Properties

 Modulus $83$ Structure $$C_{82}$$ Order $82$

Show commands for: Pari/GP / SageMath

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(83)

pari: g = idealstar(,83,2)

Character group

 sage: G.order()  pari: g.no Order = 82 sage: H.invariants()  pari: g.cyc Structure = $$C_{82}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{83}(2,\cdot)$

First 32 of 82 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{83}(1,\cdot)$$ 83.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{83}(2,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{1}{82}\right)$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{27}{82}\right)$$ $$e\left(\frac{73}{82}\right)$$ $$e\left(\frac{4}{41}\right)$$ $$e\left(\frac{3}{82}\right)$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{14}{41}\right)$$ $$e\left(\frac{12}{41}\right)$$
$$\chi_{83}(3,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{4}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{24}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$
$$\chi_{83}(4,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{2}{41}\right)$$ $$e\left(\frac{27}{41}\right)$$ $$e\left(\frac{32}{41}\right)$$ $$e\left(\frac{8}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{21}{41}\right)$$ $$e\left(\frac{28}{41}\right)$$ $$e\left(\frac{24}{41}\right)$$
$$\chi_{83}(5,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{27}{82}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{27}{41}\right)$$ $$e\left(\frac{73}{82}\right)$$ $$e\left(\frac{3}{82}\right)$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{81}{82}\right)$$ $$e\left(\frac{17}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{37}{41}\right)$$
$$\chi_{83}(6,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{73}{82}\right)$$ $$e\left(\frac{4}{41}\right)$$ $$e\left(\frac{32}{41}\right)$$ $$e\left(\frac{3}{82}\right)$$ $$e\left(\frac{81}{82}\right)$$ $$e\left(\frac{5}{41}\right)$$ $$e\left(\frac{55}{82}\right)$$ $$e\left(\frac{8}{41}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$
$$\chi_{83}(7,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{4}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{8}{41}\right)$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{5}{41}\right)$$ $$e\left(\frac{32}{41}\right)$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{2}{41}\right)$$ $$e\left(\frac{30}{41}\right)$$ $$e\left(\frac{14}{41}\right)$$
$$\chi_{83}(8,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{3}{82}\right)$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{81}{82}\right)$$ $$e\left(\frac{55}{82}\right)$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{9}{82}\right)$$ $$e\left(\frac{11}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{36}{41}\right)$$
$$\chi_{83}(9,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{21}{41}\right)$$ $$e\left(\frac{17}{41}\right)$$ $$e\left(\frac{8}{41}\right)$$ $$e\left(\frac{2}{41}\right)$$ $$e\left(\frac{11}{41}\right)$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{7}{41}\right)$$ $$e\left(\frac{6}{41}\right)$$
$$\chi_{83}(10,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{14}{41}\right)$$ $$e\left(\frac{24}{41}\right)$$ $$e\left(\frac{28}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{30}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{7}{41}\right)$$ $$e\left(\frac{23}{41}\right)$$ $$e\left(\frac{8}{41}\right)$$
$$\chi_{83}(11,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{24}{41}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{14}{41}\right)$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{6}{41}\right)$$ $$e\left(\frac{8}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$
$$\chi_{83}(12,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{40}{41}\right)$$ $$e\left(\frac{33}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{39}{41}\right)$$ $$e\left(\frac{11}{41}\right)$$ $$e\left(\frac{27}{41}\right)$$
$$\chi_{83}(13,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{77}{82}\right)$$ $$e\left(\frac{25}{41}\right)$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{29}{82}\right)$$ $$e\left(\frac{45}{82}\right)$$ $$e\left(\frac{21}{41}\right)$$ $$e\left(\frac{67}{82}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{22}{41}\right)$$
$$\chi_{83}(14,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{9}{82}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{79}{82}\right)$$ $$e\left(\frac{1}{82}\right)$$ $$e\left(\frac{36}{41}\right)$$ $$e\left(\frac{27}{82}\right)$$ $$e\left(\frac{33}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{26}{41}\right)$$
$$\chi_{83}(15,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{17}{82}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{17}{41}\right)$$ $$e\left(\frac{49}{82}\right)$$ $$e\left(\frac{11}{82}\right)$$ $$e\left(\frac{27}{41}\right)$$ $$e\left(\frac{51}{82}\right)$$ $$e\left(\frac{35}{41}\right)$$ $$e\left(\frac{33}{41}\right)$$ $$e\left(\frac{40}{41}\right)$$
$$\chi_{83}(16,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{2}{41}\right)$$ $$e\left(\frac{21}{41}\right)$$ $$e\left(\frac{4}{41}\right)$$ $$e\left(\frac{13}{41}\right)$$ $$e\left(\frac{23}{41}\right)$$ $$e\left(\frac{16}{41}\right)$$ $$e\left(\frac{6}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{7}{41}\right)$$
$$\chi_{83}(17,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{28}{41}\right)$$ $$e\left(\frac{7}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{35}{41}\right)$$ $$e\left(\frac{19}{41}\right)$$ $$e\left(\frac{2}{41}\right)$$ $$e\left(\frac{14}{41}\right)$$ $$e\left(\frac{5}{41}\right)$$ $$e\left(\frac{16}{41}\right)$$
