# Properties

 Modulus $311$ Structure $$C_{310}$$ Order $310$

Show commands for: Pari/GP / SageMath

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

sage: H = DirichletGroup_conrey(311)

pari: g = idealstar(,311,2)

## Character group

 sage: G.order()  pari: g.no Order = 310 sage: H.invariants()  pari: g.cyc Structure = $$C_{310}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{311}(17,\cdot)$

## First 32 of 310 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{311}(1,\cdot)$$ 311.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{311}(2,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{87}{155}\right)$$ $$e\left(\frac{99}{155}\right)$$ $$e\left(\frac{19}{155}\right)$$ $$e\left(\frac{141}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{29}{31}\right)$$ $$e\left(\frac{106}{155}\right)$$ $$e\left(\frac{43}{155}\right)$$ $$e\left(\frac{73}{155}\right)$$ $$e\left(\frac{18}{31}\right)$$
$$\chi_{311}(3,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{99}{155}\right)$$ $$e\left(\frac{118}{155}\right)$$ $$e\left(\frac{43}{155}\right)$$ $$e\left(\frac{107}{155}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{31}\right)$$ $$e\left(\frac{142}{155}\right)$$ $$e\left(\frac{81}{155}\right)$$ $$e\left(\frac{51}{155}\right)$$ $$e\left(\frac{13}{31}\right)$$
$$\chi_{311}(4,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{19}{155}\right)$$ $$e\left(\frac{43}{155}\right)$$ $$e\left(\frac{38}{155}\right)$$ $$e\left(\frac{127}{155}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{27}{31}\right)$$ $$e\left(\frac{57}{155}\right)$$ $$e\left(\frac{86}{155}\right)$$ $$e\left(\frac{146}{155}\right)$$ $$e\left(\frac{5}{31}\right)$$
$$\chi_{311}(5,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{141}{155}\right)$$ $$e\left(\frac{107}{155}\right)$$ $$e\left(\frac{127}{155}\right)$$ $$e\left(\frac{143}{155}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{16}{31}\right)$$ $$e\left(\frac{113}{155}\right)$$ $$e\left(\frac{59}{155}\right)$$ $$e\left(\frac{129}{155}\right)$$ $$e\left(\frac{11}{31}\right)$$
$$\chi_{311}(6,\cdot)$$ 311.c 5 yes $$1$$ $$1$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$1$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$1$$
$$\chi_{311}(7,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{29}{31}\right)$$ $$e\left(\frac{2}{31}\right)$$ $$e\left(\frac{27}{31}\right)$$ $$e\left(\frac{16}{31}\right)$$ $$1$$ $$e\left(\frac{7}{31}\right)$$ $$e\left(\frac{25}{31}\right)$$ $$e\left(\frac{4}{31}\right)$$ $$e\left(\frac{14}{31}\right)$$ $$e\left(\frac{30}{31}\right)$$
$$\chi_{311}(8,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{106}{155}\right)$$ $$e\left(\frac{142}{155}\right)$$ $$e\left(\frac{57}{155}\right)$$ $$e\left(\frac{113}{155}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{25}{31}\right)$$ $$e\left(\frac{8}{155}\right)$$ $$e\left(\frac{129}{155}\right)$$ $$e\left(\frac{64}{155}\right)$$ $$e\left(\frac{23}{31}\right)$$
$$\chi_{311}(9,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{43}{155}\right)$$ $$e\left(\frac{81}{155}\right)$$ $$e\left(\frac{86}{155}\right)$$ $$e\left(\frac{59}{155}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{31}\right)$$ $$e\left(\frac{129}{155}\right)$$ $$e\left(\frac{7}{155}\right)$$ $$e\left(\frac{102}{155}\right)$$ $$e\left(\frac{26}{31}\right)$$
$$\chi_{311}(10,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{73}{155}\right)$$ $$e\left(\frac{51}{155}\right)$$ $$e\left(\frac{146}{155}\right)$$ $$e\left(\frac{129}{155}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{14}{31}\right)$$ $$e\left(\frac{64}{155}\right)$$ $$e\left(\frac{102}{155}\right)$$ $$e\left(\frac{47}{155}\right)$$ $$e\left(\frac{29}{31}\right)$$
$$\chi_{311}(11,\cdot)$$ 311.f 62 yes $$-1$$ $$1$$ $$e\left(\frac{18}{31}\right)$$ $$e\left(\frac{13}{31}\right)$$ $$e\left(\frac{5}{31}\right)$$ $$e\left(\frac{11}{31}\right)$$ $$1$$ $$e\left(\frac{30}{31}\right)$$ $$e\left(\frac{23}{31}\right)$$ $$e\left(\frac{26}{31}\right)$$ $$e\left(\frac{29}{31}\right)$$ $$e\left(\frac{49}{62}\right)$$
