# Properties

 Modulus $208$ Structure $$C_{2}\times C_{4}\times C_{12}$$ Order $96$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(208)

pari: g = idealstar(,208,2)

## Character group

 sage: G.order()  pari: g.no Order = 96 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{4}\times C_{12}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{208}(79,\cdot)$, $\chi_{208}(53,\cdot)$, $\chi_{208}(145,\cdot)$

## First 32 of 96 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$15$$ $$17$$ $$19$$ $$21$$ $$23$$
$$\chi_{208}(1,\cdot)$$ 208.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{208}(3,\cdot)$$ 208.bg 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{208}(5,\cdot)$$ 208.r 4 yes $$-1$$ $$1$$ $$-i$$ $$1$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$-1$$ $$1$$
$$\chi_{208}(7,\cdot)$$ 208.bc 12 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{208}(9,\cdot)$$ 208.z 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{208}(11,\cdot)$$ 208.bf 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$-1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{208}(15,\cdot)$$ 208.bm 12 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{208}(17,\cdot)$$ 208.w 6 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{208}(19,\cdot)$$ 208.bf 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{208}(21,\cdot)$$ 208.m 4 yes $$-1$$ $$1$$ $$-i$$ $$-1$$ $$i$$ $$-1$$ $$1$$ $$i$$ $$-1$$ $$1$$ $$1$$ $$1$$
$$\chi_{208}(23,\cdot)$$ 208.x 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{208}(25,\cdot)$$ 208.e 2 no $$1$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{208}(27,\cdot)$$ 208.q 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$i$$ $$1$$
$$\chi_{208}(29,\cdot)$$ 208.bj 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{208}(31,\cdot)$$ 208.k 4 no $$1$$ $$1$$ $$-1$$ $$-i$$ $$-i$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$i$$ $$i$$ $$1$$
$$\chi_{208}(33,\cdot)$$ 208.bd 12 no $$-1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{208}(35,\cdot)$$ 208.bg 12 yes $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{208}(37,\cdot)$$ 208.be 12 yes $$-1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{208}(41,\cdot)$$ 208.bn 12 no $$-1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{208}(43,\cdot)$$ 208.bi 12 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{208}(45,\cdot)$$ 208.be 12 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{208}(47,\cdot)$$ 208.k 4 no $$1$$ $$1$$ $$-1$$ $$i$$ $$i$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-i$$ $$-i$$ $$1$$
$$\chi_{208}(49,\cdot)$$ 208.w 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-1$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{208}(51,\cdot)$$ 208.o 4 yes $$-1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-1$$ $$-i$$ $$1$$ $$1$$ $$i$$ $$i$$ $$1$$
$$\chi_{208}(53,\cdot)$$ 208.n 4 no $$1$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$1$$ $$1$$ $$-i$$ $$i$$ $$-1$$
$$\chi_{208}(55,\cdot)$$ 208.v 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{208}(57,\cdot)$$ 208.j 4 no $$-1$$ $$1$$ $$-1$$ $$i$$ $$i$$ $$1$$ $$-i$$ $$-i$$ $$-1$$ $$i$$ $$-i$$ $$-1$$
$$\chi_{208}(59,\cdot)$$ 208.bf 12 yes $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{208}(61,\cdot)$$ 208.bj 12 yes $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$-i$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{208}(63,\cdot)$$ 208.bm 12 no $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{208}(67,\cdot)$$ 208.bf 12 yes $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{208}(69,\cdot)$$ 208.bh 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{6}\right)$$