Properties

 Modulus $1024$ Structure $$C_{2}\times C_{256}$$ Order $512$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(1024)

pari: g = idealstar(,1024,2)

Character group

 sage: G.order()  pari: g.no Order = 512 sage: H.invariants()  pari: g.cyc Structure = $$C_{2}\times C_{256}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{1024}(1023,\cdot)$, $\chi_{1024}(5,\cdot)$

First 32 of 512 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 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$
$$\chi_{1024}(1,\cdot)$$ 1024.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{1024}(3,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{73}{256}\right)$$ $$e\left(\frac{163}{256}\right)$$ $$e\left(\frac{15}{128}\right)$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{31}{256}\right)$$ $$e\left(\frac{45}{256}\right)$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{165}{256}\right)$$ $$e\left(\frac{103}{256}\right)$$
$$\chi_{1024}(5,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{163}{256}\right)$$ $$e\left(\frac{1}{256}\right)$$ $$e\left(\frac{101}{128}\right)$$ $$e\left(\frac{35}{128}\right)$$ $$e\left(\frac{213}{256}\right)$$ $$e\left(\frac{239}{256}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{151}{256}\right)$$ $$e\left(\frac{109}{256}\right)$$
$$\chi_{1024}(7,\cdot)$$ 1024.p 128 no $$-1$$ $$1$$ $$e\left(\frac{15}{128}\right)$$ $$e\left(\frac{101}{128}\right)$$ $$e\left(\frac{57}{64}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{75}{128}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{83}{128}\right)$$ $$e\left(\frac{1}{128}\right)$$
$$\chi_{1024}(9,\cdot)$$ 1024.o 128 no $$1$$ $$1$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{35}{128}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{31}{128}\right)$$ $$e\left(\frac{45}{128}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{37}{128}\right)$$ $$e\left(\frac{103}{128}\right)$$
$$\chi_{1024}(11,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{31}{256}\right)$$ $$e\left(\frac{213}{256}\right)$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{31}{128}\right)$$ $$e\left(\frac{185}{256}\right)$$ $$e\left(\frac{219}{256}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{35}{256}\right)$$ $$e\left(\frac{177}{256}\right)$$
$$\chi_{1024}(13,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{45}{256}\right)$$ $$e\left(\frac{239}{256}\right)$$ $$e\left(\frac{75}{128}\right)$$ $$e\left(\frac{45}{128}\right)$$ $$e\left(\frac{219}{256}\right)$$ $$e\left(\frac{33}{256}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{249}{256}\right)$$ $$e\left(\frac{195}{256}\right)$$
$$\chi_{1024}(15,\cdot)$$ 1024.n 64 no $$-1$$ $$1$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{29}{32}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{7}{64}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{53}{64}\right)$$
$$\chi_{1024}(17,\cdot)$$ 1024.m 64 no $$1$$ $$1$$ $$e\left(\frac{21}{64}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{41}{64}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{27}{64}\right)$$
$$\chi_{1024}(19,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{165}{256}\right)$$ $$e\left(\frac{151}{256}\right)$$ $$e\left(\frac{83}{128}\right)$$ $$e\left(\frac{37}{128}\right)$$ $$e\left(\frac{35}{256}\right)$$ $$e\left(\frac{249}{256}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{145}{256}\right)$$ $$e\left(\frac{75}{256}\right)$$
