# Properties

 Modulus $517$ Structure $$C_{2}\times C_{230}$$ Order $460$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(517)

pari: g = idealstar(,517,2)

## Character group

 sage: G.order()  pari: g.no Order = 460 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{230}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{517}(189,\cdot)$, $\chi_{517}(287,\cdot)$

## First 32 of 460 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$
$$\chi_{517}(1,\cdot)$$ 517.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{517}(2,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{33}{230}\right)$$ $$e\left(\frac{72}{115}\right)$$ $$e\left(\frac{33}{115}\right)$$ $$e\left(\frac{91}{115}\right)$$ $$e\left(\frac{177}{230}\right)$$ $$e\left(\frac{51}{230}\right)$$ $$e\left(\frac{99}{230}\right)$$ $$e\left(\frac{29}{115}\right)$$ $$e\left(\frac{43}{46}\right)$$ $$e\left(\frac{21}{23}\right)$$
$$\chi_{517}(3,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{72}{115}\right)$$ $$e\left(\frac{11}{115}\right)$$ $$e\left(\frac{29}{115}\right)$$ $$e\left(\frac{73}{115}\right)$$ $$e\left(\frac{83}{115}\right)$$ $$e\left(\frac{59}{115}\right)$$ $$e\left(\frac{101}{115}\right)$$ $$e\left(\frac{22}{115}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$
$$\chi_{517}(4,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{33}{115}\right)$$ $$e\left(\frac{29}{115}\right)$$ $$e\left(\frac{66}{115}\right)$$ $$e\left(\frac{67}{115}\right)$$ $$e\left(\frac{62}{115}\right)$$ $$e\left(\frac{51}{115}\right)$$ $$e\left(\frac{99}{115}\right)$$ $$e\left(\frac{58}{115}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$
$$\chi_{517}(5,\cdot)$$ 517.p 230 yes $$-1$$ $$1$$ $$e\left(\frac{91}{115}\right)$$ $$e\left(\frac{73}{115}\right)$$ $$e\left(\frac{67}{115}\right)$$ $$e\left(\frac{143}{230}\right)$$ $$e\left(\frac{49}{115}\right)$$ $$e\left(\frac{57}{115}\right)$$ $$e\left(\frac{43}{115}\right)$$ $$e\left(\frac{31}{115}\right)$$ $$e\left(\frac{19}{46}\right)$$ $$e\left(\frac{5}{23}\right)$$
$$\chi_{517}(6,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{177}{230}\right)$$ $$e\left(\frac{83}{115}\right)$$ $$e\left(\frac{62}{115}\right)$$ $$e\left(\frac{49}{115}\right)$$ $$e\left(\frac{113}{230}\right)$$ $$e\left(\frac{169}{230}\right)$$ $$e\left(\frac{71}{230}\right)$$ $$e\left(\frac{51}{115}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{6}{23}\right)$$
$$\chi_{517}(7,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{51}{230}\right)$$ $$e\left(\frac{59}{115}\right)$$ $$e\left(\frac{51}{115}\right)$$ $$e\left(\frac{57}{115}\right)$$ $$e\left(\frac{169}{230}\right)$$ $$e\left(\frac{37}{230}\right)$$ $$e\left(\frac{153}{230}\right)$$ $$e\left(\frac{3}{115}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{22}{23}\right)$$
$$\chi_{517}(8,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{99}{230}\right)$$ $$e\left(\frac{101}{115}\right)$$ $$e\left(\frac{99}{115}\right)$$ $$e\left(\frac{43}{115}\right)$$ $$e\left(\frac{71}{230}\right)$$ $$e\left(\frac{153}{230}\right)$$ $$e\left(\frac{67}{230}\right)$$ $$e\left(\frac{87}{115}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{17}{23}\right)$$
$$\chi_{517}(9,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{29}{115}\right)$$ $$e\left(\frac{22}{115}\right)$$ $$e\left(\frac{58}{115}\right)$$ $$e\left(\frac{31}{115}\right)$$ $$e\left(\frac{51}{115}\right)$$ $$e\left(\frac{3}{115}\right)$$ $$e\left(\frac{87}{115}\right)$$ $$e\left(\frac{44}{115}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$
