Properties

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

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Show commands: Pari/GP / 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 \(-1\) \(1\) \(i\) \(i\) \(1\) \(-1\) \(-i\) \(-i\) \(-1\) \(1\) \(i\) \(i\)
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