# Properties

 Modulus $1444$ Structure $$C_{2}\times C_{342}$$ Order $684$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(1444)

pari: g = idealstar(,1444,2)

## Character group

 sage: G.order()  pari: g.no Order = 684 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{342}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1444}(723,\cdot)$, $\chi_{1444}(1085,\cdot)$

## First 32 of 684 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$21$$ $$23$$
$$\chi_{1444}(1,\cdot)$$ 1444.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1444}(3,\cdot)$$ 1444.w 342 yes $$1$$ $$1$$ $$e\left(\frac{170}{171}\right)$$ $$e\left(\frac{50}{171}\right)$$ $$e\left(\frac{53}{114}\right)$$ $$e\left(\frac{169}{171}\right)$$ $$e\left(\frac{109}{114}\right)$$ $$e\left(\frac{245}{342}\right)$$ $$e\left(\frac{49}{171}\right)$$ $$e\left(\frac{146}{171}\right)$$ $$e\left(\frac{157}{342}\right)$$ $$e\left(\frac{287}{342}\right)$$
$$\chi_{1444}(5,\cdot)$$ 1444.u 171 no $$1$$ $$1$$ $$e\left(\frac{50}{171}\right)$$ $$e\left(\frac{65}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{100}{171}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{115}{171}\right)$$ $$e\left(\frac{53}{171}\right)$$ $$e\left(\frac{8}{171}\right)$$ $$e\left(\frac{7}{171}\right)$$
$$\chi_{1444}(7,\cdot)$$ 1444.s 114 yes $$-1$$ $$1$$ $$e\left(\frac{53}{114}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{9}{38}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{25}{114}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{61}{114}\right)$$
$$\chi_{1444}(9,\cdot)$$ 1444.u 171 no $$1$$ $$1$$ $$e\left(\frac{169}{171}\right)$$ $$e\left(\frac{100}{171}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{167}{171}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{98}{171}\right)$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{157}{171}\right)$$ $$e\left(\frac{116}{171}\right)$$
$$\chi_{1444}(11,\cdot)$$ 1444.s 114 yes $$-1$$ $$1$$ $$e\left(\frac{109}{114}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{9}{38}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{35}{38}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{5}{114}\right)$$
$$\chi_{1444}(13,\cdot)$$ 1444.x 342 no $$-1$$ $$1$$ $$e\left(\frac{245}{342}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{169}{342}\right)$$ $$e\left(\frac{307}{342}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{5}{342}\right)$$ $$e\left(\frac{77}{171}\right)$$
$$\chi_{1444}(15,\cdot)$$ 1444.w 342 yes $$1$$ $$1$$ $$e\left(\frac{49}{171}\right)$$ $$e\left(\frac{115}{171}\right)$$ $$e\left(\frac{25}{114}\right)$$ $$e\left(\frac{98}{171}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{307}{342}\right)$$ $$e\left(\frac{164}{171}\right)$$ $$e\left(\frac{28}{171}\right)$$ $$e\left(\frac{173}{342}\right)$$ $$e\left(\frac{301}{342}\right)$$
$$\chi_{1444}(17,\cdot)$$ 1444.u 171 no $$1$$ $$1$$ $$e\left(\frac{146}{171}\right)$$ $$e\left(\frac{53}{171}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{28}{171}\right)$$ $$e\left(\frac{59}{171}\right)$$ $$e\left(\frac{167}{171}\right)$$ $$e\left(\frac{82}{171}\right)$$
$$\chi_{1444}(21,\cdot)$$ 1444.x 342 no $$-1$$ $$1$$ $$e\left(\frac{157}{342}\right)$$ $$e\left(\frac{8}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{157}{171}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{5}{342}\right)$$ $$e\left(\frac{173}{342}\right)$$ $$e\left(\frac{167}{171}\right)$$ $$e\left(\frac{73}{342}\right)$$ $$e\left(\frac{64}{171}\right)$$
$$\chi_{1444}(23,\cdot)$$ 1444.v 342 yes $$-1$$ $$1$$ $$e\left(\frac{287}{342}\right)$$ $$e\left(\frac{7}{171}\right)$$ $$e\left(\frac{61}{114}\right)$$ $$e\left(\frac{116}{171}\right)$$ $$e\left(\frac{5}{114}\right)$$ $$e\left(\frac{77}{171}\right)$$ $$e\left(\frac{301}{342}\right)$$ $$e\left(\frac{82}{171}\right)$$ $$e\left(\frac{64}{171}\right)$$ $$e\left(\frac{283}{342}\right)$$
