Properties

 Modulus $289$ Structure $$C_{272}$$ Order $272$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(289)

pari: g = idealstar(,289,2)

Character group

 sage: G.order()  pari: g.no Order = 272 sage: H.invariants()  pari: g.cyc Structure = $$C_{272}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{289}(3,\cdot)$

First 32 of 272 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{289}(1,\cdot)$$ 289.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{289}(2,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{49}{68}\right)$$ $$e\left(\frac{95}{136}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{131}{136}\right)$$ $$e\left(\frac{57}{136}\right)$$ $$e\left(\frac{37}{136}\right)$$ $$e\left(\frac{11}{68}\right)$$ $$e\left(\frac{27}{68}\right)$$ $$e\left(\frac{93}{136}\right)$$ $$e\left(\frac{9}{136}\right)$$
$$\chi_{289}(3,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{95}{136}\right)$$ $$e\left(\frac{1}{272}\right)$$ $$e\left(\frac{27}{68}\right)$$ $$e\left(\frac{229}{272}\right)$$ $$e\left(\frac{191}{272}\right)$$ $$e\left(\frac{155}{272}\right)$$ $$e\left(\frac{13}{136}\right)$$ $$e\left(\frac{1}{136}\right)$$ $$e\left(\frac{147}{272}\right)$$ $$e\left(\frac{23}{272}\right)$$
$$\chi_{289}(4,\cdot)$$ 289.h 68 yes $$1$$ $$1$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{27}{68}\right)$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{63}{68}\right)$$ $$e\left(\frac{57}{68}\right)$$ $$e\left(\frac{37}{68}\right)$$ $$e\left(\frac{11}{34}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{25}{68}\right)$$ $$e\left(\frac{9}{68}\right)$$
$$\chi_{289}(5,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{131}{136}\right)$$ $$e\left(\frac{229}{272}\right)$$ $$e\left(\frac{63}{68}\right)$$ $$e\left(\frac{217}{272}\right)$$ $$e\left(\frac{219}{272}\right)$$ $$e\left(\frac{135}{272}\right)$$ $$e\left(\frac{121}{136}\right)$$ $$e\left(\frac{93}{136}\right)$$ $$e\left(\frac{207}{272}\right)$$ $$e\left(\frac{99}{272}\right)$$
$$\chi_{289}(6,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{57}{136}\right)$$ $$e\left(\frac{191}{272}\right)$$ $$e\left(\frac{57}{68}\right)$$ $$e\left(\frac{219}{272}\right)$$ $$e\left(\frac{33}{272}\right)$$ $$e\left(\frac{229}{272}\right)$$ $$e\left(\frac{35}{136}\right)$$ $$e\left(\frac{55}{136}\right)$$ $$e\left(\frac{61}{272}\right)$$ $$e\left(\frac{41}{272}\right)$$
$$\chi_{289}(7,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{37}{136}\right)$$ $$e\left(\frac{155}{272}\right)$$ $$e\left(\frac{37}{68}\right)$$ $$e\left(\frac{135}{272}\right)$$ $$e\left(\frac{229}{272}\right)$$ $$e\left(\frac{89}{272}\right)$$ $$e\left(\frac{111}{136}\right)$$ $$e\left(\frac{19}{136}\right)$$ $$e\left(\frac{209}{272}\right)$$ $$e\left(\frac{29}{272}\right)$$
$$\chi_{289}(8,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{11}{68}\right)$$ $$e\left(\frac{13}{136}\right)$$ $$e\left(\frac{11}{34}\right)$$ $$e\left(\frac{121}{136}\right)$$ $$e\left(\frac{35}{136}\right)$$ $$e\left(\frac{111}{136}\right)$$ $$e\left(\frac{33}{68}\right)$$ $$e\left(\frac{13}{68}\right)$$ $$e\left(\frac{7}{136}\right)$$ $$e\left(\frac{27}{136}\right)$$
