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$
$$\chi_{712}(49,\cdot)$$ 712.ba 44 no $$1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{15}{44}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{19}{44}\right)$$ $$e\left(\frac{21}{22}\right)$$
$$\chi_{712}(51,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{3}{44}\right)$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{7}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{29}{88}\right)$$ $$e\left(\frac{13}{88}\right)$$ $$e\left(\frac{21}{44}\right)$$ $$e\left(\frac{69}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$
$$\chi_{712}(53,\cdot)$$ 712.z 44 yes $$1$$ $$1$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{35}{44}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{2}{11}\right)$$
$$\chi_{712}(55,\cdot)$$ 712.i 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-1$$
$$\chi_{712}(57,\cdot)$$ 712.u 22 no $$1$$ $$1$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$
$$\chi_{712}(59,\cdot)$$ 712.bf 88 yes $$1$$ $$1$$ $$e\left(\frac{43}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{7}{88}\right)$$ $$e\left(\frac{43}{44}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{65}{88}\right)$$ $$e\left(\frac{17}{88}\right)$$ $$e\left(\frac{41}{44}\right)$$ $$e\left(\frac{9}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$
$$\chi_{712}(61,\cdot)$$ 712.bc 88 yes $$-1$$ $$1$$ $$e\left(\frac{25}{88}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{45}{88}\right)$$ $$e\left(\frac{25}{44}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{47}{88}\right)$$ $$e\left(\frac{59}{88}\right)$$ $$e\left(\frac{31}{44}\right)$$ $$e\left(\frac{83}{88}\right)$$ $$e\left(\frac{35}{44}\right)$$
$$\chi_{712}(63,\cdot)$$ 712.bd 88 no $$1$$ $$1$$ $$e\left(\frac{39}{88}\right)$$ $$e\left(\frac{1}{44}\right)$$ $$e\left(\frac{79}{88}\right)$$ $$e\left(\frac{39}{44}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{61}{88}\right)$$ $$e\left(\frac{41}{88}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{45}{88}\right)$$ $$e\left(\frac{15}{44}\right)$$