# Properties

 Modulus $722$ Structure $$C_{342}$$ Order $342$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(722)

pari: g = idealstar(,722,2)

## Character group

 sage: G.order()  pari: g.no Order = 342 sage: H.invariants()  pari: g.cyc Structure = $$C_{342}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{722}(363,\cdot)$

## First 32 of 342 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$21$$ $$23$$
$$\chi_{722}(1,\cdot)$$ 722.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{722}(3,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{169}{342}\right)$$ $$e\left(\frac{50}{171}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{169}{171}\right)$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{245}{342}\right)$$ $$e\left(\frac{269}{342}\right)$$ $$e\left(\frac{146}{171}\right)$$ $$e\left(\frac{157}{342}\right)$$ $$e\left(\frac{58}{171}\right)$$
$$\chi_{722}(5,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{50}{171}\right)$$ $$e\left(\frac{65}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{100}{171}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{115}{171}\right)$$ $$e\left(\frac{53}{171}\right)$$ $$e\left(\frac{8}{171}\right)$$ $$e\left(\frac{7}{171}\right)$$
$$\chi_{722}(7,\cdot)$$ 722.i 57 no $$1$$ $$1$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{15}{19}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{2}{57}\right)$$
$$\chi_{722}(9,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{169}{171}\right)$$ $$e\left(\frac{100}{171}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{167}{171}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{98}{171}\right)$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{157}{171}\right)$$ $$e\left(\frac{116}{171}\right)$$
$$\chi_{722}(11,\cdot)$$ 722.i 57 no $$1$$ $$1$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{31}{57}\right)$$
$$\chi_{722}(13,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{245}{342}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{169}{342}\right)$$ $$e\left(\frac{307}{342}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{5}{342}\right)$$ $$e\left(\frac{77}{171}\right)$$
$$\chi_{722}(15,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{269}{342}\right)$$ $$e\left(\frac{115}{171}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{98}{171}\right)$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{307}{342}\right)$$ $$e\left(\frac{157}{342}\right)$$ $$e\left(\frac{28}{171}\right)$$ $$e\left(\frac{173}{342}\right)$$ $$e\left(\frac{65}{171}\right)$$
$$\chi_{722}(17,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{146}{171}\right)$$ $$e\left(\frac{53}{171}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{28}{171}\right)$$ $$e\left(\frac{59}{171}\right)$$ $$e\left(\frac{167}{171}\right)$$ $$e\left(\frac{82}{171}\right)$$
$$\chi_{722}(21,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{157}{342}\right)$$ $$e\left(\frac{8}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{157}{171}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{5}{342}\right)$$ $$e\left(\frac{173}{342}\right)$$ $$e\left(\frac{167}{171}\right)$$ $$e\left(\frac{73}{342}\right)$$ $$e\left(\frac{64}{171}\right)$$
$$\chi_{722}(23,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{58}{171}\right)$$ $$e\left(\frac{7}{171}\right)$$ $$e\left(\frac{2}{57}\right)$$ $$e\left(\frac{116}{171}\right)$$ $$e\left(\frac{31}{57}\right)$$ $$e\left(\frac{77}{171}\right)$$ $$e\left(\frac{65}{171}\right)$$ $$e\left(\frac{82}{171}\right)$$ $$e\left(\frac{64}{171}\right)$$ $$e\left(\frac{56}{171}\right)$$
$$\chi_{722}(25,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{100}{171}\right)$$ $$e\left(\frac{130}{171}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{29}{171}\right)$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{62}{171}\right)$$ $$e\left(\frac{59}{171}\right)$$ $$e\left(\frac{106}{171}\right)$$ $$e\left(\frac{16}{171}\right)$$ $$e\left(\frac{14}{171}\right)$$
$$\chi_{722}(27,\cdot)$$ 722.j 114 no $$-1$$ $$1$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{50}{57}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{41}{114}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{43}{114}\right)$$ $$e\left(\frac{1}{57}\right)$$
$$\chi_{722}(29,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{311}{342}\right)$$ $$e\left(\frac{91}{171}\right)$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{140}{171}\right)$$ $$e\left(\frac{4}{57}\right)$$ $$e\left(\frac{121}{342}\right)$$ $$e\left(\frac{151}{342}\right)$$ $$e\left(\frac{40}{171}\right)$$ $$e\left(\frac{125}{342}\right)$$ $$e\left(\frac{44}{171}\right)$$
$$\chi_{722}(31,\cdot)$$ 722.j 114 no $$-1$$ $$1$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{31}{57}\right)$$ $$e\left(\frac{17}{19}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{7}{19}\right)$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{79}{114}\right)$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{5}{114}\right)$$ $$e\left(\frac{20}{57}\right)$$
$$\chi_{722}(33,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{325}{342}\right)$$ $$e\left(\frac{83}{171}\right)$$ $$e\left(\frac{40}{57}\right)$$ $$e\left(\frac{154}{171}\right)$$ $$e\left(\frac{50}{57}\right)$$ $$e\left(\frac{287}{342}\right)$$ $$e\left(\frac{149}{342}\right)$$ $$e\left(\frac{44}{171}\right)$$ $$e\left(\frac{223}{342}\right)$$ $$e\left(\frac{151}{171}\right)$$