$$\chi_{1024}(21,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{103}{256}\right)$$ $$e\left(\frac{109}{256}\right)$$ $$e\left(\frac{1}{128}\right)$$ $$e\left(\frac{103}{128}\right)$$ $$e\left(\frac{177}{256}\right)$$ $$e\left(\frac{195}{256}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{75}{256}\right)$$ $$e\left(\frac{105}{256}\right)$$
$$\chi_{1024}(23,\cdot)$$ 1024.p 128 no $$-1$$ $$1$$ $$e\left(\frac{117}{128}\right)$$ $$e\left(\frac{71}{128}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{53}{64}\right)$$ $$e\left(\frac{83}{128}\right)$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{33}{128}\right)$$ $$e\left(\frac{59}{128}\right)$$
$$\chi_{1024}(25,\cdot)$$ 1024.o 128 no $$1$$ $$1$$ $$e\left(\frac{35}{128}\right)$$ $$e\left(\frac{1}{128}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{85}{128}\right)$$ $$e\left(\frac{111}{128}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{23}{128}\right)$$ $$e\left(\frac{109}{128}\right)$$
$$\chi_{1024}(27,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{219}{256}\right)$$ $$e\left(\frac{233}{256}\right)$$ $$e\left(\frac{45}{128}\right)$$ $$e\left(\frac{91}{128}\right)$$ $$e\left(\frac{93}{256}\right)$$ $$e\left(\frac{135}{256}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{63}{64}\right)$$ $$e\left(\frac{239}{256}\right)$$ $$e\left(\frac{53}{256}\right)$$
$$\chi_{1024}(29,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{81}{256}\right)$$ $$e\left(\frac{123}{256}\right)$$ $$e\left(\frac{7}{128}\right)$$ $$e\left(\frac{81}{128}\right)$$ $$e\left(\frac{87}{256}\right)$$ $$e\left(\frac{213}{256}\right)$$ $$e\left(\frac{51}{64}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{141}{256}\right)$$ $$e\left(\frac{95}{256}\right)$$
$$\chi_{1024}(31,\cdot)$$ 1024.l 32 no $$-1$$ $$1$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{29}{32}\right)$$
$$\chi_{1024}(33,\cdot)$$ 1024.k 32 no $$1$$ $$1$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$
$$\chi_{1024}(35,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{193}{256}\right)$$ $$e\left(\frac{203}{256}\right)$$ $$e\left(\frac{87}{128}\right)$$ $$e\left(\frac{65}{128}\right)$$ $$e\left(\frac{103}{256}\right)$$ $$e\left(\frac{133}{256}\right)$$ $$e\left(\frac{35}{64}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{61}{256}\right)$$ $$e\left(\frac{111}{256}\right)$$
$$\chi_{1024}(37,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{235}{256}\right)$$ $$e\left(\frac{25}{256}\right)$$ $$e\left(\frac{93}{128}\right)$$ $$e\left(\frac{107}{128}\right)$$ $$e\left(\frac{205}{256}\right)$$ $$e\left(\frac{87}{256}\right)$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{191}{256}\right)$$ $$e\left(\frac{165}{256}\right)$$
$$\chi_{1024}(39,\cdot)$$ 1024.p 128 no $$-1$$ $$1$$ $$e\left(\frac{59}{128}\right)$$ $$e\left(\frac{73}{128}\right)$$ $$e\left(\frac{45}{64}\right)$$ $$e\left(\frac{59}{64}\right)$$ $$e\left(\frac{125}{128}\right)$$ $$e\left(\frac{39}{128}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{79}{128}\right)$$ $$e\left(\frac{21}{128}\right)$$
$$\chi_{1024}(41,\cdot)$$ 1024.o 128 no $$1$$ $$1$$ $$e\left(\frac{93}{128}\right)$$ $$e\left(\frac{127}{128}\right)$$ $$e\left(\frac{27}{64}\right)$$ $$e\left(\frac{29}{64}\right)$$ $$e\left(\frac{43}{128}\right)$$ $$e\left(\frac{17}{128}\right)$$ $$e\left(\frac{23}{32}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{105}{128}\right)$$ $$e\left(\frac{19}{128}\right)$$
$$\chi_{1024}(43,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{87}{256}\right)$$ $$e\left(\frac{61}{256}\right)$$ $$e\left(\frac{81}{128}\right)$$ $$e\left(\frac{87}{128}\right)$$ $$e\left(\frac{65}{256}\right)$$ $$e\left(\frac{243}{256}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{123}{256}\right)$$ $$e\left(\frac{249}{256}\right)$$
