# Properties

 Modulus $384$ Structure $$C_{2}\times C_{2}\times C_{32}$$ Order $128$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(384)

pari: g = idealstar(,384,2)

## Character group

 sage: G.order()  pari: g.no Order = 128 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{2}\times C_{32}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{384}(127,\cdot)$, $\chi_{384}(133,\cdot)$, $\chi_{384}(257,\cdot)$

## First 32 of 128 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{384}(1,\cdot)$$ 384.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{384}(5,\cdot)$$ 384.x 32 yes $$-1$$ $$1$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$i$$
$$\chi_{384}(7,\cdot)$$ 384.t 16 no $$-1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$1$$
$$\chi_{384}(11,\cdot)$$ 384.w 32 yes $$1$$ $$1$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$-i$$
$$\chi_{384}(13,\cdot)$$ 384.v 32 no $$1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$-i$$
$$\chi_{384}(17,\cdot)$$ 384.p 8 no $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
$$\chi_{384}(19,\cdot)$$ 384.u 32 no $$-1$$ $$1$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$i$$
$$\chi_{384}(23,\cdot)$$ 384.s 16 no $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$1$$
$$\chi_{384}(25,\cdot)$$ 384.r 16 no $$1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{384}(29,\cdot)$$ 384.x 32 yes $$-1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$-i$$
$$\chi_{384}(31,\cdot)$$ 384.l 4 no $$-1$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$-i$$ $$1$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-1$$
$$\chi_{384}(35,\cdot)$$ 384.w 32 yes $$1$$ $$1$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$i$$
$$\chi_{384}(37,\cdot)$$ 384.v 32 no $$1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$i$$
$$\chi_{384}(41,\cdot)$$ 384.q 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-1$$
$$\chi_{384}(43,\cdot)$$ 384.u 32 no $$-1$$ $$1$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$-i$$
$$\chi_{384}(47,\cdot)$$ 384.o 8 no $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$-1$$
$$\chi_{384}(49,\cdot)$$ 384.n 8 no $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$1$$
$$\chi_{384}(53,\cdot)$$ 384.x 32 yes $$-1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$i$$
$$\chi_{384}(55,\cdot)$$ 384.t 16 no $$-1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$1$$
$$\chi_{384}(59,\cdot)$$ 384.w 32 yes $$1$$ $$1$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$-i$$
$$\chi_{384}(61,\cdot)$$ 384.v 32 no $$1$$ $$1$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$-i$$
$$\chi_{384}(65,\cdot)$$ 384.h 2 no $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$
$$\chi_{384}(67,\cdot)$$ 384.u 32 no $$-1$$ $$1$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$i$$
$$\chi_{384}(71,\cdot)$$ 384.s 16 no $$1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$1$$
$$\chi_{384}(73,\cdot)$$ 384.r 16 no $$1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-1$$
$$\chi_{384}(77,\cdot)$$ 384.x 32 yes $$-1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$-i$$
$$\chi_{384}(79,\cdot)$$ 384.m 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{7}{8}\right)$$ $$-1$$
$$\chi_{384}(83,\cdot)$$ 384.w 32 yes $$1$$ $$1$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$i$$
$$\chi_{384}(85,\cdot)$$ 384.v 32 no $$1$$ $$1$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$i$$
$$\chi_{384}(89,\cdot)$$ 384.q 16 no $$-1$$ $$1$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-1$$
$$\chi_{384}(91,\cdot)$$ 384.u 32 no $$-1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$-i$$
$$\chi_{384}(95,\cdot)$$ 384.k 4 no $$1$$ $$1$$ $$i$$ $$1$$ $$-i$$ $$i$$ $$-1$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$-1$$