# Properties

 Modulus $1013$ Structure $$C_{1012}$$ Order $1012$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(1013)

pari: g = idealstar(,1013,2)

## Character group

 sage: G.order()  pari: g.no Order = 1012 sage: H.invariants()  pari: g.cyc Structure = $$C_{1012}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1013}(3,\cdot)$

## First 32 of 1012 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{1013}(1,\cdot)$$ 1013.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1013}(2,\cdot)$$ 1013.i 92 yes $$-1$$ $$1$$ $$e\left(\frac{19}{92}\right)$$ $$e\left(\frac{33}{92}\right)$$ $$e\left(\frac{19}{46}\right)$$ $$e\left(\frac{49}{92}\right)$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{45}{92}\right)$$ $$e\left(\frac{57}{92}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$
$$\chi_{1013}(3,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{33}{92}\right)$$ $$e\left(\frac{1}{1012}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{757}{1012}\right)$$ $$e\left(\frac{91}{253}\right)$$ $$e\left(\frac{821}{1012}\right)$$ $$e\left(\frac{7}{92}\right)$$ $$e\left(\frac{1}{506}\right)$$ $$e\left(\frac{27}{253}\right)$$ $$e\left(\frac{27}{46}\right)$$
$$\chi_{1013}(4,\cdot)$$ 1013.h 46 yes $$1$$ $$1$$ $$e\left(\frac{19}{46}\right)$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$
$$\chi_{1013}(5,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{49}{92}\right)$$ $$e\left(\frac{757}{1012}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{257}{1012}\right)$$ $$e\left(\frac{71}{253}\right)$$ $$e\left(\frac{129}{1012}\right)$$ $$e\left(\frac{55}{92}\right)$$ $$e\left(\frac{251}{506}\right)$$ $$e\left(\frac{199}{253}\right)$$ $$e\left(\frac{15}{46}\right)$$
$$\chi_{1013}(6,\cdot)$$ 1013.j 253 yes $$1$$ $$1$$ $$e\left(\frac{13}{23}\right)$$ $$e\left(\frac{91}{253}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{71}{253}\right)$$ $$e\left(\frac{234}{253}\right)$$ $$e\left(\frac{76}{253}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{182}{253}\right)$$ $$e\left(\frac{214}{253}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{1013}(7,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{45}{92}\right)$$ $$e\left(\frac{821}{1012}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{129}{1012}\right)$$ $$e\left(\frac{76}{253}\right)$$ $$e\left(\frac{49}{1012}\right)$$ $$e\left(\frac{43}{92}\right)$$ $$e\left(\frac{315}{506}\right)$$ $$e\left(\frac{156}{253}\right)$$ $$e\left(\frac{41}{46}\right)$$
$$\chi_{1013}(8,\cdot)$$ 1013.i 92 yes $$-1$$ $$1$$ $$e\left(\frac{57}{92}\right)$$ $$e\left(\frac{7}{92}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{55}{92}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{43}{92}\right)$$ $$e\left(\frac{79}{92}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{9}{46}\right)$$
$$\chi_{1013}(9,\cdot)$$ 1013.k 506 yes $$1$$ $$1$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{1}{506}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{251}{506}\right)$$ $$e\left(\frac{182}{253}\right)$$ $$e\left(\frac{315}{506}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{1}{253}\right)$$ $$e\left(\frac{54}{253}\right)$$ $$e\left(\frac{4}{23}\right)$$
$$\chi_{1013}(10,\cdot)$$ 1013.j 253 yes $$1$$ $$1$$ $$e\left(\frac{17}{23}\right)$$ $$e\left(\frac{27}{253}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{199}{253}\right)$$ $$e\left(\frac{214}{253}\right)$$ $$e\left(\frac{156}{253}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{54}{253}\right)$$ $$e\left(\frac{133}{253}\right)$$ $$e\left(\frac{9}{23}\right)$$
$$\chi_{1013}(11,\cdot)$$ 1013.h 46 yes $$1$$ $$1$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{15}{46}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{4}{23}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{1013}(12,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{71}{92}\right)$$ $$e\left(\frac{727}{1012}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{823}{1012}\right)$$ $$e\left(\frac{124}{253}\right)$$ $$e\left(\frac{799}{1012}\right)$$ $$e\left(\frac{29}{92}\right)$$ $$e\left(\frac{221}{506}\right)$$ $$e\left(\frac{148}{253}\right)$$ $$e\left(\frac{33}{46}\right)$$