$$\chi_{517}(10,\cdot)$$ 517.l 46 yes $$1$$ $$1$$ $$e\left(\frac{43}{46}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{19}{46}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{12}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$
$$\chi_{517}(12,\cdot)$$ 517.i 23 no $$1$$ $$1$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$
$$\chi_{517}(13,\cdot)$$ 517.n 230 yes $$1$$ $$1$$ $$e\left(\frac{93}{230}\right)$$ $$e\left(\frac{67}{115}\right)$$ $$e\left(\frac{93}{115}\right)$$ $$e\left(\frac{147}{230}\right)$$ $$e\left(\frac{227}{230}\right)$$ $$e\left(\frac{81}{230}\right)$$ $$e\left(\frac{49}{230}\right)$$ $$e\left(\frac{19}{115}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$
$$\chi_{517}(14,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{42}{115}\right)$$ $$e\left(\frac{16}{115}\right)$$ $$e\left(\frac{84}{115}\right)$$ $$e\left(\frac{33}{115}\right)$$ $$e\left(\frac{58}{115}\right)$$ $$e\left(\frac{44}{115}\right)$$ $$e\left(\frac{11}{115}\right)$$ $$e\left(\frac{32}{115}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$
$$\chi_{517}(15,\cdot)$$ 517.p 230 yes $$-1$$ $$1$$ $$e\left(\frac{48}{115}\right)$$ $$e\left(\frac{84}{115}\right)$$ $$e\left(\frac{96}{115}\right)$$ $$e\left(\frac{59}{230}\right)$$ $$e\left(\frac{17}{115}\right)$$ $$e\left(\frac{1}{115}\right)$$ $$e\left(\frac{29}{115}\right)$$ $$e\left(\frac{53}{115}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{13}{23}\right)$$
$$\chi_{517}(16,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{66}{115}\right)$$ $$e\left(\frac{58}{115}\right)$$ $$e\left(\frac{17}{115}\right)$$ $$e\left(\frac{19}{115}\right)$$ $$e\left(\frac{9}{115}\right)$$ $$e\left(\frac{102}{115}\right)$$ $$e\left(\frac{83}{115}\right)$$ $$e\left(\frac{1}{115}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{517}(17,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{37}{230}\right)$$ $$e\left(\frac{18}{115}\right)$$ $$e\left(\frac{37}{115}\right)$$ $$e\left(\frac{109}{115}\right)$$ $$e\left(\frac{73}{230}\right)$$ $$e\left(\frac{99}{230}\right)$$ $$e\left(\frac{111}{230}\right)$$ $$e\left(\frac{36}{115}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{11}{23}\right)$$
$$\chi_{517}(18,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{91}{230}\right)$$ $$e\left(\frac{94}{115}\right)$$ $$e\left(\frac{91}{115}\right)$$ $$e\left(\frac{7}{115}\right)$$ $$e\left(\frac{49}{230}\right)$$ $$e\left(\frac{57}{230}\right)$$ $$e\left(\frac{43}{230}\right)$$ $$e\left(\frac{73}{115}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{14}{23}\right)$$
$$\chi_{517}(19,\cdot)$$ 517.n 230 yes $$1$$ $$1$$ $$e\left(\frac{209}{230}\right)$$ $$e\left(\frac{111}{115}\right)$$ $$e\left(\frac{94}{115}\right)$$ $$e\left(\frac{41}{230}\right)$$ $$e\left(\frac{201}{230}\right)$$ $$e\left(\frac{93}{230}\right)$$ $$e\left(\frac{167}{230}\right)$$ $$e\left(\frac{107}{115}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$
$$\chi_{517}(20,\cdot)$$ 517.p 230 yes $$-1$$ $$1$$ $$e\left(\frac{9}{115}\right)$$ $$e\left(\frac{102}{115}\right)$$ $$e\left(\frac{18}{115}\right)$$ $$e\left(\frac{47}{230}\right)$$ $$e\left(\frac{111}{115}\right)$$ $$e\left(\frac{108}{115}\right)$$ $$e\left(\frac{27}{115}\right)$$ $$e\left(\frac{89}{115}\right)$$ $$e\left(\frac{13}{46}\right)$$ $$e\left(\frac{1}{23}\right)$$
$$\chi_{517}(21,\cdot)$$ 517.k 46 yes $$-1$$ $$1$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{7}{23}\right)$$
$$\chi_{517}(23,\cdot)$$ 517.j 46 no $$-1$$ $$1$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{2}{23}\right)$$