$$\chi_{289}(9,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{27}{68}\right)$$ $$e\left(\frac{1}{136}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{93}{136}\right)$$ $$e\left(\frac{55}{136}\right)$$ $$e\left(\frac{19}{136}\right)$$ $$e\left(\frac{13}{68}\right)$$ $$e\left(\frac{1}{68}\right)$$ $$e\left(\frac{11}{136}\right)$$ $$e\left(\frac{23}{136}\right)$$
$$\chi_{289}(10,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{93}{136}\right)$$ $$e\left(\frac{147}{272}\right)$$ $$e\left(\frac{25}{68}\right)$$ $$e\left(\frac{207}{272}\right)$$ $$e\left(\frac{61}{272}\right)$$ $$e\left(\frac{209}{272}\right)$$ $$e\left(\frac{7}{136}\right)$$ $$e\left(\frac{11}{136}\right)$$ $$e\left(\frac{121}{272}\right)$$ $$e\left(\frac{117}{272}\right)$$
$$\chi_{289}(11,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{9}{136}\right)$$ $$e\left(\frac{23}{272}\right)$$ $$e\left(\frac{9}{68}\right)$$ $$e\left(\frac{99}{272}\right)$$ $$e\left(\frac{41}{272}\right)$$ $$e\left(\frac{29}{272}\right)$$ $$e\left(\frac{27}{136}\right)$$ $$e\left(\frac{23}{136}\right)$$ $$e\left(\frac{117}{272}\right)$$ $$e\left(\frac{257}{272}\right)$$
$$\chi_{289}(12,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{19}{136}\right)$$ $$e\left(\frac{109}{272}\right)$$ $$e\left(\frac{19}{68}\right)$$ $$e\left(\frac{209}{272}\right)$$ $$e\left(\frac{147}{272}\right)$$ $$e\left(\frac{31}{272}\right)$$ $$e\left(\frac{57}{136}\right)$$ $$e\left(\frac{109}{136}\right)$$ $$e\left(\frac{247}{272}\right)$$ $$e\left(\frac{59}{272}\right)$$
$$\chi_{289}(13,\cdot)$$ 289.h 68 yes $$1$$ $$1$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{49}{68}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{1}{68}\right)$$ $$e\left(\frac{43}{68}\right)$$ $$e\left(\frac{47}{68}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{15}{34}\right)$$ $$e\left(\frac{63}{68}\right)$$ $$e\left(\frac{39}{68}\right)$$
$$\chi_{289}(14,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{135}{136}\right)$$ $$e\left(\frac{73}{272}\right)$$ $$e\left(\frac{67}{68}\right)$$ $$e\left(\frac{125}{272}\right)$$ $$e\left(\frac{71}{272}\right)$$ $$e\left(\frac{163}{272}\right)$$ $$e\left(\frac{133}{136}\right)$$ $$e\left(\frac{73}{136}\right)$$ $$e\left(\frac{123}{272}\right)$$ $$e\left(\frac{47}{272}\right)$$
$$\chi_{289}(15,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{45}{68}\right)$$ $$e\left(\frac{115}{136}\right)$$ $$e\left(\frac{11}{34}\right)$$ $$e\left(\frac{87}{136}\right)$$ $$e\left(\frac{69}{136}\right)$$ $$e\left(\frac{9}{136}\right)$$ $$e\left(\frac{67}{68}\right)$$ $$e\left(\frac{47}{68}\right)$$ $$e\left(\frac{41}{136}\right)$$ $$e\left(\frac{61}{136}\right)$$
$$\chi_{289}(16,\cdot)$$ 289.g 34 yes $$1$$ $$1$$ $$e\left(\frac{15}{17}\right)$$ $$e\left(\frac{27}{34}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{10}{17}\right)$$ $$e\left(\frac{25}{34}\right)$$ $$e\left(\frac{9}{34}\right)$$
