# Properties

 Modulus $192$ Structure $$C_{2}\times C_{2}\times C_{16}$$ Order $64$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(192)

pari: g = idealstar(,192,2)

## Character group

 sage: G.order()  pari: g.no Order = 64 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{16}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{192}(127,\cdot)$, $\chi_{192}(133,\cdot)$, $\chi_{192}(65,\cdot)$

## First 32 of 64 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{192}(1,\cdot)$$ 192.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{192}(5,\cdot)$$ 192.q 16 yes $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{192}(7,\cdot)$$ 192.m 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$
$$\chi_{192}(11,\cdot)$$ 192.s 16 yes $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$1$$
$$\chi_{192}(13,\cdot)$$ 192.r 16 no $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-1$$
$$\chi_{192}(17,\cdot)$$ 192.i 4 no $$-1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{192}(19,\cdot)$$ 192.t 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$1$$
$$\chi_{192}(23,\cdot)$$ 192.o 8 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$
$$\chi_{192}(25,\cdot)$$ 192.n 8 no $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$1$$
$$\chi_{192}(29,\cdot)$$ 192.q 16 yes $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-1$$
$$\chi_{192}(31,\cdot)$$ 192.b 2 no $$-1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{192}(35,\cdot)$$ 192.s 16 yes $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$1$$
$$\chi_{192}(37,\cdot)$$ 192.r 16 no $$1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-1$$
$$\chi_{192}(41,\cdot)$$ 192.p 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{192}(43,\cdot)$$ 192.t 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$1$$
$$\chi_{192}(47,\cdot)$$ 192.k 4 no $$1$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$
$$\chi_{192}(49,\cdot)$$ 192.j 4 no $$1$$ $$1$$ $$i$$ $$-1$$ $$i$$ $$-i$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$1$$
$$\chi_{192}(53,\cdot)$$ 192.q 16 yes $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{192}(55,\cdot)$$ 192.m 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$
$$\chi_{192}(59,\cdot)$$ 192.s 16 yes $$1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$1$$
$$\chi_{192}(61,\cdot)$$ 192.r 16 no $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-1$$
$$\chi_{192}(65,\cdot)$$ 192.e 2 no $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$
$$\chi_{192}(67,\cdot)$$ 192.t 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$1$$
$$\chi_{192}(71,\cdot)$$ 192.o 8 no $$1$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$
$$\chi_{192}(73,\cdot)$$ 192.n 8 no $$1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{192}(77,\cdot)$$ 192.q 16 yes $$-1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-1$$
$$\chi_{192}(79,\cdot)$$ 192.l 4 no $$-1$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-1$$
$$\chi_{192}(83,\cdot)$$ 192.s 16 yes $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$1$$
$$\chi_{192}(85,\cdot)$$ 192.r 16 no $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-1$$
$$\chi_{192}(89,\cdot)$$ 192.p 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{192}(91,\cdot)$$ 192.t 16 no $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$1$$
$$\chi_{192}(95,\cdot)$$ 192.f 2 no $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$