# Properties

 Modulus $608$ Structure $$C_{2}\times C_{2}\times C_{72}$$ Order $288$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(608)

pari: g = idealstar(,608,2)

## Character group

 sage: G.order()  pari: g.no Order = 288 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{72}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{608}(191,\cdot)$, $\chi_{608}(229,\cdot)$, $\chi_{608}(97,\cdot)$

## First 32 of 288 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$21$$ $$23$$
$$\chi_{608}(1,\cdot)$$ 608.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{608}(3,\cdot)$$ 608.bt 72 yes $$1$$ $$1$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{7}{36}\right)$$
$$\chi_{608}(5,\cdot)$$ 608.bs 72 yes $$1$$ $$1$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{25}{72}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{23}{72}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{37}{72}\right)$$ $$e\left(\frac{19}{36}\right)$$
$$\chi_{608}(7,\cdot)$$ 608.bc 12 no $$-1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{608}(9,\cdot)$$ 608.bq 36 no $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{608}(11,\cdot)$$ 608.bk 24 yes $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{608}(13,\cdot)$$ 608.bv 72 yes $$-1$$ $$1$$ $$e\left(\frac{17}{72}\right)$$ $$e\left(\frac{23}{72}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{37}{72}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{29}{36}\right)$$
$$\chi_{608}(15,\cdot)$$ 608.bh 18 no $$1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{608}(17,\cdot)$$ 608.bf 18 no $$1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{608}(21,\cdot)$$ 608.bv 72 yes $$-1$$ $$1$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{37}{72}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{31}{36}\right)$$
$$\chi_{608}(23,\cdot)$$ 608.bp 36 no $$-1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{608}(25,\cdot)$$ 608.bq 36 no $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{608}(27,\cdot)$$ 608.bn 24 yes $$1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{7}{12}\right)$$
$$\chi_{608}(29,\cdot)$$ 608.bv 72 yes $$-1$$ $$1$$ $$e\left(\frac{29}{72}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{25}{72}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{5}{36}\right)$$
$$\chi_{608}(31,\cdot)$$ 608.n 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{608}(33,\cdot)$$ 608.bd 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{608}(35,\cdot)$$ 608.bu 72 yes $$-1$$ $$1$$ $$e\left(\frac{37}{72}\right)$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{53}{72}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{72}\right)$$ $$e\left(\frac{7}{36}\right)$$
$$\chi_{608}(37,\cdot)$$ 608.w 8 yes $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$
$$\chi_{608}(39,\cdot)$$ 608.l 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$1$$
$$\chi_{608}(41,\cdot)$$ 608.br 36 no $$-1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$
$$\chi_{608}(43,\cdot)$$ 608.bu 72 yes $$-1$$ $$1$$ $$e\left(\frac{67}{72}\right)$$ $$e\left(\frac{61}{72}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{1}{36}\right)$$
$$\chi_{608}(45,\cdot)$$ 608.bm 24 yes $$1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{11}{12}\right)$$
$$\chi_{608}(47,\cdot)$$ 608.bg 18 no $$-1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{608}(49,\cdot)$$ 608.t 6 no $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{608}(51,\cdot)$$ 608.bt 72 yes $$1$$ $$1$$ $$e\left(\frac{53}{72}\right)$$ $$e\left(\frac{23}{72}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{37}{72}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{47}{72}\right)$$ $$e\left(\frac{11}{36}\right)$$
$$\chi_{608}(53,\cdot)$$ 608.bv 72 yes $$-1$$ $$1$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{29}{72}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{31}{72}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{53}{72}\right)$$ $$e\left(\frac{35}{36}\right)$$
$$\chi_{608}(55,\cdot)$$ 608.bp 36 no $$-1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{608}(59,\cdot)$$ 608.bt 72 yes $$1$$ $$1$$ $$e\left(\frac{43}{72}\right)$$ $$e\left(\frac{1}{72}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{49}{72}\right)$$ $$e\left(\frac{13}{36}\right)$$
$$\chi_{608}(61,\cdot)$$ 608.bs 72 yes $$1$$ $$1$$ $$e\left(\frac{41}{72}\right)$$ $$e\left(\frac{11}{72}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{13}{72}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{71}{72}\right)$$ $$e\left(\frac{17}{36}\right)$$
$$\chi_{608}(63,\cdot)$$ 608.bj 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{608}(65,\cdot)$$ 608.r 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{608}(67,\cdot)$$ 608.bt 72 yes $$1$$ $$1$$ $$e\left(\frac{65}{72}\right)$$ $$e\left(\frac{35}{72}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{25}{72}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{59}{72}\right)$$ $$e\left(\frac{23}{36}\right)$$