$$\chi_{311}(12,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{118}{155}\right)$$ $$e\left(\frac{6}{155}\right)$$ $$e\left(\frac{81}{155}\right)$$ $$e\left(\frac{79}{155}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{29}{31}\right)$$ $$e\left(\frac{44}{155}\right)$$ $$e\left(\frac{12}{155}\right)$$ $$e\left(\frac{42}{155}\right)$$ $$e\left(\frac{18}{31}\right)$$
$$\chi_{311}(13,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{21}{31}\right)$$ $$e\left(\frac{10}{31}\right)$$ $$e\left(\frac{11}{31}\right)$$ $$e\left(\frac{18}{31}\right)$$ $$1$$ $$e\left(\frac{4}{31}\right)$$ $$e\left(\frac{1}{31}\right)$$ $$e\left(\frac{20}{31}\right)$$ $$e\left(\frac{8}{31}\right)$$ $$e\left(\frac{26}{31}\right)$$
$$\chi_{311}(14,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{77}{155}\right)$$ $$e\left(\frac{109}{155}\right)$$ $$e\left(\frac{154}{155}\right)$$ $$e\left(\frac{66}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{5}{31}\right)$$ $$e\left(\frac{76}{155}\right)$$ $$e\left(\frac{63}{155}\right)$$ $$e\left(\frac{143}{155}\right)$$ $$e\left(\frac{17}{31}\right)$$
$$\chi_{311}(15,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{17}{31}\right)$$ $$e\left(\frac{14}{31}\right)$$ $$e\left(\frac{3}{31}\right)$$ $$e\left(\frac{19}{31}\right)$$ $$1$$ $$e\left(\frac{18}{31}\right)$$ $$e\left(\frac{20}{31}\right)$$ $$e\left(\frac{28}{31}\right)$$ $$e\left(\frac{5}{31}\right)$$ $$e\left(\frac{24}{31}\right)$$
$$\chi_{311}(16,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{38}{155}\right)$$ $$e\left(\frac{86}{155}\right)$$ $$e\left(\frac{76}{155}\right)$$ $$e\left(\frac{99}{155}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{23}{31}\right)$$ $$e\left(\frac{114}{155}\right)$$ $$e\left(\frac{17}{155}\right)$$ $$e\left(\frac{137}{155}\right)$$ $$e\left(\frac{10}{31}\right)$$
$$\chi_{311}(17,\cdot)$$ 311.h 310 yes $$-1$$ $$1$$ $$e\left(\frac{11}{155}\right)$$ $$e\left(\frac{82}{155}\right)$$ $$e\left(\frac{22}{155}\right)$$ $$e\left(\frac{98}{155}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{14}{31}\right)$$ $$e\left(\frac{33}{155}\right)$$ $$e\left(\frac{9}{155}\right)$$ $$e\left(\frac{109}{155}\right)$$ $$e\left(\frac{27}{62}\right)$$
$$\chi_{311}(18,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{26}{31}\right)$$ $$e\left(\frac{5}{31}\right)$$ $$e\left(\frac{21}{31}\right)$$ $$e\left(\frac{9}{31}\right)$$ $$1$$ $$e\left(\frac{2}{31}\right)$$ $$e\left(\frac{16}{31}\right)$$ $$e\left(\frac{10}{31}\right)$$ $$e\left(\frac{4}{31}\right)$$ $$e\left(\frac{13}{31}\right)$$
$$\chi_{311}(19,\cdot)$$ 311.h 310 yes $$-1$$ $$1$$ $$e\left(\frac{107}{155}\right)$$ $$e\left(\frac{79}{155}\right)$$ $$e\left(\frac{59}{155}\right)$$ $$e\left(\frac{136}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{15}{31}\right)$$ $$e\left(\frac{11}{155}\right)$$ $$e\left(\frac{3}{155}\right)$$ $$e\left(\frac{88}{155}\right)$$ $$e\left(\frac{9}{62}\right)$$
$$\chi_{311}(20,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{1}{31}\right)$$ $$e\left(\frac{30}{31}\right)$$ $$e\left(\frac{2}{31}\right)$$ $$e\left(\frac{23}{31}\right)$$ $$1$$ $$e\left(\frac{12}{31}\right)$$ $$e\left(\frac{3}{31}\right)$$ $$e\left(\frac{29}{31}\right)$$ $$e\left(\frac{24}{31}\right)$$ $$e\left(\frac{16}{31}\right)$$
$$\chi_{311}(21,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{89}{155}\right)$$ $$e\left(\frac{128}{155}\right)$$ $$e\left(\frac{23}{155}\right)$$ $$e\left(\frac{32}{155}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{31}\right)$$ $$e\left(\frac{112}{155}\right)$$ $$e\left(\frac{101}{155}\right)$$ $$e\left(\frac{121}{155}\right)$$ $$e\left(\frac{12}{31}\right)$$
