# Properties

 Modulus $720$ Structure $$C_{2}\times C_{2}\times C_{4}\times C_{12}$$ Order $192$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(720)

pari: g = idealstar(,720,2)

## Character group

 sage: G.order()  pari: g.no Order = 192 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{4}\times C_{12}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{720}(271,\cdot)$, $\chi_{720}(181,\cdot)$, $\chi_{720}(641,\cdot)$, $\chi_{720}(577,\cdot)$

## First 32 of 192 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 $$-1$$ $$1$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$
$$\chi_{720}(1,\cdot)$$ 720.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{720}(7,\cdot)$$ 720.ci 12 no $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(11,\cdot)$$ 720.cf 12 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(13,\cdot)$$ 720.cr 12 yes $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{720}(17,\cdot)$$ 720.w 4 no $$1$$ $$1$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-1$$
$$\chi_{720}(19,\cdot)$$ 720.r 4 no $$-1$$ $$1$$ $$-1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-1$$ $$i$$ $$-1$$ $$i$$ $$-1$$
$$\chi_{720}(23,\cdot)$$ 720.cv 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{720}(29,\cdot)$$ 720.ch 12 yes $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$1$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(31,\cdot)$$ 720.cb 6 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{720}(37,\cdot)$$ 720.bb 4 no $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{720}(41,\cdot)$$ 720.bq 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{720}(43,\cdot)$$ 720.cp 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{720}(47,\cdot)$$ 720.ck 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{720}(49,\cdot)$$ 720.by 6 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{720}(53,\cdot)$$ 720.bg 4 no $$1$$ $$1$$ $$i$$ $$-i$$ $$1$$ $$i$$ $$i$$ $$i$$ $$-i$$ $$1$$ $$1$$ $$1$$
$$\chi_{720}(59,\cdot)$$ 720.da 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{720}(61,\cdot)$$ 720.db 12 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{720}(67,\cdot)$$ 720.cp 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{720}(71,\cdot)$$ 720.b 2 no $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$
$$\chi_{720}(73,\cdot)$$ 720.y 4 no $$-1$$ $$1$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$1$$ $$i$$ $$1$$
$$\chi_{720}(77,\cdot)$$ 720.cm 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{720}(79,\cdot)$$ 720.bu 6 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(83,\cdot)$$ 720.cs 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(89,\cdot)$$ 720.i 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$
$$\chi_{720}(91,\cdot)$$ 720.bo 4 no $$-1$$ $$1$$ $$1$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$-1$$ $$i$$ $$-1$$
$$\chi_{720}(97,\cdot)$$ 720.cj 12 no $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(101,\cdot)$$ 720.cy 12 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{720}(103,\cdot)$$ 720.ci 12 no $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{720}(107,\cdot)$$ 720.ba 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-i$$ $$-i$$ $$i$$ $$-i$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{720}(109,\cdot)$$ 720.bm 4 no $$1$$ $$1$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$1$$ $$i$$ $$1$$ $$i$$ $$-1$$
$$\chi_{720}(113,\cdot)$$ 720.cu 12 no $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{720}(119,\cdot)$$ 720.bt 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$