# Properties

 Modulus $512$ Structure $$C_{2}\times C_{128}$$ Order $256$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(512)

pari: g = idealstar(,512,2)

## Character group

 sage: G.order()  pari: g.no Order = 256 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{128}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{512}(511,\cdot)$, $\chi_{512}(5,\cdot)$

## First 32 of 256 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{512}(1,\cdot)$$ 512.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{512}(3,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{9}{128}\right)$$ $$e\left(\frac{35}{128}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{95}{128}\right)$$ $$e\left(\frac{45}{128}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{101}{128}\right)$$ $$e\left(\frac{103}{128}\right)$$
$$\chi_{512}(5,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{35}{128}\right)$$ $$e\left(\frac{1}{128}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{85}{128}\right)$$ $$e\left(\frac{111}{128}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{23}{128}\right)$$ $$e\left(\frac{109}{128}\right)$$
$$\chi_{512}(7,\cdot)$$ 512.n 64 no $$-1$$ $$1$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{1}{64}\right)$$
$$\chi_{512}(9,\cdot)$$ 512.m 64 no $$1$$ $$1$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{31}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{39}{64}\right)$$
$$\chi_{512}(11,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{95}{128}\right)$$ $$e\left(\frac{85}{128}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{31}{64}\right)$$ $$e\left(\frac{121}{128}\right)$$ $$e\left(\frac{91}{128}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{99}{128}\right)$$ $$e\left(\frac{49}{128}\right)$$
$$\chi_{512}(13,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{45}{128}\right)$$ $$e\left(\frac{111}{128}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{91}{128}\right)$$ $$e\left(\frac{33}{128}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{121}{128}\right)$$ $$e\left(\frac{67}{128}\right)$$
$$\chi_{512}(15,\cdot)$$ 512.l 32 no $$-1$$ $$1$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$
$$\chi_{512}(17,\cdot)$$ 512.k 32 no $$1$$ $$1$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$
$$\chi_{512}(19,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{101}{128}\right)$$ $$e\left(\frac{23}{128}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{99}{128}\right)$$ $$e\left(\frac{121}{128}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{81}{128}\right)$$ $$e\left(\frac{75}{128}\right)$$
$$\chi_{512}(21,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{103}{128}\right)$$ $$e\left(\frac{109}{128}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{49}{128}\right)$$ $$e\left(\frac{67}{128}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{75}{128}\right)$$ $$e\left(\frac{105}{128}\right)$$
$$\chi_{512}(23,\cdot)$$ 512.n 64 no $$-1$$ $$1$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{59}{64}\right)$$
$$\chi_{512}(25,\cdot)$$ 512.m 64 no $$1$$ $$1$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$
$$\chi_{512}(27,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{27}{128}\right)$$ $$e\left(\frac{105}{128}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{29}{128}\right)$$ $$e\left(\frac{7}{128}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{47}{128}\right)$$ $$e\left(\frac{53}{128}\right)$$
$$\chi_{512}(29,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{81}{128}\right)$$ $$e\left(\frac{123}{128}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{87}{128}\right)$$ $$e\left(\frac{85}{128}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{13}{128}\right)$$ $$e\left(\frac{95}{128}\right)$$
$$\chi_{512}(31,\cdot)$$ 512.j 16 no $$-1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$
$$\chi_{512}(33,\cdot)$$ 512.i 16 no $$1$$ $$1$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$
$$\chi_{512}(35,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{1}{128}\right)$$ $$e\left(\frac{75}{128}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{39}{128}\right)$$ $$e\left(\frac{5}{128}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{125}{128}\right)$$ $$e\left(\frac{111}{128}\right)$$
$$\chi_{512}(37,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{107}{128}\right)$$ $$e\left(\frac{25}{128}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{77}{128}\right)$$ $$e\left(\frac{87}{128}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{63}{128}\right)$$ $$e\left(\frac{37}{128}\right)$$
$$\chi_{512}(39,\cdot)$$ 512.n 64 no $$-1$$ $$1$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{21}{64}\right)$$
$$\chi_{512}(41,\cdot)$$ 512.m 64 no $$1$$ $$1$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{19}{64}\right)$$
$$\chi_{512}(43,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{23}{128}\right)$$ $$e\left(\frac{61}{128}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{1}{128}\right)$$ $$e\left(\frac{115}{128}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{59}{128}\right)$$ $$e\left(\frac{121}{128}\right)$$
$$\chi_{512}(45,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{53}{128}\right)$$ $$e\left(\frac{71}{128}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{19}{128}\right)$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{97}{128}\right)$$ $$e\left(\frac{59}{128}\right)$$
$$\chi_{512}(47,\cdot)$$ 512.l 32 no $$-1$$ $$1$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$
$$\chi_{512}(49,\cdot)$$ 512.k 32 no $$1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$
$$\chi_{512}(51,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{93}{128}\right)$$ $$e\left(\frac{63}{128}\right)$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{43}{128}\right)$$ $$e\left(\frac{81}{128}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{105}{128}\right)$$ $$e\left(\frac{83}{128}\right)$$
$$\chi_{512}(53,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{47}{128}\right)$$ $$e\left(\frac{5}{128}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{47}{64}\right)$$ $$e\left(\frac{41}{128}\right)$$ $$e\left(\frac{43}{128}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{115}{128}\right)$$ $$e\left(\frac{33}{128}\right)$$
$$\chi_{512}(55,\cdot)$$ 512.n 64 no $$-1$$ $$1$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{15}{64}\right)$$
$$\chi_{512}(57,\cdot)$$ 512.m 64 no $$1$$ $$1$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{25}{64}\right)$$
$$\chi_{512}(59,\cdot)$$ 512.p 128 yes $$-1$$ $$1$$ $$e\left(\frac{83}{128}\right)$$ $$e\left(\frac{81}{128}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{37}{128}\right)$$ $$e\left(\frac{31}{128}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{7}{128}\right)$$ $$e\left(\frac{125}{128}\right)$$
$$\chi_{512}(61,\cdot)$$ 512.o 128 yes $$1$$ $$1$$ $$e\left(\frac{89}{128}\right)$$ $$e\left(\frac{83}{128}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{15}{128}\right)$$ $$e\left(\frac{125}{128}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{117}{128}\right)$$ $$e\left(\frac{87}{128}\right)$$
$$\chi_{512}(63,\cdot)$$ 512.h 8 no $$-1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$1$$ $$-1$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$