Properties

Modulus $116$
Structure \(C_{28}\times C_{2}\)
Order $56$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(116)
 
pari: g = idealstar(,116,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 56
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{28}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{116}(59,\cdot)$, $\chi_{116}(89,\cdot)$

First 32 of 56 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 \(-1\) \(1\) \(3\) \(5\) \(7\) \(9\) \(11\) \(13\) \(15\) \(17\) \(19\) \(21\)
\(\chi_{116}(1,\cdot)\) 116.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{116}(3,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(-i\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{28}\right)\)
\(\chi_{116}(5,\cdot)\) 116.i 14 no \(1\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(-1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{116}(7,\cdot)\) 116.j 14 yes \(-1\) \(1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{116}(9,\cdot)\) 116.i 14 no \(1\) \(1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(-1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{14}\right)\)
\(\chi_{116}(11,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(-i\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{28}\right)\)
\(\chi_{116}(13,\cdot)\) 116.i 14 no \(1\) \(1\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(-1\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{14}\right)\)
\(\chi_{116}(15,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(i\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{11}{28}\right)\)
\(\chi_{116}(17,\cdot)\) 116.f 4 no \(-1\) \(1\) \(-i\) \(-1\) \(1\) \(-1\) \(-i\) \(-1\) \(i\) \(-i\) \(-i\) \(-i\)
\(\chi_{116}(19,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(-i\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{13}{28}\right)\)
\(\chi_{116}(21,\cdot)\) 116.k 28 no \(-1\) \(1\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(-i\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{9}{28}\right)\)
\(\chi_{116}(23,\cdot)\) 116.j 14 yes \(-1\) \(1\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(1\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{116}(25,\cdot)\) 116.g 7 no \(1\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{116}(27,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{5}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{27}{28}\right)\) \(i\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{3}{28}\right)\)
\(\chi_{116}(31,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{11}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(-i\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{17}{28}\right)\)
\(\chi_{116}(33,\cdot)\) 116.i 14 no \(1\) \(1\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(-1\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{14}\right)\)
\(\chi_{116}(35,\cdot)\) 116.h 14 yes \(-1\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(-1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\)
\(\chi_{116}(37,\cdot)\) 116.k 28 no \(-1\) \(1\) \(e\left(\frac{15}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{25}{28}\right)\) \(i\) \(e\left(\frac{27}{28}\right)\) \(e\left(\frac{23}{28}\right)\)
\(\chi_{116}(39,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{19}{28}\right)\) \(i\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{27}{28}\right)\)
\(\chi_{116}(41,\cdot)\) 116.f 4 no \(-1\) \(1\) \(i\) \(-1\) \(1\) \(-1\) \(i\) \(-1\) \(-i\) \(i\) \(i\) \(i\)
\(\chi_{116}(43,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{23}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(e\left(\frac{5}{14}\right)\) \(e\left(\frac{1}{28}\right)\) \(-i\) \(e\left(\frac{19}{28}\right)\) \(e\left(\frac{25}{28}\right)\)
\(\chi_{116}(45,\cdot)\) 116.g 7 no \(1\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{116}(47,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{28}\right)\) \(e\left(\frac{1}{14}\right)\) \(e\left(\frac{3}{28}\right)\) \(i\) \(e\left(\frac{1}{28}\right)\) \(e\left(\frac{19}{28}\right)\)
\(\chi_{116}(49,\cdot)\) 116.g 7 no \(1\) \(1\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(1\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{116}(51,\cdot)\) 116.h 14 yes \(-1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(-1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{11}{14}\right)\)
\(\chi_{116}(53,\cdot)\) 116.g 7 no \(1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{116}(55,\cdot)\) 116.l 28 yes \(1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{9}{14}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(i\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{15}{28}\right)\)
\(\chi_{116}(57,\cdot)\) 116.c 2 no \(1\) \(1\) \(-1\) \(1\) \(1\) \(1\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\) \(-1\)
\(\chi_{116}(59,\cdot)\) 116.b 2 no \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\) \(-1\) \(1\)
\(\chi_{116}(61,\cdot)\) 116.k 28 no \(-1\) \(1\) \(e\left(\frac{25}{28}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{11}{14}\right)\) \(e\left(\frac{13}{28}\right)\) \(e\left(\frac{3}{14}\right)\) \(e\left(\frac{23}{28}\right)\) \(-i\) \(e\left(\frac{17}{28}\right)\) \(e\left(\frac{1}{28}\right)\)
\(\chi_{116}(63,\cdot)\) 116.h 14 yes \(-1\) \(1\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{13}{14}\right)\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(-1\) \(e\left(\frac{4}{7}\right)\) \(e\left(\frac{5}{14}\right)\)
\(\chi_{116}(65,\cdot)\) 116.g 7 no \(1\) \(1\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{3}{7}\right)\) \(e\left(\frac{1}{7}\right)\) \(e\left(\frac{2}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{5}{7}\right)\) \(e\left(\frac{4}{7}\right)\) \(1\) \(e\left(\frac{6}{7}\right)\) \(e\left(\frac{2}{7}\right)\)