Properties

Modulus $500$
Structure \(C_{2}\times C_{100}\)
Order $200$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(500)
 
pari: g = idealstar(,500,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 200
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{2}\times C_{100}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{500}(251,\cdot)$, $\chi_{500}(377,\cdot)$

First 32 of 200 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 \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{500}(1,\cdot)\) 500.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{500}(3,\cdot)\) 500.r 100 yes \(1\) \(1\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{97}{100}\right)\)
\(\chi_{500}(7,\cdot)\) 500.l 20 no \(1\) \(1\) \(e\left(\frac{9}{20}\right)\) \(-i\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{7}{20}\right)\)
\(\chi_{500}(9,\cdot)\) 500.o 50 no \(1\) \(1\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{47}{50}\right)\)
\(\chi_{500}(11,\cdot)\) 500.p 50 yes \(-1\) \(1\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{23}{50}\right)\)
\(\chi_{500}(13,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{19}{100}\right)\)
\(\chi_{500}(17,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{33}{100}\right)\)
\(\chi_{500}(19,\cdot)\) 500.n 50 yes \(-1\) \(1\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{7}{25}\right)\)
\(\chi_{500}(21,\cdot)\) 500.m 25 no \(1\) \(1\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{8}{25}\right)\)
\(\chi_{500}(23,\cdot)\) 500.r 100 yes \(1\) \(1\) \(e\left(\frac{67}{100}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{1}{100}\right)\)
\(\chi_{500}(27,\cdot)\) 500.r 100 yes \(1\) \(1\) \(e\left(\frac{97}{100}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{91}{100}\right)\)
\(\chi_{500}(29,\cdot)\) 500.o 50 no \(1\) \(1\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{1}{50}\right)\)
\(\chi_{500}(31,\cdot)\) 500.p 50 yes \(-1\) \(1\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{29}{50}\right)\)
\(\chi_{500}(33,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{59}{100}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{43}{100}\right)\)
\(\chi_{500}(37,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{99}{100}\right)\) \(e\left(\frac{9}{100}\right)\)
\(\chi_{500}(39,\cdot)\) 500.n 50 yes \(-1\) \(1\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{4}{25}\right)\)
\(\chi_{500}(41,\cdot)\) 500.m 25 no \(1\) \(1\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{6}{25}\right)\)
\(\chi_{500}(43,\cdot)\) 500.l 20 no \(1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(i\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{9}{20}\right)\)
\(\chi_{500}(47,\cdot)\) 500.r 100 yes \(1\) \(1\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{87}{100}\right)\)
\(\chi_{500}(49,\cdot)\) 500.i 10 no \(1\) \(1\) \(e\left(\frac{9}{10}\right)\) \(-1\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{500}(51,\cdot)\) 500.j 10 no \(-1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(-1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{3}{10}\right)\)
\(\chi_{500}(53,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{69}{100}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{13}{100}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{7}{100}\right)\)
\(\chi_{500}(57,\cdot)\) 500.f 4 no \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(-i\) \(i\) \(-1\) \(1\) \(-i\) \(i\)
\(\chi_{500}(59,\cdot)\) 500.n 50 yes \(-1\) \(1\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{16}{25}\right)\)
\(\chi_{500}(61,\cdot)\) 500.m 25 no \(1\) \(1\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{19}{25}\right)\)
\(\chi_{500}(63,\cdot)\) 500.r 100 yes \(1\) \(1\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{29}{100}\right)\)
\(\chi_{500}(67,\cdot)\) 500.r 100 yes \(1\) \(1\) \(e\left(\frac{41}{100}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{23}{100}\right)\)
\(\chi_{500}(69,\cdot)\) 500.o 50 no \(1\) \(1\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{49}{50}\right)\)
\(\chi_{500}(71,\cdot)\) 500.p 50 yes \(-1\) \(1\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{31}{50}\right)\)
\(\chi_{500}(73,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{37}{100}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{43}{100}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{11}{100}\right)\)
\(\chi_{500}(77,\cdot)\) 500.q 100 no \(-1\) \(1\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{53}{100}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{91}{100}\right)\) \(e\left(\frac{81}{100}\right)\)
\(\chi_{500}(79,\cdot)\) 500.n 50 yes \(-1\) \(1\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{41}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{18}{25}\right)\)
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