# Properties

 Modulus $51984$ Structure $$C_{2}\times C_{2}\times C_{6}\times C_{684}$$ Order $16416$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(51984)

pari: g = idealstar(,51984,2)

## Character group

 sage: G.order()  pari: g.no Order = 16416 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{6}\times C_{684}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{51984}(45487,\cdot)$, $\chi_{51984}(12997,\cdot)$, $\chi_{51984}(23105,\cdot)$, $\chi_{51984}(25633,\cdot)$

## First 32 of 16416 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$
$$\chi_{51984}(1,\cdot)$$ 51984.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{51984}(5,\cdot)$$ 51984.ob 684 yes $$-1$$ $$1$$ $$e\left(\frac{545}{684}\right)$$ $$e\left(\frac{67}{114}\right)$$ $$e\left(\frac{21}{76}\right)$$ $$e\left(\frac{409}{684}\right)$$ $$e\left(\frac{277}{342}\right)$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{203}{342}\right)$$ $$e\left(\frac{79}{684}\right)$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{263}{684}\right)$$
$$\chi_{51984}(7,\cdot)$$ 51984.je 114 no $$-1$$ $$1$$ $$e\left(\frac{67}{114}\right)$$ $$e\left(\frac{109}{114}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{5}{38}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{10}{57}\right)$$ $$e\left(\frac{71}{114}\right)$$ $$e\left(\frac{83}{114}\right)$$ $$e\left(\frac{31}{57}\right)$$
$$\chi_{51984}(11,\cdot)$$ 51984.lp 228 yes $$1$$ $$1$$ $$e\left(\frac{21}{76}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{77}{228}\right)$$ $$e\left(\frac{47}{228}\right)$$ $$e\left(\frac{103}{114}\right)$$ $$e\left(\frac{43}{114}\right)$$ $$e\left(\frac{21}{38}\right)$$ $$e\left(\frac{75}{76}\right)$$ $$e\left(\frac{23}{114}\right)$$ $$e\left(\frac{155}{228}\right)$$
$$\chi_{51984}(13,\cdot)$$ 51984.oq 684 yes $$-1$$ $$1$$ $$e\left(\frac{409}{684}\right)$$ $$e\left(\frac{5}{38}\right)$$ $$e\left(\frac{47}{228}\right)$$ $$e\left(\frac{281}{684}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{211}{342}\right)$$ $$e\left(\frac{67}{342}\right)$$ $$e\left(\frac{641}{684}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{499}{684}\right)$$
$$\chi_{51984}(17,\cdot)$$ 51984.nf 342 no $$-1$$ $$1$$ $$e\left(\frac{277}{342}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{103}{114}\right)$$ $$e\left(\frac{70}{171}\right)$$ $$e\left(\frac{289}{342}\right)$$ $$e\left(\frac{335}{342}\right)$$ $$e\left(\frac{106}{171}\right)$$ $$e\left(\frac{251}{342}\right)$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{319}{342}\right)$$
$$\chi_{51984}(23,\cdot)$$ 51984.np 342 no $$1$$ $$1$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{43}{114}\right)$$ $$e\left(\frac{211}{342}\right)$$ $$e\left(\frac{335}{342}\right)$$ $$e\left(\frac{170}{171}\right)$$ $$e\left(\frac{71}{171}\right)$$ $$e\left(\frac{101}{171}\right)$$ $$e\left(\frac{59}{114}\right)$$ $$e\left(\frac{197}{342}\right)$$
$$\chi_{51984}(25,\cdot)$$ 51984.nl 342 no $$1$$ $$1$$ $$e\left(\frac{203}{342}\right)$$ $$e\left(\frac{10}{57}\right)$$ $$e\left(\frac{21}{38}\right)$$ $$e\left(\frac{67}{342}\right)$$ $$e\left(\frac{106}{171}\right)$$ $$e\left(\frac{71}{171}\right)$$ $$e\left(\frac{32}{171}\right)$$ $$e\left(\frac{79}{342}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{263}{342}\right)$$