$$\chi_{311}(22,\cdot)$$ 311.h 310 yes $$-1$$ $$1$$ $$e\left(\frac{22}{155}\right)$$ $$e\left(\frac{9}{155}\right)$$ $$e\left(\frac{44}{155}\right)$$ $$e\left(\frac{41}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{28}{31}\right)$$ $$e\left(\frac{66}{155}\right)$$ $$e\left(\frac{18}{155}\right)$$ $$e\left(\frac{63}{155}\right)$$ $$e\left(\frac{23}{62}\right)$$
$$\chi_{311}(23,\cdot)$$ 311.h 310 yes $$-1$$ $$1$$ $$e\left(\frac{13}{155}\right)$$ $$e\left(\frac{111}{155}\right)$$ $$e\left(\frac{26}{155}\right)$$ $$e\left(\frac{144}{155}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{25}{31}\right)$$ $$e\left(\frac{39}{155}\right)$$ $$e\left(\frac{67}{155}\right)$$ $$e\left(\frac{2}{155}\right)$$ $$e\left(\frac{15}{62}\right)$$
$$\chi_{311}(24,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{10}{31}\right)$$ $$e\left(\frac{21}{31}\right)$$ $$e\left(\frac{20}{31}\right)$$ $$e\left(\frac{13}{31}\right)$$ $$1$$ $$e\left(\frac{27}{31}\right)$$ $$e\left(\frac{30}{31}\right)$$ $$e\left(\frac{11}{31}\right)$$ $$e\left(\frac{23}{31}\right)$$ $$e\left(\frac{5}{31}\right)$$
$$\chi_{311}(25,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{127}{155}\right)$$ $$e\left(\frac{59}{155}\right)$$ $$e\left(\frac{99}{155}\right)$$ $$e\left(\frac{131}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{31}\right)$$ $$e\left(\frac{71}{155}\right)$$ $$e\left(\frac{118}{155}\right)$$ $$e\left(\frac{103}{155}\right)$$ $$e\left(\frac{22}{31}\right)$$
$$\chi_{311}(26,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{37}{155}\right)$$ $$e\left(\frac{149}{155}\right)$$ $$e\left(\frac{74}{155}\right)$$ $$e\left(\frac{76}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{31}\right)$$ $$e\left(\frac{111}{155}\right)$$ $$e\left(\frac{143}{155}\right)$$ $$e\left(\frac{113}{155}\right)$$ $$e\left(\frac{13}{31}\right)$$
$$\chi_{311}(27,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{142}{155}\right)$$ $$e\left(\frac{44}{155}\right)$$ $$e\left(\frac{129}{155}\right)$$ $$e\left(\frac{11}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{6}{31}\right)$$ $$e\left(\frac{116}{155}\right)$$ $$e\left(\frac{88}{155}\right)$$ $$e\left(\frac{153}{155}\right)$$ $$e\left(\frac{8}{31}\right)$$
$$\chi_{311}(28,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{9}{155}\right)$$ $$e\left(\frac{53}{155}\right)$$ $$e\left(\frac{18}{155}\right)$$ $$e\left(\frac{52}{155}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{3}{31}\right)$$ $$e\left(\frac{27}{155}\right)$$ $$e\left(\frac{106}{155}\right)$$ $$e\left(\frac{61}{155}\right)$$ $$e\left(\frac{4}{31}\right)$$
$$\chi_{311}(29,\cdot)$$ 311.h 310 yes $$-1$$ $$1$$ $$e\left(\frac{16}{155}\right)$$ $$e\left(\frac{77}{155}\right)$$ $$e\left(\frac{32}{155}\right)$$ $$e\left(\frac{58}{155}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{26}{31}\right)$$ $$e\left(\frac{48}{155}\right)$$ $$e\left(\frac{154}{155}\right)$$ $$e\left(\frac{74}{155}\right)$$ $$e\left(\frac{59}{62}\right)$$
$$\chi_{311}(30,\cdot)$$ 311.g 155 yes $$1$$ $$1$$ $$e\left(\frac{17}{155}\right)$$ $$e\left(\frac{14}{155}\right)$$ $$e\left(\frac{34}{155}\right)$$ $$e\left(\frac{81}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{16}{31}\right)$$ $$e\left(\frac{51}{155}\right)$$ $$e\left(\frac{28}{155}\right)$$ $$e\left(\frac{98}{155}\right)$$ $$e\left(\frac{11}{31}\right)$$
$$\chi_{311}(31,\cdot)$$ 311.h 310 yes $$-1$$ $$1$$ $$e\left(\frac{82}{155}\right)$$ $$e\left(\frac{104}{155}\right)$$ $$e\left(\frac{9}{155}\right)$$ $$e\left(\frac{26}{155}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{17}{31}\right)$$ $$e\left(\frac{91}{155}\right)$$ $$e\left(\frac{53}{155}\right)$$ $$e\left(\frac{108}{155}\right)$$ $$e\left(\frac{35}{62}\right)$$
$$\chi_{311}(32,\cdot)$$ 311.e 31 yes $$1$$ $$1$$ $$e\left(\frac{25}{31}\right)$$ $$e\left(\frac{6}{31}\right)$$ $$e\left(\frac{19}{31}\right)$$ $$e\left(\frac{17}{31}\right)$$ $$1$$ $$e\left(\frac{21}{31}\right)$$ $$e\left(\frac{13}{31}\right)$$ $$e\left(\frac{12}{31}\right)$$ $$e\left(\frac{11}{31}\right)$$ $$e\left(\frac{28}{31}\right)$$