$$\chi_{1024}(45,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{53}{256}\right)$$ $$e\left(\frac{71}{256}\right)$$ $$e\left(\frac{3}{128}\right)$$ $$e\left(\frac{53}{128}\right)$$ $$e\left(\frac{19}{256}\right)$$ $$e\left(\frac{73}{256}\right)$$ $$e\left(\frac{31}{64}\right)$$ $$e\left(\frac{17}{64}\right)$$ $$e\left(\frac{225}{256}\right)$$ $$e\left(\frac{59}{256}\right)$$
$$\chi_{1024}(47,\cdot)$$ 1024.n 64 no $$-1$$ $$1$$ $$e\left(\frac{1}{64}\right)$$ $$e\left(\frac{43}{64}\right)$$ $$e\left(\frac{7}{32}\right)$$ $$e\left(\frac{1}{32}\right)$$ $$e\left(\frac{39}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{61}{64}\right)$$ $$e\left(\frac{15}{64}\right)$$
$$\chi_{1024}(49,\cdot)$$ 1024.m 64 no $$1$$ $$1$$ $$e\left(\frac{15}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{25}{32}\right)$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{9}{64}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{19}{64}\right)$$ $$e\left(\frac{1}{64}\right)$$
$$\chi_{1024}(51,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{157}{256}\right)$$ $$e\left(\frac{63}{256}\right)$$ $$e\left(\frac{27}{128}\right)$$ $$e\left(\frac{29}{128}\right)$$ $$e\left(\frac{235}{256}\right)$$ $$e\left(\frac{209}{256}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{169}{256}\right)$$ $$e\left(\frac{211}{256}\right)$$
$$\chi_{1024}(53,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{47}{256}\right)$$ $$e\left(\frac{5}{256}\right)$$ $$e\left(\frac{121}{128}\right)$$ $$e\left(\frac{47}{128}\right)$$ $$e\left(\frac{41}{256}\right)$$ $$e\left(\frac{171}{256}\right)$$ $$e\left(\frac{13}{64}\right)$$ $$e\left(\frac{3}{64}\right)$$ $$e\left(\frac{243}{256}\right)$$ $$e\left(\frac{33}{256}\right)$$
$$\chi_{1024}(55,\cdot)$$ 1024.p 128 no $$-1$$ $$1$$ $$e\left(\frac{97}{128}\right)$$ $$e\left(\frac{107}{128}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{33}{64}\right)$$ $$e\left(\frac{71}{128}\right)$$ $$e\left(\frac{101}{128}\right)$$ $$e\left(\frac{19}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{93}{128}\right)$$ $$e\left(\frac{15}{128}\right)$$
$$\chi_{1024}(57,\cdot)$$ 1024.o 128 no $$1$$ $$1$$ $$e\left(\frac{119}{128}\right)$$ $$e\left(\frac{29}{128}\right)$$ $$e\left(\frac{49}{64}\right)$$ $$e\left(\frac{55}{64}\right)$$ $$e\left(\frac{33}{128}\right)$$ $$e\left(\frac{19}{128}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{11}{32}\right)$$ $$e\left(\frac{27}{128}\right)$$ $$e\left(\frac{89}{128}\right)$$
$$\chi_{1024}(59,\cdot)$$ 1024.r 256 yes $$-1$$ $$1$$ $$e\left(\frac{147}{256}\right)$$ $$e\left(\frac{209}{256}\right)$$ $$e\left(\frac{53}{128}\right)$$ $$e\left(\frac{19}{128}\right)$$ $$e\left(\frac{101}{256}\right)$$ $$e\left(\frac{31}{256}\right)$$ $$e\left(\frac{25}{64}\right)$$ $$e\left(\frac{23}{64}\right)$$ $$e\left(\frac{199}{256}\right)$$ $$e\left(\frac{253}{256}\right)$$
$$\chi_{1024}(61,\cdot)$$ 1024.q 256 yes $$1$$ $$1$$ $$e\left(\frac{217}{256}\right)$$ $$e\left(\frac{83}{256}\right)$$ $$e\left(\frac{63}{128}\right)$$ $$e\left(\frac{89}{128}\right)$$ $$e\left(\frac{15}{256}\right)$$ $$e\left(\frac{125}{256}\right)$$ $$e\left(\frac{11}{64}\right)$$ $$e\left(\frac{37}{64}\right)$$ $$e\left(\frac{245}{256}\right)$$ $$e\left(\frac{87}{256}\right)$$
$$\chi_{1024}(63,\cdot)$$ 1024.j 16 no $$-1$$ $$1$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$
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