# Properties

 Modulus $304$ Structure $$C_{2}\times C_{2}\times C_{36}$$ Order $144$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(304)

pari: g = idealstar(,304,2)

## Character group

 sage: G.order()  pari: g.no Order = 144 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{36}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{304}(191,\cdot)$, $\chi_{304}(229,\cdot)$, $\chi_{304}(97,\cdot)$

## First 32 of 144 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$21$$ $$23$$
$$\chi_{304}(1,\cdot)$$ 304.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{304}(3,\cdot)$$ 304.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{304}(5,\cdot)$$ 304.bi 36 yes $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$
$$\chi_{304}(7,\cdot)$$ 304.o 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{304}(9,\cdot)$$ 304.bb 18 no $$1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{304}(11,\cdot)$$ 304.y 12 yes $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{304}(13,\cdot)$$ 304.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{304}(15,\cdot)$$ 304.be 18 no $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{304}(17,\cdot)$$ 304.u 9 no $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{304}(21,\cdot)$$ 304.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{304}(23,\cdot)$$ 304.bc 18 no $$-1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{304}(25,\cdot)$$ 304.bb 18 no $$1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{304}(27,\cdot)$$ 304.x 12 yes $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$-i$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{304}(29,\cdot)$$ 304.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{304}(31,\cdot)$$ 304.n 6 no $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-1$$ $$e\left(\frac{2}{3}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{304}(33,\cdot)$$ 304.z 18 no $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{304}(35,\cdot)$$ 304.bh 36 yes $$-1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{304}(37,\cdot)$$ 304.j 4 yes $$-1$$ $$1$$ $$i$$ $$i$$ $$-1$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$1$$ $$-i$$ $$-1$$
$$\chi_{304}(39,\cdot)$$ 304.f 2 no $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$
$$\chi_{304}(41,\cdot)$$ 304.ba 18 no $$-1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{304}(43,\cdot)$$ 304.bh 36 yes $$-1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{304}(45,\cdot)$$ 304.v 12 yes $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$-1$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{304}(47,\cdot)$$ 304.bf 18 no $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$
$$\chi_{304}(49,\cdot)$$ 304.i 3 no $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{304}(51,\cdot)$$ 304.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{304}(53,\cdot)$$ 304.bj 36 yes $$-1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{304}(55,\cdot)$$ 304.bc 18 no $$-1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{11}{18}\right)$$
$$\chi_{304}(59,\cdot)$$ 304.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{304}(61,\cdot)$$ 304.bi 36 yes $$1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$
$$\chi_{304}(63,\cdot)$$ 304.bf 18 no $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$
$$\chi_{304}(65,\cdot)$$ 304.r 6 no $$-1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{304}(67,\cdot)$$ 304.bg 36 yes $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$