$$\chi_{1444}(25,\cdot)$$ 1444.u 171 no $$1$$ $$1$$ $$e\left(\frac{100}{171}\right)$$ $$e\left(\frac{130}{171}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{29}{171}\right)$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{62}{171}\right)$$ $$e\left(\frac{59}{171}\right)$$ $$e\left(\frac{106}{171}\right)$$ $$e\left(\frac{16}{171}\right)$$ $$e\left(\frac{14}{171}\right)$$
$$\chi_{1444}(27,\cdot)$$ 1444.t 114 yes $$1$$ $$1$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{50}{57}\right)$$ $$e\left(\frac{15}{38}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{49}{57}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{43}{114}\right)$$ $$e\left(\frac{59}{114}\right)$$
$$\chi_{1444}(29,\cdot)$$ 1444.x 342 no $$-1$$ $$1$$ $$e\left(\frac{311}{342}\right)$$ $$e\left(\frac{91}{171}\right)$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{140}{171}\right)$$ $$e\left(\frac{4}{57}\right)$$ $$e\left(\frac{121}{342}\right)$$ $$e\left(\frac{151}{342}\right)$$ $$e\left(\frac{40}{171}\right)$$ $$e\left(\frac{125}{342}\right)$$ $$e\left(\frac{44}{171}\right)$$
$$\chi_{1444}(31,\cdot)$$ 1444.t 114 yes $$1$$ $$1$$ $$e\left(\frac{37}{57}\right)$$ $$e\left(\frac{31}{57}\right)$$ $$e\left(\frac{15}{38}\right)$$ $$e\left(\frac{17}{57}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{11}{57}\right)$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{5}{114}\right)$$ $$e\left(\frac{97}{114}\right)$$
$$\chi_{1444}(33,\cdot)$$ 1444.x 342 no $$-1$$ $$1$$ $$e\left(\frac{325}{342}\right)$$ $$e\left(\frac{83}{171}\right)$$ $$e\left(\frac{40}{57}\right)$$ $$e\left(\frac{154}{171}\right)$$ $$e\left(\frac{50}{57}\right)$$ $$e\left(\frac{287}{342}\right)$$ $$e\left(\frac{149}{342}\right)$$ $$e\left(\frac{44}{171}\right)$$ $$e\left(\frac{223}{342}\right)$$ $$e\left(\frac{151}{171}\right)$$
$$\chi_{1444}(35,\cdot)$$ 1444.v 342 yes $$-1$$ $$1$$ $$e\left(\frac{259}{342}\right)$$ $$e\left(\frac{23}{171}\right)$$ $$e\left(\frac{5}{114}\right)$$ $$e\left(\frac{88}{171}\right)$$ $$e\left(\frac{49}{114}\right)$$ $$e\left(\frac{82}{171}\right)$$ $$e\left(\frac{305}{342}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{137}{171}\right)$$ $$e\left(\frac{197}{342}\right)$$
$$\chi_{1444}(37,\cdot)$$ 1444.o 38 no $$-1$$ $$1$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{10}{19}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{11}{38}\right)$$ $$e\left(\frac{31}{38}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{1}{38}\right)$$ $$e\left(\frac{4}{19}\right)$$
$$\chi_{1444}(39,\cdot)$$ 1444.p 38 yes $$-1$$ $$1$$ $$e\left(\frac{27}{38}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{29}{38}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{3}{38}\right)$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{7}{38}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{11}{38}\right)$$
$$\chi_{1444}(41,\cdot)$$ 1444.x 342 no $$-1$$ $$1$$ $$e\left(\frac{79}{342}\right)$$ $$e\left(\frac{77}{171}\right)$$ $$e\left(\frac{22}{57}\right)$$ $$e\left(\frac{79}{171}\right)$$ $$e\left(\frac{56}{57}\right)$$ $$e\left(\frac{155}{342}\right)$$ $$e\left(\frac{233}{342}\right)$$ $$e\left(\frac{47}{171}\right)$$ $$e\left(\frac{211}{342}\right)$$ $$e\left(\frac{103}{171}\right)$$
$$\chi_{1444}(43,\cdot)$$ 1444.v 342 yes $$-1$$ $$1$$ $$e\left(\frac{217}{342}\right)$$ $$e\left(\frac{47}{171}\right)$$ $$e\left(\frac{35}{114}\right)$$ $$e\left(\frac{46}{171}\right)$$ $$e\left(\frac{1}{114}\right)$$ $$e\left(\frac{4}{171}\right)$$ $$e\left(\frac{311}{342}\right)$$ $$e\left(\frac{62}{171}\right)$$ $$e\left(\frac{161}{171}\right)$$ $$e\left(\frac{239}{342}\right)$$
$$\chi_{1444}(45,\cdot)$$ 1444.q 57 no $$1$$ $$1$$ $$e\left(\frac{16}{57}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{2}{19}\right)$$ $$e\left(\frac{35}{57}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{1}{57}\right)$$ $$e\left(\frac{55}{57}\right)$$ $$e\left(\frac{41}{57}\right)$$