$$\chi_{51984}(29,\cdot)$$ 51984.og 684 yes $$1$$ $$1$$ $$e\left(\frac{79}{684}\right)$$ $$e\left(\frac{71}{114}\right)$$ $$e\left(\frac{75}{76}\right)$$ $$e\left(\frac{641}{684}\right)$$ $$e\left(\frac{251}{342}\right)$$ $$e\left(\frac{101}{171}\right)$$ $$e\left(\frac{79}{342}\right)$$ $$e\left(\frac{179}{684}\right)$$ $$e\left(\frac{15}{38}\right)$$ $$e\left(\frac{505}{684}\right)$$
$$\chi_{51984}(31,\cdot)$$ 51984.in 114 no $$1$$ $$1$$ $$e\left(\frac{4}{19}\right)$$ $$e\left(\frac{83}{114}\right)$$ $$e\left(\frac{23}{114}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{59}{114}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{15}{38}\right)$$ $$e\left(\frac{35}{57}\right)$$ $$e\left(\frac{107}{114}\right)$$
$$\chi_{51984}(35,\cdot)$$ 51984.ok 684 no $$1$$ $$1$$ $$e\left(\frac{263}{684}\right)$$ $$e\left(\frac{31}{57}\right)$$ $$e\left(\frac{155}{228}\right)$$ $$e\left(\frac{499}{684}\right)$$ $$e\left(\frac{319}{342}\right)$$ $$e\left(\frac{197}{342}\right)$$ $$e\left(\frac{263}{342}\right)$$ $$e\left(\frac{505}{684}\right)$$ $$e\left(\frac{107}{114}\right)$$ $$e\left(\frac{635}{684}\right)$$
$$\chi_{51984}(37,\cdot)$$ 51984.ij 76 no $$-1$$ $$1$$ $$e\left(\frac{59}{76}\right)$$ $$e\left(\frac{9}{38}\right)$$ $$e\left(\frac{51}{76}\right)$$ $$e\left(\frac{3}{76}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{27}{38}\right)$$ $$e\left(\frac{21}{38}\right)$$ $$e\left(\frac{75}{76}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{1}{76}\right)$$
$$\chi_{51984}(41,\cdot)$$ 51984.nb 342 no $$1$$ $$1$$ $$e\left(\frac{20}{171}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{106}{171}\right)$$ $$e\left(\frac{265}{342}\right)$$ $$e\left(\frac{263}{342}\right)$$ $$e\left(\frac{40}{171}\right)$$ $$e\left(\frac{227}{342}\right)$$ $$e\left(\frac{1}{38}\right)$$ $$e\left(\frac{143}{171}\right)$$
$$\chi_{51984}(43,\cdot)$$ 51984.nx 684 yes $$-1$$ $$1$$ $$e\left(\frac{587}{684}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{211}{228}\right)$$ $$e\left(\frac{73}{684}\right)$$ $$e\left(\frac{62}{171}\right)$$ $$e\left(\frac{91}{171}\right)$$ $$e\left(\frac{245}{342}\right)$$ $$e\left(\frac{1}{684}\right)$$ $$e\left(\frac{103}{114}\right)$$ $$e\left(\frac{227}{684}\right)$$
$$\chi_{51984}(47,\cdot)$$ 51984.mm 342 no $$1$$ $$1$$ $$e\left(\frac{161}{342}\right)$$ $$e\left(\frac{65}{114}\right)$$ $$e\left(\frac{8}{19}\right)$$ $$e\left(\frac{59}{171}\right)$$ $$e\left(\frac{233}{342}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{161}{171}\right)$$ $$e\left(\frac{157}{342}\right)$$ $$e\left(\frac{33}{38}\right)$$ $$e\left(\frac{7}{171}\right)$$
$$\chi_{51984}(49,\cdot)$$ 51984.ic 57 no $$1$$ $$1$$ $$e\left(\frac{10}{57}\right)$$ $$e\left(\frac{52}{57}\right)$$ $$e\left(\frac{46}{57}\right)$$ $$e\left(\frac{5}{19}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{14}{19}\right)$$ $$e\left(\frac{20}{57}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{26}{57}\right)$$ $$e\left(\frac{5}{57}\right)$$