$$\chi_{517}(24,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{13}{230}\right)$$ $$e\left(\frac{112}{115}\right)$$ $$e\left(\frac{13}{115}\right)$$ $$e\left(\frac{1}{115}\right)$$ $$e\left(\frac{7}{230}\right)$$ $$e\left(\frac{41}{230}\right)$$ $$e\left(\frac{39}{230}\right)$$ $$e\left(\frac{109}{115}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{2}{23}\right)$$
$$\chi_{517}(25,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{67}{115}\right)$$ $$e\left(\frac{31}{115}\right)$$ $$e\left(\frac{19}{115}\right)$$ $$e\left(\frac{28}{115}\right)$$ $$e\left(\frac{98}{115}\right)$$ $$e\left(\frac{114}{115}\right)$$ $$e\left(\frac{86}{115}\right)$$ $$e\left(\frac{62}{115}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$
$$\chi_{517}(26,\cdot)$$ 517.p 230 yes $$-1$$ $$1$$ $$e\left(\frac{63}{115}\right)$$ $$e\left(\frac{24}{115}\right)$$ $$e\left(\frac{11}{115}\right)$$ $$e\left(\frac{99}{230}\right)$$ $$e\left(\frac{87}{115}\right)$$ $$e\left(\frac{66}{115}\right)$$ $$e\left(\frac{74}{115}\right)$$ $$e\left(\frac{48}{115}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{7}{23}\right)$$
$$\chi_{517}(27,\cdot)$$ 517.m 115 yes $$1$$ $$1$$ $$e\left(\frac{101}{115}\right)$$ $$e\left(\frac{33}{115}\right)$$ $$e\left(\frac{87}{115}\right)$$ $$e\left(\frac{104}{115}\right)$$ $$e\left(\frac{19}{115}\right)$$ $$e\left(\frac{62}{115}\right)$$ $$e\left(\frac{73}{115}\right)$$ $$e\left(\frac{66}{115}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$
$$\chi_{517}(28,\cdot)$$ 517.o 230 yes $$-1$$ $$1$$ $$e\left(\frac{117}{230}\right)$$ $$e\left(\frac{88}{115}\right)$$ $$e\left(\frac{2}{115}\right)$$ $$e\left(\frac{9}{115}\right)$$ $$e\left(\frac{63}{230}\right)$$ $$e\left(\frac{139}{230}\right)$$ $$e\left(\frac{121}{230}\right)$$ $$e\left(\frac{61}{115}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{18}{23}\right)$$
$$\chi_{517}(29,\cdot)$$ 517.n 230 yes $$1$$ $$1$$ $$e\left(\frac{91}{230}\right)$$ $$e\left(\frac{94}{115}\right)$$ $$e\left(\frac{91}{115}\right)$$ $$e\left(\frac{129}{230}\right)$$ $$e\left(\frac{49}{230}\right)$$ $$e\left(\frac{57}{230}\right)$$ $$e\left(\frac{43}{230}\right)$$ $$e\left(\frac{73}{115}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$
$$\chi_{517}(30,\cdot)$$ 517.n 230 yes $$1$$ $$1$$ $$e\left(\frac{129}{230}\right)$$ $$e\left(\frac{41}{115}\right)$$ $$e\left(\frac{14}{115}\right)$$ $$e\left(\frac{11}{230}\right)$$ $$e\left(\frac{211}{230}\right)$$ $$e\left(\frac{53}{230}\right)$$ $$e\left(\frac{157}{230}\right)$$ $$e\left(\frac{82}{115}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$
$$\chi_{517}(31,\cdot)$$ 517.p 230 yes $$-1$$ $$1$$ $$e\left(\frac{89}{115}\right)$$ $$e\left(\frac{12}{115}\right)$$ $$e\left(\frac{63}{115}\right)$$ $$e\left(\frac{107}{230}\right)$$ $$e\left(\frac{101}{115}\right)$$ $$e\left(\frac{33}{115}\right)$$ $$e\left(\frac{37}{115}\right)$$ $$e\left(\frac{24}{115}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{517}(32,\cdot)$$ 517.k 46 yes $$-1$$ $$1$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{13}{23}\right)$$
$$\chi_{517}(34,\cdot)$$ 517.i 23 no $$1$$ $$1$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{14}{23}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{21}{23}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$
$$\chi_{517}(35,\cdot)$$ 517.n 230 yes $$1$$ $$1$$ $$e\left(\frac{3}{230}\right)$$ $$e\left(\frac{17}{115}\right)$$ $$e\left(\frac{3}{115}\right)$$ $$e\left(\frac{27}{230}\right)$$ $$e\left(\frac{37}{230}\right)$$ $$e\left(\frac{151}{230}\right)$$ $$e\left(\frac{9}{230}\right)$$ $$e\left(\frac{34}{115}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{4}{23}\right)$$