Properties

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

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Show commands for: 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 \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\)