Properties

Modulus $115$
Structure \(C_{44}\times C_{2}\)
Order $88$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(115)
 
pari: g = idealstar(,115,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 88
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{44}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{115}(47,\cdot)$, $\chi_{115}(51,\cdot)$

First 32 of 88 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{115}(1,\cdot)\) 115.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{115}(2,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{1}{44}\right)\)
\(\chi_{115}(3,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{19}{44}\right)\)
\(\chi_{115}(4,\cdot)\) 115.j 22 yes \(1\) \(1\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{115}(6,\cdot)\) 115.g 11 no \(1\) \(1\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{115}(7,\cdot)\) 115.l 44 yes \(1\) \(1\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{25}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{37}{44}\right)\)
\(\chi_{115}(8,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{41}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{115}(9,\cdot)\) 115.j 22 yes \(1\) \(1\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{115}(11,\cdot)\) 115.h 22 no \(-1\) \(1\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{115}(12,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{21}{44}\right)\)
\(\chi_{115}(13,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{115}(14,\cdot)\) 115.i 22 yes \(-1\) \(1\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{19}{22}\right)\)
\(\chi_{115}(16,\cdot)\) 115.g 11 no \(1\) \(1\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{115}(17,\cdot)\) 115.l 44 yes \(1\) \(1\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{9}{44}\right)\)
\(\chi_{115}(18,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{5}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{39}{44}\right)\)
\(\chi_{115}(19,\cdot)\) 115.i 22 yes \(-1\) \(1\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{1}{22}\right)\)
\(\chi_{115}(21,\cdot)\) 115.h 22 no \(-1\) \(1\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{115}(22,\cdot)\) 115.e 4 yes \(1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(-i\) \(-i\) \(-1\) \(-1\) \(i\) \(-i\)
\(\chi_{115}(24,\cdot)\) 115.b 2 no \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\) \(1\) \(1\) \(-1\) \(-1\)
\(\chi_{115}(26,\cdot)\) 115.g 11 no \(1\) \(1\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{115}(27,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{31}{44}\right)\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{13}{44}\right)\)
\(\chi_{115}(28,\cdot)\) 115.l 44 yes \(1\) \(1\) \(e\left(\frac{37}{44}\right)\) \(e\left(\frac{43}{44}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{23}{44}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{29}{44}\right)\) \(e\left(\frac{39}{44}\right)\)
\(\chi_{115}(29,\cdot)\) 115.j 22 yes \(1\) \(1\) \(e\left(\frac{3}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{21}{22}\right)\)
\(\chi_{115}(31,\cdot)\) 115.g 11 no \(1\) \(1\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{115}(32,\cdot)\) 115.k 44 yes \(-1\) \(1\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{5}{44}\right)\)
\(\chi_{115}(33,\cdot)\) 115.l 44 yes \(1\) \(1\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{3}{44}\right)\) \(e\left(\frac{19}{22}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{7}{44}\right)\)
\(\chi_{115}(34,\cdot)\) 115.i 22 yes \(-1\) \(1\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{5}{22}\right)\)
\(\chi_{115}(36,\cdot)\) 115.g 11 no \(1\) \(1\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{115}(37,\cdot)\) 115.l 44 yes \(1\) \(1\) \(e\left(\frac{7}{44}\right)\) \(e\left(\frac{1}{44}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{17}{44}\right)\) \(e\left(\frac{21}{44}\right)\) \(e\left(\frac{1}{22}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{15}{44}\right)\) \(e\left(\frac{5}{44}\right)\)
\(\chi_{115}(38,\cdot)\) 115.l 44 yes \(1\) \(1\) \(e\left(\frac{13}{44}\right)\) \(e\left(\frac{27}{44}\right)\) \(e\left(\frac{13}{22}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{19}{44}\right)\) \(e\left(\frac{39}{44}\right)\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{21}{22}\right)\) \(e\left(\frac{9}{44}\right)\) \(e\left(\frac{3}{44}\right)\)
\(\chi_{115}(39,\cdot)\) 115.j 22 yes \(1\) \(1\) \(e\left(\frac{5}{22}\right)\) \(e\left(\frac{7}{22}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{9}{22}\right)\) \(e\left(\frac{15}{22}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{17}{22}\right)\) \(e\left(\frac{13}{22}\right)\)
\(\chi_{115}(41,\cdot)\) 115.g 11 no \(1\) \(1\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{7}{11}\right)\)