$$\chi_{289}(18,\cdot)$$ 289.f 17 yes $$1$$ $$1$$ $$e\left(\frac{2}{17}\right)$$ $$e\left(\frac{12}{17}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{11}{17}\right)$$ $$e\left(\frac{14}{17}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{6}{17}\right)$$ $$e\left(\frac{7}{17}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{4}{17}\right)$$
$$\chi_{289}(19,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{53}{68}\right)$$ $$e\left(\frac{7}{136}\right)$$ $$e\left(\frac{19}{34}\right)$$ $$e\left(\frac{107}{136}\right)$$ $$e\left(\frac{113}{136}\right)$$ $$e\left(\frac{133}{136}\right)$$ $$e\left(\frac{23}{68}\right)$$ $$e\left(\frac{7}{68}\right)$$ $$e\left(\frac{77}{136}\right)$$ $$e\left(\frac{25}{136}\right)$$
$$\chi_{289}(20,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{55}{136}\right)$$ $$e\left(\frac{65}{272}\right)$$ $$e\left(\frac{55}{68}\right)$$ $$e\left(\frac{197}{272}\right)$$ $$e\left(\frac{175}{272}\right)$$ $$e\left(\frac{11}{272}\right)$$ $$e\left(\frac{29}{136}\right)$$ $$e\left(\frac{65}{136}\right)$$ $$e\left(\frac{35}{272}\right)$$ $$e\left(\frac{135}{272}\right)$$
$$\chi_{289}(21,\cdot)$$ 289.h 68 yes $$1$$ $$1$$ $$e\left(\frac{33}{34}\right)$$ $$e\left(\frac{39}{68}\right)$$ $$e\left(\frac{16}{17}\right)$$ $$e\left(\frac{23}{68}\right)$$ $$e\left(\frac{37}{68}\right)$$ $$e\left(\frac{61}{68}\right)$$ $$e\left(\frac{31}{34}\right)$$ $$e\left(\frac{5}{34}\right)$$ $$e\left(\frac{21}{68}\right)$$ $$e\left(\frac{13}{68}\right)$$
$$\chi_{289}(22,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{107}{136}\right)$$ $$e\left(\frac{213}{272}\right)$$ $$e\left(\frac{39}{68}\right)$$ $$e\left(\frac{89}{272}\right)$$ $$e\left(\frac{155}{272}\right)$$ $$e\left(\frac{103}{272}\right)$$ $$e\left(\frac{49}{136}\right)$$ $$e\left(\frac{77}{136}\right)$$ $$e\left(\frac{31}{272}\right)$$ $$e\left(\frac{3}{272}\right)$$
$$\chi_{289}(23,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{105}{136}\right)$$ $$e\left(\frac{223}{272}\right)$$ $$e\left(\frac{37}{68}\right)$$ $$e\left(\frac{203}{272}\right)$$ $$e\left(\frac{161}{272}\right)$$ $$e\left(\frac{21}{272}\right)$$ $$e\left(\frac{43}{136}\right)$$ $$e\left(\frac{87}{136}\right)$$ $$e\left(\frac{141}{272}\right)$$ $$e\left(\frac{233}{272}\right)$$
$$\chi_{289}(24,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{117}{136}\right)$$ $$e\left(\frac{27}{272}\right)$$ $$e\left(\frac{49}{68}\right)$$ $$e\left(\frac{199}{272}\right)$$ $$e\left(\frac{261}{272}\right)$$ $$e\left(\frac{105}{272}\right)$$ $$e\left(\frac{79}{136}\right)$$ $$e\left(\frac{27}{136}\right)$$ $$e\left(\frac{161}{272}\right)$$ $$e\left(\frac{77}{272}\right)$$
$$\chi_{289}(25,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{63}{68}\right)$$ $$e\left(\frac{93}{136}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{81}{136}\right)$$ $$e\left(\frac{83}{136}\right)$$ $$e\left(\frac{135}{136}\right)$$ $$e\left(\frac{53}{68}\right)$$ $$e\left(\frac{25}{68}\right)$$ $$e\left(\frac{71}{136}\right)$$ $$e\left(\frac{99}{136}\right)$$