$$\chi_{1013}(13,\cdot)$$ 1013.k 506 yes $$1$$ $$1$$ $$e\left(\frac{33}{46}\right)$$ $$e\left(\frac{415}{506}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{435}{506}\right)$$ $$e\left(\frac{136}{253}\right)$$ $$e\left(\frac{177}{506}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{162}{253}\right)$$ $$e\left(\frac{146}{253}\right)$$ $$e\left(\frac{4}{23}\right)$$
$$\chi_{1013}(14,\cdot)$$ 1013.j 253 yes $$1$$ $$1$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{43}{253}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{167}{253}\right)$$ $$e\left(\frac{219}{253}\right)$$ $$e\left(\frac{136}{253}\right)$$ $$e\left(\frac{2}{23}\right)$$ $$e\left(\frac{86}{253}\right)$$ $$e\left(\frac{90}{253}\right)$$ $$e\left(\frac{22}{23}\right)$$
$$\chi_{1013}(15,\cdot)$$ 1013.k 506 yes $$1$$ $$1$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{379}{506}\right)$$ $$e\left(\frac{18}{23}\right)$$ $$e\left(\frac{1}{506}\right)$$ $$e\left(\frac{162}{253}\right)$$ $$e\left(\frac{475}{506}\right)$$ $$e\left(\frac{31}{46}\right)$$ $$e\left(\frac{126}{253}\right)$$ $$e\left(\frac{226}{253}\right)$$ $$e\left(\frac{21}{23}\right)$$
$$\chi_{1013}(16,\cdot)$$ 1013.f 23 yes $$1$$ $$1$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{15}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{11}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{6}{23}\right)$$
$$\chi_{1013}(17,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{17}{92}\right)$$ $$e\left(\frac{349}{1012}\right)$$ $$e\left(\frac{17}{46}\right)$$ $$e\left(\frac{61}{1012}\right)$$ $$e\left(\frac{134}{253}\right)$$ $$e\left(\frac{133}{1012}\right)$$ $$e\left(\frac{51}{92}\right)$$ $$e\left(\frac{349}{506}\right)$$ $$e\left(\frac{62}{253}\right)$$ $$e\left(\frac{39}{46}\right)$$
$$\chi_{1013}(18,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{85}{92}\right)$$ $$e\left(\frac{365}{1012}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{29}{1012}\right)$$ $$e\left(\frac{72}{253}\right)$$ $$e\left(\frac{113}{1012}\right)$$ $$e\left(\frac{71}{92}\right)$$ $$e\left(\frac{365}{506}\right)$$ $$e\left(\frac{241}{253}\right)$$ $$e\left(\frac{11}{46}\right)$$
$$\chi_{1013}(19,\cdot)$$ 1013.j 253 yes $$1$$ $$1$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{116}{253}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{21}{253}\right)$$ $$e\left(\frac{226}{253}\right)$$ $$e\left(\frac{108}{253}\right)$$ $$e\left(\frac{7}{23}\right)$$ $$e\left(\frac{232}{253}\right)$$ $$e\left(\frac{131}{253}\right)$$ $$e\left(\frac{8}{23}\right)$$
$$\chi_{1013}(20,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{87}{92}\right)$$ $$e\left(\frac{471}{1012}\right)$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{323}{1012}\right)$$ $$e\left(\frac{104}{253}\right)$$ $$e\left(\frac{107}{1012}\right)$$ $$e\left(\frac{77}{92}\right)$$ $$e\left(\frac{471}{506}\right)$$ $$e\left(\frac{67}{253}\right)$$ $$e\left(\frac{21}{46}\right)$$
$$\chi_{1013}(21,\cdot)$$ 1013.k 506 yes $$1$$ $$1$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{411}{506}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{443}{506}\right)$$ $$e\left(\frac{167}{253}\right)$$ $$e\left(\frac{435}{506}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{158}{253}\right)$$ $$e\left(\frac{183}{253}\right)$$ $$e\left(\frac{11}{23}\right)$$
$$\chi_{1013}(22,\cdot)$$ 1013.i 92 yes $$-1$$ $$1$$ $$e\left(\frac{25}{92}\right)$$ $$e\left(\frac{87}{92}\right)$$ $$e\left(\frac{25}{46}\right)$$ $$e\left(\frac{79}{92}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{35}{92}\right)$$ $$e\left(\frac{75}{92}\right)$$ $$e\left(\frac{41}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{33}{46}\right)$$
