Properties

Modulus $125$
Structure \(C_{100}\)
Order $100$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(125)
 
pari: g = idealstar(,125,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 100
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{100}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{125}(2,\cdot)$

First 32 of 100 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{125}(1,\cdot)\) 125.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{125}(2,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{1}{100}\right)\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{39}{100}\right)\)
\(\chi_{125}(3,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{7}{100}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{73}{100}\right)\)
\(\chi_{125}(4,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{39}{50}\right)\)
\(\chi_{125}(6,\cdot)\) 125.g 25 yes \(1\) \(1\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{3}{25}\right)\)
\(\chi_{125}(7,\cdot)\) 125.f 20 no \(-1\) \(1\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(i\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{3}{20}\right)\)
\(\chi_{125}(8,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{17}{100}\right)\)
\(\chi_{125}(9,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{49}{50}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{23}{50}\right)\)
\(\chi_{125}(11,\cdot)\) 125.g 25 yes \(1\) \(1\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{16}{25}\right)\)
\(\chi_{125}(12,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{27}{100}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{51}{100}\right)\)
\(\chi_{125}(13,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{51}{100}\right)\) \(e\left(\frac{21}{100}\right)\)
\(\chi_{125}(14,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{1}{50}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{27}{50}\right)\)
\(\chi_{125}(16,\cdot)\) 125.g 25 yes \(1\) \(1\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{14}{25}\right)\)
\(\chi_{125}(17,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{73}{100}\right)\) \(e\left(\frac{11}{100}\right)\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{47}{100}\right)\)
\(\chi_{125}(18,\cdot)\) 125.f 20 no \(-1\) \(1\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(-i\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{17}{20}\right)\)
\(\chi_{125}(19,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{13}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{1}{50}\right)\)
\(\chi_{125}(21,\cdot)\) 125.g 25 yes \(1\) \(1\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{19}{25}\right)\) \(e\left(\frac{22}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{22}{25}\right)\)
\(\chi_{125}(22,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{77}{100}\right)\) \(e\left(\frac{39}{100}\right)\) \(e\left(\frac{27}{50}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{39}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{3}{100}\right)\)
\(\chi_{125}(23,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{31}{100}\right)\) \(e\left(\frac{17}{100}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{93}{100}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{14}{25}\right)\) \(e\left(\frac{79}{100}\right)\) \(e\left(\frac{9}{100}\right)\)
\(\chi_{125}(24,\cdot)\) 125.e 10 no \(1\) \(1\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(-1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{9}{10}\right)\)
\(\chi_{125}(26,\cdot)\) 125.d 5 no \(1\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{3}{5}\right)\)
\(\chi_{125}(27,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{21}{100}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{21}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{63}{100}\right)\) \(e\left(\frac{47}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{89}{100}\right)\) \(e\left(\frac{19}{100}\right)\)
\(\chi_{125}(28,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{9}{100}\right)\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{93}{100}\right)\)
\(\chi_{125}(29,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{17}{50}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{43}{50}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{9}{50}\right)\)
\(\chi_{125}(31,\cdot)\) 125.g 25 yes \(1\) \(1\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{21}{25}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{18}{25}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{18}{25}\right)\)
\(\chi_{125}(32,\cdot)\) 125.f 20 no \(-1\) \(1\) \(e\left(\frac{1}{20}\right)\) \(e\left(\frac{7}{20}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{2}{5}\right)\) \(i\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{9}{20}\right)\) \(e\left(\frac{19}{20}\right)\)
\(\chi_{125}(33,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{83}{100}\right)\) \(e\left(\frac{81}{100}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{16}{25}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{49}{100}\right)\) \(e\left(\frac{31}{50}\right)\) \(e\left(\frac{2}{25}\right)\) \(e\left(\frac{47}{100}\right)\) \(e\left(\frac{37}{100}\right)\)
\(\chi_{125}(34,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{37}{50}\right)\) \(e\left(\frac{9}{50}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{9}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{43}{50}\right)\)
\(\chi_{125}(36,\cdot)\) 125.g 25 yes \(1\) \(1\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{3}{25}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{7}{25}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{12}{25}\right)\) \(e\left(\frac{6}{25}\right)\) \(e\left(\frac{4}{25}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{6}{25}\right)\)
\(\chi_{125}(37,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{29}{100}\right)\) \(e\left(\frac{3}{100}\right)\) \(e\left(\frac{29}{50}\right)\) \(e\left(\frac{8}{25}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{87}{100}\right)\) \(e\left(\frac{3}{50}\right)\) \(e\left(\frac{1}{25}\right)\) \(e\left(\frac{61}{100}\right)\) \(e\left(\frac{31}{100}\right)\)
\(\chi_{125}(38,\cdot)\) 125.i 100 yes \(-1\) \(1\) \(e\left(\frac{19}{100}\right)\) \(e\left(\frac{33}{100}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{13}{25}\right)\) \(e\left(\frac{3}{20}\right)\) \(e\left(\frac{57}{100}\right)\) \(e\left(\frac{33}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{71}{100}\right)\) \(e\left(\frac{41}{100}\right)\)
\(\chi_{125}(39,\cdot)\) 125.h 50 yes \(1\) \(1\) \(e\left(\frac{23}{50}\right)\) \(e\left(\frac{11}{50}\right)\) \(e\left(\frac{23}{25}\right)\) \(e\left(\frac{17}{25}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{19}{50}\right)\) \(e\left(\frac{11}{25}\right)\) \(e\left(\frac{24}{25}\right)\) \(e\left(\frac{7}{50}\right)\) \(e\left(\frac{47}{50}\right)\)
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