$$\chi_{51984}(53,\cdot)$$ 51984.od 684 no $$1$$ $$1$$ $$e\left(\frac{37}{684}\right)$$ $$e\left(\frac{103}{114}\right)$$ $$e\left(\frac{115}{228}\right)$$ $$e\left(\frac{407}{684}\right)$$ $$e\left(\frac{119}{342}\right)$$ $$e\left(\frac{74}{171}\right)$$ $$e\left(\frac{37}{342}\right)$$ $$e\left(\frac{257}{684}\right)$$ $$e\left(\frac{61}{114}\right)$$ $$e\left(\frac{655}{684}\right)$$
$$\chi_{51984}(55,\cdot)$$ 51984.mt 342 no $$-1$$ $$1$$ $$e\left(\frac{25}{342}\right)$$ $$e\left(\frac{113}{114}\right)$$ $$e\left(\frac{35}{57}\right)$$ $$e\left(\frac{275}{342}\right)$$ $$e\left(\frac{122}{171}\right)$$ $$e\left(\frac{29}{342}\right)$$ $$e\left(\frac{25}{171}\right)$$ $$e\left(\frac{35}{342}\right)$$ $$e\left(\frac{47}{114}\right)$$ $$e\left(\frac{11}{171}\right)$$
$$\chi_{51984}(59,\cdot)$$ 51984.oi 684 yes $$-1$$ $$1$$ $$e\left(\frac{137}{684}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{43}{76}\right)$$ $$e\left(\frac{367}{684}\right)$$ $$e\left(\frac{37}{342}\right)$$ $$e\left(\frac{149}{342}\right)$$ $$e\left(\frac{137}{342}\right)$$ $$e\left(\frac{397}{684}\right)$$ $$e\left(\frac{10}{19}\right)$$ $$e\left(\frac{629}{684}\right)$$
$$\chi_{51984}(61,\cdot)$$ 51984.oa 684 yes $$1$$ $$1$$ $$e\left(\frac{337}{684}\right)$$ $$e\left(\frac{59}{114}\right)$$ $$e\left(\frac{65}{76}\right)$$ $$e\left(\frac{59}{684}\right)$$ $$e\left(\frac{136}{171}\right)$$ $$e\left(\frac{37}{342}\right)$$ $$e\left(\frac{337}{342}\right)$$ $$e\left(\frac{335}{684}\right)$$ $$e\left(\frac{16}{19}\right)$$ $$e\left(\frac{7}{684}\right)$$
$$\chi_{51984}(65,\cdot)$$ 51984.kq 114 no $$1$$ $$1$$ $$e\left(\frac{15}{38}\right)$$ $$e\left(\frac{41}{57}\right)$$ $$e\left(\frac{55}{114}\right)$$ $$e\left(\frac{1}{114}\right)$$ $$e\left(\frac{25}{114}\right)$$ $$e\left(\frac{37}{114}\right)$$ $$e\left(\frac{15}{19}\right)$$ $$e\left(\frac{1}{19}\right)$$ $$e\left(\frac{41}{114}\right)$$ $$e\left(\frac{13}{114}\right)$$
$$\chi_{51984}(67,\cdot)$$ 51984.ny 684 yes $$1$$ $$1$$ $$e\left(\frac{613}{684}\right)$$ $$e\left(\frac{6}{19}\right)$$ $$e\left(\frac{185}{228}\right)$$ $$e\left(\frac{473}{684}\right)$$ $$e\left(\frac{130}{171}\right)$$ $$e\left(\frac{86}{171}\right)$$ $$e\left(\frac{271}{342}\right)$$ $$e\left(\frac{653}{684}\right)$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{145}{684}\right)$$
$$\chi_{51984}(71,\cdot)$$ 51984.mq 342 no $$-1$$ $$1$$ $$e\left(\frac{101}{171}\right)$$ $$e\left(\frac{17}{114}\right)$$ $$e\left(\frac{7}{114}\right)$$ $$e\left(\frac{85}{171}\right)$$ $$e\left(\frac{241}{342}\right)$$ $$e\left(\frac{124}{171}\right)$$ $$e\left(\frac{31}{171}\right)$$ $$e\left(\frac{317}{342}\right)$$ $$e\left(\frac{28}{57}\right)$$ $$e\left(\frac{253}{342}\right)$$
$$\chi_{51984}(73,\cdot)$$ 51984.ng 342 no $$1$$ $$1$$ $$e\left(\frac{163}{342}\right)$$ $$e\left(\frac{7}{57}\right)$$ $$e\left(\frac{103}{114}\right)$$ $$e\left(\frac{83}{342}\right)$$ $$e\left(\frac{2}{171}\right)$$ $$e\left(\frac{139}{171}\right)$$ $$e\left(\frac{163}{171}\right)$$ $$e\left(\frac{23}{342}\right)$$ $$e\left(\frac{13}{57}\right)$$ $$e\left(\frac{205}{342}\right)$$