$$\chi_{83}(18,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{63}{82}\right)$$ $$e\left(\frac{13}{41}\right)$$ $$e\left(\frac{22}{41}\right)$$ $$e\left(\frac{61}{82}\right)$$ $$e\left(\frac{7}{82}\right)$$ $$e\left(\frac{6}{41}\right)$$ $$e\left(\frac{25}{82}\right)$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{21}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$
$$\chi_{83}(19,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{47}{82}\right)$$ $$e\left(\frac{11}{41}\right)$$ $$e\left(\frac{6}{41}\right)$$ $$e\left(\frac{39}{82}\right)$$ $$e\left(\frac{69}{82}\right)$$ $$e\left(\frac{24}{41}\right)$$ $$e\left(\frac{59}{82}\right)$$ $$e\left(\frac{22}{41}\right)$$ $$e\left(\frac{2}{41}\right)$$ $$e\left(\frac{31}{41}\right)$$
$$\chi_{83}(20,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{29}{82}\right)$$ $$e\left(\frac{19}{41}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{45}{82}\right)$$ $$e\left(\frac{67}{82}\right)$$ $$e\left(\frac{34}{41}\right)$$ $$e\left(\frac{5}{82}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{20}{41}\right)$$
$$\chi_{83}(21,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{40}{41}\right)$$ $$e\left(\frac{10}{41}\right)$$ $$e\left(\frac{39}{41}\right)$$ $$e\left(\frac{14}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{33}{41}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{20}{41}\right)$$ $$e\left(\frac{13}{41}\right)$$ $$e\left(\frac{17}{41}\right)$$
$$\chi_{83}(22,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{25}{82}\right)$$ $$e\left(\frac{39}{41}\right)$$ $$e\left(\frac{25}{41}\right)$$ $$e\left(\frac{19}{82}\right)$$ $$e\left(\frac{21}{82}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{75}{82}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{22}{41}\right)$$ $$e\left(\frac{13}{41}\right)$$
$$\chi_{83}(23,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{30}{41}\right)$$ $$e\left(\frac{28}{41}\right)$$ $$e\left(\frac{19}{41}\right)$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{17}{41}\right)$$ $$e\left(\frac{35}{41}\right)$$ $$e\left(\frac{8}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{20}{41}\right)$$ $$e\left(\frac{23}{41}\right)$$
$$\chi_{83}(24,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{75}{82}\right)$$ $$e\left(\frac{35}{41}\right)$$ $$e\left(\frac{34}{41}\right)$$ $$e\left(\frac{57}{82}\right)$$ $$e\left(\frac{63}{82}\right)$$ $$e\left(\frac{13}{41}\right)$$ $$e\left(\frac{61}{82}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{25}{41}\right)$$ $$e\left(\frac{39}{41}\right)$$
$$\chi_{83}(25,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{27}{41}\right)$$ $$e\left(\frac{17}{41}\right)$$ $$e\left(\frac{13}{41}\right)$$ $$e\left(\frac{32}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{11}{41}\right)$$ $$e\left(\frac{40}{41}\right)$$ $$e\left(\frac{34}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{33}{41}\right)$$
$$\chi_{83}(26,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{39}{41}\right)$$ $$e\left(\frac{20}{41}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{28}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{25}{41}\right)$$ $$e\left(\frac{35}{41}\right)$$ $$e\left(\frac{40}{41}\right)$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{34}{41}\right)$$
$$\chi_{83}(27,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{26}{41}\right)$$ $$e\left(\frac{27}{41}\right)$$ $$e\left(\frac{11}{41}\right)$$ $$e\left(\frac{5}{41}\right)$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{13}{41}\right)$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{9}{41}\right)$$
$$\chi_{83}(28,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{5}{41}\right)$$ $$e\left(\frac{32}{41}\right)$$ $$e\left(\frac{10}{41}\right)$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{37}{41}\right)$$ $$e\left(\frac{40}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{23}{41}\right)$$ $$e\left(\frac{17}{41}\right)$$ $$e\left(\frac{38}{41}\right)$$
$$\chi_{83}(29,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{6}{41}\right)$$ $$e\left(\frac{22}{41}\right)$$ $$e\left(\frac{12}{41}\right)$$ $$e\left(\frac{39}{41}\right)$$ $$e\left(\frac{28}{41}\right)$$ $$e\left(\frac{7}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{3}{41}\right)$$ $$e\left(\frac{4}{41}\right)$$ $$e\left(\frac{21}{41}\right)$$
$$\chi_{83}(30,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{9}{41}\right)$$ $$e\left(\frac{33}{41}\right)$$ $$e\left(\frac{18}{41}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{1}{41}\right)$$ $$e\left(\frac{31}{41}\right)$$ $$e\left(\frac{27}{41}\right)$$ $$e\left(\frac{25}{41}\right)$$ $$e\left(\frac{6}{41}\right)$$ $$e\left(\frac{11}{41}\right)$$
$$\chi_{83}(31,\cdot)$$ 83.c 41 yes $$1$$ $$1$$ $$e\left(\frac{19}{41}\right)$$ $$e\left(\frac{15}{41}\right)$$ $$e\left(\frac{38}{41}\right)$$ $$e\left(\frac{21}{41}\right)$$ $$e\left(\frac{34}{41}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{16}{41}\right)$$ $$e\left(\frac{30}{41}\right)$$ $$e\left(\frac{40}{41}\right)$$ $$e\left(\frac{5}{41}\right)$$
$$\chi_{83}(32,\cdot)$$ 83.d 82 yes $$-1$$ $$1$$ $$e\left(\frac{5}{82}\right)$$ $$e\left(\frac{16}{41}\right)$$ $$e\left(\frac{5}{41}\right)$$ $$e\left(\frac{53}{82}\right)$$ $$e\left(\frac{37}{82}\right)$$ $$e\left(\frac{20}{41}\right)$$ $$e\left(\frac{15}{82}\right)$$ $$e\left(\frac{32}{41}\right)$$ $$e\left(\frac{29}{41}\right)$$ $$e\left(\frac{19}{41}\right)$$