$$\chi_{1013}(23,\cdot)$$ 1013.h 46 yes $$1$$ $$1$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{1}{23}\right)$$ $$e\left(\frac{5}{46}\right)$$ $$e\left(\frac{5}{23}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{9}{23}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{5}{23}\right)$$
$$\chi_{1013}(24,\cdot)$$ 1013.k 506 yes $$1$$ $$1$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{39}{506}\right)$$ $$e\left(\frac{22}{23}\right)$$ $$e\left(\frac{175}{506}\right)$$ $$e\left(\frac{14}{253}\right)$$ $$e\left(\frac{141}{506}\right)$$ $$e\left(\frac{43}{46}\right)$$ $$e\left(\frac{39}{253}\right)$$ $$e\left(\frac{82}{253}\right)$$ $$e\left(\frac{18}{23}\right)$$
$$\chi_{1013}(25,\cdot)$$ 1013.k 506 yes $$1$$ $$1$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{251}{506}\right)$$ $$e\left(\frac{3}{23}\right)$$ $$e\left(\frac{257}{506}\right)$$ $$e\left(\frac{142}{253}\right)$$ $$e\left(\frac{129}{506}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{251}{253}\right)$$ $$e\left(\frac{145}{253}\right)$$ $$e\left(\frac{15}{23}\right)$$
$$\chi_{1013}(26,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{85}{92}\right)$$ $$e\left(\frac{181}{1012}\right)$$ $$e\left(\frac{39}{46}\right)$$ $$e\left(\frac{397}{1012}\right)$$ $$e\left(\frac{26}{253}\right)$$ $$e\left(\frac{849}{1012}\right)$$ $$e\left(\frac{71}{92}\right)$$ $$e\left(\frac{181}{506}\right)$$ $$e\left(\frac{80}{253}\right)$$ $$e\left(\frac{11}{46}\right)$$
$$\chi_{1013}(27,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{7}{92}\right)$$ $$e\left(\frac{3}{1012}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{247}{1012}\right)$$ $$e\left(\frac{20}{253}\right)$$ $$e\left(\frac{439}{1012}\right)$$ $$e\left(\frac{21}{92}\right)$$ $$e\left(\frac{3}{506}\right)$$ $$e\left(\frac{81}{253}\right)$$ $$e\left(\frac{35}{46}\right)$$
$$\chi_{1013}(28,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{83}{92}\right)$$ $$e\left(\frac{535}{1012}\right)$$ $$e\left(\frac{37}{46}\right)$$ $$e\left(\frac{195}{1012}\right)$$ $$e\left(\frac{109}{253}\right)$$ $$e\left(\frac{27}{1012}\right)$$ $$e\left(\frac{65}{92}\right)$$ $$e\left(\frac{29}{506}\right)$$ $$e\left(\frac{24}{253}\right)$$ $$e\left(\frac{1}{46}\right)$$
$$\chi_{1013}(29,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{17}{92}\right)$$ $$e\left(\frac{809}{1012}\right)$$ $$e\left(\frac{17}{46}\right)$$ $$e\left(\frac{153}{1012}\right)$$ $$e\left(\frac{249}{253}\right)$$ $$e\left(\frac{317}{1012}\right)$$ $$e\left(\frac{51}{92}\right)$$ $$e\left(\frac{303}{506}\right)$$ $$e\left(\frac{85}{253}\right)$$ $$e\left(\frac{39}{46}\right)$$
$$\chi_{1013}(30,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{9}{92}\right)$$ $$e\left(\frac{109}{1012}\right)$$ $$e\left(\frac{9}{46}\right)$$ $$e\left(\frac{541}{1012}\right)$$ $$e\left(\frac{52}{253}\right)$$ $$e\left(\frac{433}{1012}\right)$$ $$e\left(\frac{27}{92}\right)$$ $$e\left(\frac{109}{506}\right)$$ $$e\left(\frac{160}{253}\right)$$ $$e\left(\frac{45}{46}\right)$$
$$\chi_{1013}(31,\cdot)$$ 1013.l 1012 yes $$-1$$ $$1$$ $$e\left(\frac{29}{92}\right)$$ $$e\left(\frac{525}{1012}\right)$$ $$e\left(\frac{29}{46}\right)$$ $$e\left(\frac{721}{1012}\right)$$ $$e\left(\frac{211}{253}\right)$$ $$e\left(\frac{925}{1012}\right)$$ $$e\left(\frac{87}{92}\right)$$ $$e\left(\frac{19}{506}\right)$$ $$e\left(\frac{7}{253}\right)$$ $$e\left(\frac{7}{46}\right)$$
$$\chi_{1013}(32,\cdot)$$ 1013.i 92 yes $$-1$$ $$1$$ $$e\left(\frac{3}{92}\right)$$ $$e\left(\frac{73}{92}\right)$$ $$e\left(\frac{3}{46}\right)$$ $$e\left(\frac{61}{92}\right)$$ $$e\left(\frac{19}{23}\right)$$ $$e\left(\frac{41}{92}\right)$$ $$e\left(\frac{9}{92}\right)$$ $$e\left(\frac{27}{46}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{15}{46}\right)$$