$$\chi_{722}(35,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{44}{171}\right)$$ $$e\left(\frac{23}{171}\right)$$ $$e\left(\frac{31}{57}\right)$$ $$e\left(\frac{88}{171}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{82}{171}\right)$$ $$e\left(\frac{67}{171}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{137}{171}\right)$$ $$e\left(\frac{13}{171}\right)$$
$$\chi_{722}(37,\cdot)$$ 722.h 38 no $$-1$$ $$1$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{10}{19}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{31}{38}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{1}{38}\right)$$ $$e\left(\frac{4}{19}\right)$$
$$\chi_{722}(39,\cdot)$$ 722.g 19 no $$1$$ $$1$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{15}{19}\right)$$
$$\chi_{722}(41,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{79}{342}\right)$$ $$e\left(\frac{77}{171}\right)$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{79}{171}\right)$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{155}{342}\right)$$ $$e\left(\frac{233}{342}\right)$$ $$e\left(\frac{47}{171}\right)$$ $$e\left(\frac{211}{342}\right)$$ $$e\left(\frac{103}{171}\right)$$
$$\chi_{722}(43,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{23}{171}\right)$$ $$e\left(\frac{47}{171}\right)$$ $$e\left(\frac{46}{57}\right)$$ $$e\left(\frac{46}{171}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{4}{171}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{62}{171}\right)$$ $$e\left(\frac{161}{171}\right)$$ $$e\left(\frac{34}{171}\right)$$
$$\chi_{722}(45,\cdot)$$ 722.i 57 no $$1$$ $$1$$ $$e\left(\frac{16}{57}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{35}{57}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{1}{57}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{41}{57}\right)$$
$$\chi_{722}(47,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{97}{171}\right)$$ $$e\left(\frac{109}{171}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{23}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{2}{171}\right)$$ $$e\left(\frac{35}{171}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{166}{171}\right)$$ $$e\left(\frac{17}{171}\right)$$
$$\chi_{722}(49,\cdot)$$ 722.i 57 no $$1$$ $$1$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{49}{57}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{34}{57}\right)$$ $$e\left(\frac{25}{57}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{4}{57}\right)$$
$$\chi_{722}(51,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{119}{342}\right)$$ $$e\left(\frac{103}{171}\right)$$ $$e\left(\frac{5}{57}\right)$$ $$e\left(\frac{119}{171}\right)$$ $$e\left(\frac{49}{57}\right)$$ $$e\left(\frac{43}{342}\right)$$ $$e\left(\frac{325}{342}\right)$$ $$e\left(\frac{34}{171}\right)$$ $$e\left(\frac{149}{342}\right)$$ $$e\left(\frac{140}{171}\right)$$
$$\chi_{722}(53,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{251}{342}\right)$$ $$e\left(\frac{52}{171}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{80}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{289}{342}\right)$$ $$e\left(\frac{13}{342}\right)$$ $$e\left(\frac{145}{171}\right)$$ $$e\left(\frac{47}{342}\right)$$ $$e\left(\frac{74}{171}\right)$$
$$\chi_{722}(55,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{128}{171}\right)$$ $$e\left(\frac{98}{171}\right)$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{85}{171}\right)$$ $$e\left(\frac{35}{57}\right)$$ $$e\left(\frac{52}{171}\right)$$ $$e\left(\frac{55}{171}\right)$$ $$e\left(\frac{122}{171}\right)$$ $$e\left(\frac{41}{171}\right)$$ $$e\left(\frac{100}{171}\right)$$
$$\chi_{722}(59,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{193}{342}\right)$$ $$e\left(\frac{134}{171}\right)$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{22}{171}\right)$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{41}{342}\right)$$ $$e\left(\frac{119}{342}\right)$$ $$e\left(\frac{104}{171}\right)$$ $$e\left(\frac{325}{342}\right)$$ $$e\left(\frac{46}{171}\right)$$
$$\chi_{722}(61,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{67}{171}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{20}{57}\right)$$ $$e\left(\frac{134}{171}\right)$$ $$e\left(\frac{25}{57}\right)$$ $$e\left(\frac{86}{171}\right)$$ $$e\left(\frac{137}{171}\right)$$ $$e\left(\frac{136}{171}\right)$$ $$e\left(\frac{127}{171}\right)$$ $$e\left(\frac{47}{171}\right)$$
$$\chi_{722}(63,\cdot)$$ 722.k 171 no $$1$$ $$1$$ $$e\left(\frac{163}{171}\right)$$ $$e\left(\frac{58}{171}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{155}{171}\right)$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{125}{171}\right)$$ $$e\left(\frac{50}{171}\right)$$ $$e\left(\frac{142}{171}\right)$$ $$e\left(\frac{115}{171}\right)$$ $$e\left(\frac{122}{171}\right)$$
$$\chi_{722}(65,\cdot)$$ 722.j 114 no $$-1$$ $$1$$ $$e\left(\frac{1}{114}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{1}{57}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{77}{114}\right)$$ $$e\left(\frac{65}{114}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{7}{114}\right)$$ $$e\left(\frac{28}{57}\right)$$
$$\chi_{722}(67,\cdot)$$ 722.l 342 no $$-1$$ $$1$$ $$e\left(\frac{113}{342}\right)$$ $$e\left(\frac{82}{171}\right)$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{113}{171}\right)$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{265}{342}\right)$$ $$e\left(\frac{277}{342}\right)$$ $$e\left(\frac{130}{171}\right)$$ $$e\left(\frac{107}{342}\right)$$ $$e\left(\frac{143}{171}\right)$$