# Properties

 Modulus $128$ Structure $$C_{32}\times C_{2}$$ Order $64$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(128)

pari: g = idealstar(,128,2)

## Character group

 sage: G.order()  pari: g.no Order = 64 sage: H.invariants()  pari: g.cyc Structure = $$C_{32}\times C_{2}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{128}(127,\cdot)$, $\chi_{128}(5,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{128}(1,\cdot)$$ 128.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{128}(3,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$
$$\chi_{128}(5,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$
$$\chi_{128}(7,\cdot)$$ 128.j 16 no $$-1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{128}(9,\cdot)$$ 128.i 16 no $$1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$
$$\chi_{128}(11,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$
$$\chi_{128}(13,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$
$$\chi_{128}(15,\cdot)$$ 128.h 8 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{128}(17,\cdot)$$ 128.g 8 no $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{128}(19,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$
$$\chi_{128}(21,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$
$$\chi_{128}(23,\cdot)$$ 128.j 16 no $$-1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$i$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{128}(25,\cdot)$$ 128.i 16 no $$1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{128}(27,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$
$$\chi_{128}(29,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$
$$\chi_{128}(31,\cdot)$$ 128.f 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$i$$
$$\chi_{128}(33,\cdot)$$ 128.e 4 no $$1$$ $$1$$ $$i$$ $$-i$$ $$-1$$ $$-1$$ $$-i$$ $$i$$ $$1$$ $$1$$ $$i$$ $$-i$$
$$\chi_{128}(35,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$
$$\chi_{128}(37,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$
$$\chi_{128}(39,\cdot)$$ 128.j 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$
$$\chi_{128}(41,\cdot)$$ 128.i 16 no $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{128}(43,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$
$$\chi_{128}(45,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$
$$\chi_{128}(47,\cdot)$$ 128.h 8 no $$-1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$
$$\chi_{128}(49,\cdot)$$ 128.g 8 no $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$
$$\chi_{128}(51,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$
$$\chi_{128}(53,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$
$$\chi_{128}(55,\cdot)$$ 128.j 16 no $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{128}(57,\cdot)$$ 128.i 16 no $$1$$ $$1$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$
$$\chi_{128}(59,\cdot)$$ 128.l 32 yes $$-1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$
$$\chi_{128}(61,\cdot)$$ 128.k 32 yes $$1$$ $$1$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$
$$\chi_{128}(63,\cdot)$$ 128.d 2 no $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$-1$$