Properties

Modulus $205$
Structure \(C_{40}\times C_{4}\)
Order $160$

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Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(205)
 
pari: g = idealstar(,205,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 160
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{40}\times C_{4}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{205}(42,\cdot)$, $\chi_{205}(6,\cdot)$

First 32 of 160 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{205}(1,\cdot)\) 205.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{205}(2,\cdot)\) 205.t 20 yes \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(-1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{9}{20}\right)\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{205}(3,\cdot)\) 205.l 8 yes \(1\) \(1\) \(-1\) \(e\left(\frac{7}{8}\right)\) \(1\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{205}(4,\cdot)\) 205.r 10 yes \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{4}{5}\right)\)
\(\chi_{205}(6,\cdot)\) 205.z 40 no \(-1\) \(1\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{19}{20}\right)\) \(-i\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{31}{40}\right)\)
\(\chi_{205}(7,\cdot)\) 205.y 40 yes \(1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{4}{5}\right)\) \(-i\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{39}{40}\right)\)
\(\chi_{205}(8,\cdot)\) 205.t 20 yes \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(-1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{20}\right)\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{205}(9,\cdot)\) 205.j 4 yes \(1\) \(1\) \(1\) \(-i\) \(1\) \(-i\) \(-i\) \(1\) \(-1\) \(i\) \(-i\) \(-i\)
\(\chi_{205}(11,\cdot)\) 205.z 40 no \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{40}\right)\) \(e\left(\frac{37}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(i\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{13}{40}\right)\)
\(\chi_{205}(12,\cdot)\) 205.y 40 yes \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{2}{5}\right)\) \(-i\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{27}{40}\right)\)
\(\chi_{205}(13,\cdot)\) 205.bb 40 yes \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{31}{40}\right)\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(-i\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{11}{40}\right)\)
\(\chi_{205}(14,\cdot)\) 205.m 8 yes \(-1\) \(1\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(i\) \(-i\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{7}{8}\right)\)
\(\chi_{205}(16,\cdot)\) 205.k 5 no \(1\) \(1\) \(e\left(\frac{3}{5}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{205}(17,\cdot)\) 205.bb 40 yes \(1\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{1}{10}\right)\) \(i\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{13}{40}\right)\)
\(\chi_{205}(18,\cdot)\) 205.u 20 yes \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(i\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(-1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{13}{20}\right)\)
\(\chi_{205}(19,\cdot)\) 205.ba 40 yes \(-1\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{1}{20}\right)\) \(-i\) \(e\left(\frac{27}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{19}{40}\right)\)
\(\chi_{205}(21,\cdot)\) 205.x 20 no \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(i\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(-1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{17}{20}\right)\)
\(\chi_{205}(22,\cdot)\) 205.bb 40 yes \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{29}{40}\right)\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{3}{10}\right)\) \(i\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{9}{40}\right)\)
\(\chi_{205}(23,\cdot)\) 205.v 20 yes \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(-i\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{9}{20}\right)\) \(-1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{20}\right)\)
\(\chi_{205}(24,\cdot)\) 205.ba 40 yes \(-1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{17}{20}\right)\) \(-i\) \(e\left(\frac{39}{40}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{23}{40}\right)\)
\(\chi_{205}(26,\cdot)\) 205.z 40 no \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{23}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(-i\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{7}{40}\right)\)
\(\chi_{205}(27,\cdot)\) 205.l 8 yes \(1\) \(1\) \(-1\) \(e\left(\frac{5}{8}\right)\) \(1\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{8}\right)\) \(-1\) \(i\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{5}{8}\right)\)
\(\chi_{205}(28,\cdot)\) 205.bb 40 yes \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{40}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{7}{10}\right)\) \(-i\) \(e\left(\frac{33}{40}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{31}{40}\right)\)
\(\chi_{205}(29,\cdot)\) 205.ba 40 yes \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{1}{8}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{3}{20}\right)\) \(i\) \(e\left(\frac{21}{40}\right)\) \(e\left(\frac{9}{40}\right)\) \(e\left(\frac{37}{40}\right)\)
\(\chi_{205}(31,\cdot)\) 205.p 10 no \(1\) \(1\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{205}(32,\cdot)\) 205.f 4 yes \(-1\) \(1\) \(-i\) \(-1\) \(-1\) \(i\) \(1\) \(i\) \(1\) \(-i\) \(1\) \(-1\)
\(\chi_{205}(33,\cdot)\) 205.w 20 yes \(-1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{20}\right)\) \(1\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{5}\right)\)
\(\chi_{205}(34,\cdot)\) 205.ba 40 yes \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{19}{40}\right)\) \(e\left(\frac{1}{40}\right)\) \(e\left(\frac{11}{20}\right)\) \(i\) \(e\left(\frac{17}{40}\right)\) \(e\left(\frac{13}{40}\right)\) \(e\left(\frac{9}{40}\right)\)
\(\chi_{205}(36,\cdot)\) 205.x 20 no \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(-i\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(-1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{11}{20}\right)\)
\(\chi_{205}(37,\cdot)\) 205.u 20 yes \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(-i\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{3}{20}\right)\) \(-1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{11}{20}\right)\)
\(\chi_{205}(38,\cdot)\) 205.l 8 yes \(1\) \(1\) \(-1\) \(e\left(\frac{3}{8}\right)\) \(1\) \(e\left(\frac{7}{8}\right)\) \(e\left(\frac{7}{8}\right)\) \(-1\) \(-i\) \(e\left(\frac{5}{8}\right)\) \(e\left(\frac{3}{8}\right)\) \(e\left(\frac{3}{8}\right)\)
\(\chi_{205}(39,\cdot)\) 205.s 20 yes \(1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(-i\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{5}\right)\) \(-1\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{3}{20}\right)\)