$$\chi_{1444}(47,\cdot)$$ 1444.v 342 yes $$-1$$ $$1$$ $$e\left(\frac{23}{342}\right)$$ $$e\left(\frac{109}{171}\right)$$ $$e\left(\frac{103}{114}\right)$$ $$e\left(\frac{23}{171}\right)$$ $$e\left(\frac{29}{114}\right)$$ $$e\left(\frac{2}{171}\right)$$ $$e\left(\frac{241}{342}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{166}{171}\right)$$ $$e\left(\frac{205}{342}\right)$$
$$\chi_{1444}(49,\cdot)$$ 1444.q 57 no $$1$$ $$1$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{11}{19}\right)$$ $$e\left(\frac{49}{57}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{34}{57}\right)$$ $$e\left(\frac{25}{57}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{29}{57}\right)$$ $$e\left(\frac{4}{57}\right)$$
$$\chi_{1444}(51,\cdot)$$ 1444.w 342 yes $$1$$ $$1$$ $$e\left(\frac{145}{171}\right)$$ $$e\left(\frac{103}{171}\right)$$ $$e\left(\frac{67}{114}\right)$$ $$e\left(\frac{119}{171}\right)$$ $$e\left(\frac{41}{114}\right)$$ $$e\left(\frac{43}{342}\right)$$ $$e\left(\frac{77}{171}\right)$$ $$e\left(\frac{34}{171}\right)$$ $$e\left(\frac{149}{342}\right)$$ $$e\left(\frac{109}{342}\right)$$
$$\chi_{1444}(53,\cdot)$$ 1444.x 342 no $$-1$$ $$1$$ $$e\left(\frac{251}{342}\right)$$ $$e\left(\frac{52}{171}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{80}{171}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{289}{342}\right)$$ $$e\left(\frac{13}{342}\right)$$ $$e\left(\frac{145}{171}\right)$$ $$e\left(\frac{47}{342}\right)$$ $$e\left(\frac{74}{171}\right)$$
$$\chi_{1444}(55,\cdot)$$ 1444.v 342 yes $$-1$$ $$1$$ $$e\left(\frac{85}{342}\right)$$ $$e\left(\frac{98}{171}\right)$$ $$e\left(\frac{113}{114}\right)$$ $$e\left(\frac{85}{171}\right)$$ $$e\left(\frac{13}{114}\right)$$ $$e\left(\frac{52}{171}\right)$$ $$e\left(\frac{281}{342}\right)$$ $$e\left(\frac{122}{171}\right)$$ $$e\left(\frac{41}{171}\right)$$ $$e\left(\frac{29}{342}\right)$$
$$\chi_{1444}(59,\cdot)$$ 1444.w 342 yes $$1$$ $$1$$ $$e\left(\frac{11}{171}\right)$$ $$e\left(\frac{134}{171}\right)$$ $$e\left(\frac{101}{114}\right)$$ $$e\left(\frac{22}{171}\right)$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{41}{342}\right)$$ $$e\left(\frac{145}{171}\right)$$ $$e\left(\frac{104}{171}\right)$$ $$e\left(\frac{325}{342}\right)$$ $$e\left(\frac{263}{342}\right)$$
$$\chi_{1444}(61,\cdot)$$ 1444.u 171 no $$1$$ $$1$$ $$e\left(\frac{67}{171}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{20}{57}\right)$$ $$e\left(\frac{134}{171}\right)$$ $$e\left(\frac{25}{57}\right)$$ $$e\left(\frac{86}{171}\right)$$ $$e\left(\frac{137}{171}\right)$$ $$e\left(\frac{136}{171}\right)$$ $$e\left(\frac{127}{171}\right)$$ $$e\left(\frac{47}{171}\right)$$
$$\chi_{1444}(63,\cdot)$$ 1444.v 342 yes $$-1$$ $$1$$ $$e\left(\frac{155}{342}\right)$$ $$e\left(\frac{58}{171}\right)$$ $$e\left(\frac{25}{114}\right)$$ $$e\left(\frac{155}{171}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{125}{171}\right)$$ $$e\left(\frac{271}{342}\right)$$ $$e\left(\frac{142}{171}\right)$$ $$e\left(\frac{115}{171}\right)$$ $$e\left(\frac{73}{342}\right)$$
$$\chi_{1444}(65,\cdot)$$ 1444.r 114 no $$-1$$ $$1$$ $$e\left(\frac{1}{114}\right)$$ $$e\left(\frac{32}{57}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{1}{57}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{77}{114}\right)$$ $$e\left(\frac{65}{114}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{7}{114}\right)$$ $$e\left(\frac{28}{57}\right)$$
$$\chi_{1444}(67,\cdot)$$ 1444.w 342 yes $$1$$ $$1$$ $$e\left(\frac{142}{171}\right)$$ $$e\left(\frac{82}{171}\right)$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{113}{171}\right)$$ $$e\left(\frac{83}{114}\right)$$ $$e\left(\frac{265}{342}\right)$$ $$e\left(\frac{53}{171}\right)$$ $$e\left(\frac{130}{171}\right)$$ $$e\left(\frac{107}{342}\right)$$ $$e\left(\frac{115}{342}\right)$$