# Properties

 Modulus $256$ Structure $$C_{2}\times C_{64}$$ Order $128$

Show commands: PariGP / SageMath

sage: H = DirichletGroup(256)

pari: g = idealstar(,256,2)

## Character group

 sage: G.order()  pari: g.no Order = 128 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{64}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{256}(255,\cdot)$, $\chi_{256}(5,\cdot)$

## 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{256}(1,\cdot)$$ 256.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{256}(3,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{39}{64}\right)$$
$$\chi_{256}(5,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$
$$\chi_{256}(7,\cdot)$$ 256.l 32 no $$-1$$ $$1$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$
$$\chi_{256}(9,\cdot)$$ 256.k 32 no $$1$$ $$1$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$
$$\chi_{256}(11,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{49}{64}\right)$$
$$\chi_{256}(13,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{3}{64}\right)$$
$$\chi_{256}(15,\cdot)$$ 256.j 16 no $$-1$$ $$1$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$
$$\chi_{256}(17,\cdot)$$ 256.i 16 no $$1$$ $$1$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$
$$\chi_{256}(19,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{11}{64}\right)$$
$$\chi_{256}(21,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{41}{64}\right)$$
$$\chi_{256}(23,\cdot)$$ 256.l 32 no $$-1$$ $$1$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$
$$\chi_{256}(25,\cdot)$$ 256.k 32 no $$1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$
$$\chi_{256}(27,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{53}{64}\right)$$
$$\chi_{256}(29,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{31}{64}\right)$$
$$\chi_{256}(31,\cdot)$$ 256.h 8 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
$$\chi_{256}(33,\cdot)$$ 256.g 8 no $$1$$ $$1$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
$$\chi_{256}(35,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$
$$\chi_{256}(37,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$
$$\chi_{256}(39,\cdot)$$ 256.l 32 no $$-1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$
$$\chi_{256}(41,\cdot)$$ 256.k 32 no $$1$$ $$1$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$
$$\chi_{256}(43,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{57}{64}\right)$$
$$\chi_{256}(45,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{59}{64}\right)$$
$$\chi_{256}(47,\cdot)$$ 256.j 16 no $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$i$$ $$i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$
$$\chi_{256}(49,\cdot)$$ 256.i 16 no $$1$$ $$1$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$i$$ $$-i$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$
$$\chi_{256}(51,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{19}{64}\right)$$
$$\chi_{256}(53,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{33}{64}\right)$$
$$\chi_{256}(55,\cdot)$$ 256.l 32 no $$-1$$ $$1$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$
$$\chi_{256}(57,\cdot)$$ 256.k 32 no $$1$$ $$1$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$
$$\chi_{256}(59,\cdot)$$ 256.n 64 yes $$-1$$ $$1$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{5}{64}\right)$$ $$e\left(\frac{31}{64}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{61}{64}\right)$$
$$\chi_{256}(61,\cdot)$$ 256.m 64 yes $$1$$ $$1$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{23}{64}\right)$$
$$\chi_{256}(63,\cdot)$$ 256.f 4 no $$-1$$ $$1$$ $$i$$ $$i$$ $$1$$ $$-1$$ $$-i$$ $$-i$$ $$-1$$ $$1$$ $$i$$ $$i$$