$$\chi_{51984}(77,\cdot)$$ 51984.lm 228 yes $$-1$$ $$1$$ $$e\left(\frac{197}{228}\right)$$ $$e\left(\frac{41}{114}\right)$$ $$e\left(\frac{169}{228}\right)$$ $$e\left(\frac{77}{228}\right)$$ $$e\left(\frac{1}{38}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{83}{114}\right)$$ $$e\left(\frac{139}{228}\right)$$ $$e\left(\frac{53}{57}\right)$$ $$e\left(\frac{17}{76}\right)$$
$$\chi_{51984}(79,\cdot)$$ 51984.nr 342 no $$1$$ $$1$$ $$e\left(\frac{161}{171}\right)$$ $$e\left(\frac{37}{38}\right)$$ $$e\left(\frac{77}{114}\right)$$ $$e\left(\frac{179}{342}\right)$$ $$e\left(\frac{119}{171}\right)$$ $$e\left(\frac{239}{342}\right)$$ $$e\left(\frac{151}{171}\right)$$ $$e\left(\frac{143}{342}\right)$$ $$e\left(\frac{23}{57}\right)$$ $$e\left(\frac{313}{342}\right)$$
$$\chi_{51984}(83,\cdot)$$ 51984.lr 228 yes $$1$$ $$1$$ $$e\left(\frac{173}{228}\right)$$ $$e\left(\frac{14}{57}\right)$$ $$e\left(\frac{203}{228}\right)$$ $$e\left(\frac{39}{76}\right)$$ $$e\left(\frac{47}{114}\right)$$ $$e\left(\frac{9}{38}\right)$$ $$e\left(\frac{59}{114}\right)$$ $$e\left(\frac{151}{228}\right)$$ $$e\left(\frac{71}{114}\right)$$ $$e\left(\frac{1}{228}\right)$$
$$\chi_{51984}(85,\cdot)$$ 51984.ot 684 yes $$1$$ $$1$$ $$e\left(\frac{415}{684}\right)$$ $$e\left(\frac{27}{38}\right)$$ $$e\left(\frac{41}{228}\right)$$ $$e\left(\frac{5}{684}\right)$$ $$e\left(\frac{112}{171}\right)$$ $$e\left(\frac{235}{342}\right)$$ $$e\left(\frac{73}{342}\right)$$ $$e\left(\frac{581}{684}\right)$$ $$e\left(\frac{25}{57}\right)$$ $$e\left(\frac{217}{684}\right)$$
$$\chi_{51984}(89,\cdot)$$ 51984.nh 342 no $$1$$ $$1$$ $$e\left(\frac{4}{171}\right)$$ $$e\left(\frac{50}{57}\right)$$ $$e\left(\frac{34}{57}\right)$$ $$e\left(\frac{44}{171}\right)$$ $$e\left(\frac{167}{342}\right)$$ $$e\left(\frac{235}{342}\right)$$ $$e\left(\frac{8}{171}\right)$$ $$e\left(\frac{319}{342}\right)$$ $$e\left(\frac{31}{114}\right)$$ $$e\left(\frac{154}{171}\right)$$
$$\chi_{51984}(91,\cdot)$$ 51984.om 684 no $$1$$ $$1$$ $$e\left(\frac{127}{684}\right)$$ $$e\left(\frac{5}{57}\right)$$ $$e\left(\frac{139}{228}\right)$$ $$e\left(\frac{371}{684}\right)$$ $$e\left(\frac{91}{171}\right)$$ $$e\left(\frac{83}{171}\right)$$ $$e\left(\frac{127}{342}\right)$$ $$e\left(\frac{383}{684}\right)$$ $$e\left(\frac{50}{57}\right)$$ $$e\left(\frac{187}{684}\right)$$
$$\chi_{51984}(97,\cdot)$$ 51984.mi 342 no $$-1$$ $$1$$ $$e\left(\frac{146}{171}\right)$$ $$e\left(\frac{13}{19}\right)$$ $$e\left(\frac{25}{57}\right)$$ $$e\left(\frac{191}{342}\right)$$ $$e\left(\frac{41}{171}\right)$$ $$e\left(\frac{85}{171}\right)$$ $$e\left(\frac{121}{171}\right)$$ $$e\left(\frac{101}{342}\right)$$ $$e\left(\frac{1}{114}\right)$$ $$e\left(\frac{92}{171}\right)$$
$$\chi_{51984}(101,\cdot)$$ 51984.ob 684 yes $$-1$$ $$1$$ $$e\left(\frac{1}{684}\right)$$ $$e\left(\frac{11}{114}\right)$$ $$e\left(\frac{25}{76}\right)$$ $$e\left(\frac{125}{684}\right)$$ $$e\left(\frac{185}{342}\right)$$ $$e\left(\frac{116}{171}\right)$$ $$e\left(\frac{1}{342}\right)$$ $$e\left(\frac{275}{684}\right)$$ $$e\left(\frac{12}{19}\right)$$ $$e\left(\frac{67}{684}\right)$$