# Properties

 Modulus $1728$ Structure $$C_{2}\times C_{2}\times C_{144}$$ Order $576$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(1728)

pari: g = idealstar(,1728,2)

## Character group

 sage: G.order()  pari: g.no Order = 576 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{144}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1728}(703,\cdot)$, $\chi_{1728}(325,\cdot)$, $\chi_{1728}(1217,\cdot)$

## First 32 of 576 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{1728}(1,\cdot)$$ 1728.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1728}(5,\cdot)$$ 1728.cf 144 yes $$-1$$ $$1$$ $$e\left(\frac{65}{144}\right)$$ $$e\left(\frac{5}{72}\right)$$ $$e\left(\frac{133}{144}\right)$$ $$e\left(\frac{23}{144}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{139}{144}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{1728}(7,\cdot)$$ 1728.cd 72 no $$-1$$ $$1$$ $$e\left(\frac{5}{72}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{55}{72}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{1728}(11,\cdot)$$ 1728.ch 144 yes $$1$$ $$1$$ $$e\left(\frac{133}{144}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{65}{144}\right)$$ $$e\left(\frac{67}{144}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{61}{72}\right)$$ $$e\left(\frac{23}{144}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{1728}(13,\cdot)$$ 1728.ce 144 yes $$1$$ $$1$$ $$e\left(\frac{23}{144}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$e\left(\frac{67}{144}\right)$$ $$e\left(\frac{89}{144}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{23}{72}\right)$$ $$e\left(\frac{109}{144}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{1728}(17,\cdot)$$ 1728.bb 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1728}(19,\cdot)$$ 1728.bw 48 no $$-1$$ $$1$$ $$e\left(\frac{37}{48}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{17}{48}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1728}(23,\cdot)$$ 1728.cb 72 no $$1$$ $$1$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{1728}(25,\cdot)$$ 1728.cc 72 no $$1$$ $$1$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{61}{72}\right)$$ $$e\left(\frac{23}{72}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{1728}(29,\cdot)$$ 1728.cf 144 yes $$-1$$ $$1$$ $$e\left(\frac{139}{144}\right)$$ $$e\left(\frac{55}{72}\right)$$ $$e\left(\frac{23}{144}\right)$$ $$e\left(\frac{109}{144}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{48}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{89}{144}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{1728}(31,\cdot)$$ 1728.bh 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{1728}(35,\cdot)$$ 1728.bx 48 no $$1$$ $$1$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{35}{48}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1728}(37,\cdot)$$ 1728.by 48 no $$1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(41,\cdot)$$ 1728.ca 72 no $$-1$$ $$1$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{49}{72}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{1728}(43,\cdot)$$ 1728.cg 144 yes $$-1$$ $$1$$ $$e\left(\frac{133}{144}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{65}{144}\right)$$ $$e\left(\frac{139}{144}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{41}{48}\right)$$ $$e\left(\frac{23}{72}\right)$$ $$e\left(\frac{61}{72}\right)$$ $$e\left(\frac{23}{144}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{1728}(47,\cdot)$$ 1728.bv 36 no $$1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{1728}(49,\cdot)$$ 1728.bs 36 no $$1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{1728}(53,\cdot)$$ 1728.bd 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{1728}(55,\cdot)$$ 1728.u 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$
$$\chi_{1728}(59,\cdot)$$ 1728.ch 144 yes $$1$$ $$1$$ $$e\left(\frac{65}{144}\right)$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{61}{144}\right)$$ $$e\left(\frac{23}{144}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{48}\right)$$ $$e\left(\frac{31}{72}\right)$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{139}{144}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{1728}(61,\cdot)$$ 1728.ce 144 yes $$1$$ $$1$$ $$e\left(\frac{91}{144}\right)$$ $$e\left(\frac{7}{72}\right)$$ $$e\left(\frac{71}{144}\right)$$ $$e\left(\frac{133}{144}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{47}{48}\right)$$ $$e\left(\frac{29}{72}\right)$$ $$e\left(\frac{19}{72}\right)$$ $$e\left(\frac{137}{144}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{1728}(65,\cdot)$$ 1728.bk 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{1728}(67,\cdot)$$ 1728.cg 144 yes $$-1$$ $$1$$ $$e\left(\frac{59}{144}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$e\left(\frac{31}{144}\right)$$ $$e\left(\frac{53}{144}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{48}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{73}{144}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{1728}(71,\cdot)$$ 1728.br 24 no $$1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{1728}(73,\cdot)$$ 1728.bo 24 no $$1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1728}(77,\cdot)$$ 1728.cf 144 yes $$-1$$ $$1$$ $$e\left(\frac{143}{144}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$e\left(\frac{91}{144}\right)$$ $$e\left(\frac{137}{144}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{43}{48}\right)$$ $$e\left(\frac{61}{72}\right)$$ $$e\left(\frac{71}{72}\right)$$ $$e\left(\frac{133}{144}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{1728}(79,\cdot)$$ 1728.bt 36 no $$-1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{1728}(83,\cdot)$$ 1728.ch 144 yes $$1$$ $$1$$ $$e\left(\frac{103}{144}\right)$$ $$e\left(\frac{55}{72}\right)$$ $$e\left(\frac{59}{144}\right)$$ $$e\left(\frac{1}{144}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{31}{72}\right)$$ $$e\left(\frac{125}{144}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{1728}(85,\cdot)$$ 1728.ce 144 yes $$1$$ $$1$$ $$e\left(\frac{53}{144}\right)$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{73}{144}\right)$$ $$e\left(\frac{11}{144}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{48}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{53}{72}\right)$$ $$e\left(\frac{7}{144}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{1728}(89,\cdot)$$ 1728.bq 24 no $$-1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{1728}(91,\cdot)$$ 1728.bw 48 no $$-1$$ $$1$$ $$e\left(\frac{11}{48}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{31}{48}\right)$$ $$e\left(\frac{5}{48}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{25}{48}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{1728}(95,\cdot)$$ 1728.bl 18 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$