$$\chi_{289}(26,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{43}{68}\right)$$ $$e\left(\frac{57}{136}\right)$$ $$e\left(\frac{9}{34}\right)$$ $$e\left(\frac{133}{136}\right)$$ $$e\left(\frac{7}{136}\right)$$ $$e\left(\frac{131}{136}\right)$$ $$e\left(\frac{61}{68}\right)$$ $$e\left(\frac{57}{68}\right)$$ $$e\left(\frac{83}{136}\right)$$ $$e\left(\frac{87}{136}\right)$$
$$\chi_{289}(27,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{13}{136}\right)$$ $$e\left(\frac{3}{272}\right)$$ $$e\left(\frac{13}{68}\right)$$ $$e\left(\frac{143}{272}\right)$$ $$e\left(\frac{29}{272}\right)$$ $$e\left(\frac{193}{272}\right)$$ $$e\left(\frac{39}{136}\right)$$ $$e\left(\frac{3}{136}\right)$$ $$e\left(\frac{169}{272}\right)$$ $$e\left(\frac{69}{272}\right)$$
$$\chi_{289}(28,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{97}{136}\right)$$ $$e\left(\frac{263}{272}\right)$$ $$e\left(\frac{29}{68}\right)$$ $$e\left(\frac{115}{272}\right)$$ $$e\left(\frac{185}{272}\right)$$ $$e\left(\frac{237}{272}\right)$$ $$e\left(\frac{19}{136}\right)$$ $$e\left(\frac{127}{136}\right)$$ $$e\left(\frac{37}{272}\right)$$ $$e\left(\frac{65}{272}\right)$$
$$\chi_{289}(29,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{43}{136}\right)$$ $$e\left(\frac{125}{272}\right)$$ $$e\left(\frac{43}{68}\right)$$ $$e\left(\frac{65}{272}\right)$$ $$e\left(\frac{211}{272}\right)$$ $$e\left(\frac{63}{272}\right)$$ $$e\left(\frac{129}{136}\right)$$ $$e\left(\frac{125}{136}\right)$$ $$e\left(\frac{151}{272}\right)$$ $$e\left(\frac{155}{272}\right)$$
$$\chi_{289}(30,\cdot)$$ 289.h 68 yes $$1$$ $$1$$ $$e\left(\frac{13}{34}\right)$$ $$e\left(\frac{37}{68}\right)$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{41}{68}\right)$$ $$e\left(\frac{63}{68}\right)$$ $$e\left(\frac{23}{68}\right)$$ $$e\left(\frac{5}{34}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{67}{68}\right)$$ $$e\left(\frac{35}{68}\right)$$
$$\chi_{289}(31,\cdot)$$ 289.j 272 yes $$-1$$ $$1$$ $$e\left(\frac{39}{136}\right)$$ $$e\left(\frac{9}{272}\right)$$ $$e\left(\frac{39}{68}\right)$$ $$e\left(\frac{157}{272}\right)$$ $$e\left(\frac{87}{272}\right)$$ $$e\left(\frac{35}{272}\right)$$ $$e\left(\frac{117}{136}\right)$$ $$e\left(\frac{9}{136}\right)$$ $$e\left(\frac{235}{272}\right)$$ $$e\left(\frac{207}{272}\right)$$
$$\chi_{289}(32,\cdot)$$ 289.i 136 yes $$1$$ $$1$$ $$e\left(\frac{41}{68}\right)$$ $$e\left(\frac{67}{136}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{111}{136}\right)$$ $$e\left(\frac{13}{136}\right)$$ $$e\left(\frac{49}{136}\right)$$ $$e\left(\frac{55}{68}\right)$$ $$e\left(\frac{67}{68}\right)$$ $$e\left(\frac{57}{136}\right)$$ $$e\left(\frac{45}{136}\right)$$
$$\chi_{289}(33,\cdot)$$ 289.g 34 yes $$1$$ $$1$$ $$e\left(\frac{13}{17}\right)$$ $$e\left(\frac{3}{34}\right)$$ $$e\left(\frac{9}{17}\right)$$ $$e\left(\frac{7}{34}\right)$$ $$e\left(\frac{29}{34}\right)$$ $$e\left(\frac{23}{34}\right)$$ $$e\left(\frac{5}{17}\right)$$ $$e\left(\frac{3}{17}\right)$$ $$e\left(\frac{33}{34}\right)$$ $$e\left(\frac{1}{34}\right)$$