Properties

Modulus $475$
Structure \(C_{180}\times C_{2}\)
Order $360$

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Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(475)
 
pari: g = idealstar(,475,2)
 

Character group

sage: G.order()
 
pari: g.no
 
Order = 360
sage: H.invariants()
 
pari: g.cyc
 
Structure = \(C_{180}\times C_{2}\)
sage: H.gens()
 
pari: g.gen
 
Generators = $\chi_{475}(77,\cdot)$, $\chi_{475}(401,\cdot)$

First 32 of 360 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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{475}(1,\cdot)\) 475.a 1 no \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(\chi_{475}(2,\cdot)\) 475.bi 180 yes \(1\) \(1\) \(e\left(\frac{19}{180}\right)\) \(e\left(\frac{13}{180}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{41}{180}\right)\)
\(\chi_{475}(3,\cdot)\) 475.bi 180 yes \(1\) \(1\) \(e\left(\frac{13}{180}\right)\) \(e\left(\frac{151}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{47}{180}\right)\)
\(\chi_{475}(4,\cdot)\) 475.bg 90 yes \(1\) \(1\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{41}{90}\right)\)
\(\chi_{475}(6,\cdot)\) 475.bc 45 yes \(1\) \(1\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{22}{45}\right)\)
\(\chi_{475}(7,\cdot)\) 475.q 12 no \(-1\) \(1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{3}\right)\) \(i\) \(-i\) \(e\left(\frac{1}{6}\right)\) \(1\) \(i\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{475}(8,\cdot)\) 475.be 60 yes \(1\) \(1\) \(e\left(\frac{19}{60}\right)\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{8}{15}\right)\) \(-i\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{41}{60}\right)\)
\(\chi_{475}(9,\cdot)\) 475.bg 90 yes \(1\) \(1\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{47}{90}\right)\)
\(\chi_{475}(11,\cdot)\) 475.r 15 yes \(1\) \(1\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{11}{15}\right)\) \(1\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{8}{15}\right)\)
\(\chi_{475}(12,\cdot)\) 475.be 60 yes \(1\) \(1\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{17}{30}\right)\) \(e\left(\frac{4}{15}\right)\) \(i\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{43}{60}\right)\)
\(\chi_{475}(13,\cdot)\) 475.bi 180 yes \(1\) \(1\) \(e\left(\frac{41}{180}\right)\) \(e\left(\frac{47}{180}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{22}{45}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{41}{60}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{79}{180}\right)\)
\(\chi_{475}(14,\cdot)\) 475.bh 90 yes \(-1\) \(1\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{17}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{29}{45}\right)\)
\(\chi_{475}(16,\cdot)\) 475.bc 45 yes \(1\) \(1\) \(e\left(\frac{19}{45}\right)\) \(e\left(\frac{13}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{41}{45}\right)\)
\(\chi_{475}(17,\cdot)\) 475.bj 180 yes \(-1\) \(1\) \(e\left(\frac{37}{180}\right)\) \(e\left(\frac{139}{180}\right)\) \(e\left(\frac{37}{90}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{49}{90}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{23}{180}\right)\)
\(\chi_{475}(18,\cdot)\) 475.g 4 no \(1\) \(1\) \(i\) \(-i\) \(-1\) \(1\) \(-i\) \(-i\) \(-1\) \(1\) \(i\) \(-i\)
\(\chi_{475}(21,\cdot)\) 475.bf 90 yes \(-1\) \(1\) \(e\left(\frac{59}{90}\right)\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{30}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{61}{90}\right)\)
\(\chi_{475}(22,\cdot)\) 475.bi 180 yes \(1\) \(1\) \(e\left(\frac{103}{180}\right)\) \(e\left(\frac{61}{180}\right)\) \(e\left(\frac{13}{90}\right)\) \(e\left(\frac{41}{45}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{43}{60}\right)\) \(e\left(\frac{61}{90}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{29}{60}\right)\) \(e\left(\frac{137}{180}\right)\)
\(\chi_{475}(23,\cdot)\) 475.bj 180 yes \(-1\) \(1\) \(e\left(\frac{119}{180}\right)\) \(e\left(\frac{53}{180}\right)\) \(e\left(\frac{29}{90}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{59}{60}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{1}{180}\right)\)
\(\chi_{475}(24,\cdot)\) 475.u 18 no \(1\) \(1\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{17}{18}\right)\)
\(\chi_{475}(26,\cdot)\) 475.e 3 no \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)
\(\chi_{475}(27,\cdot)\) 475.be 60 yes \(1\) \(1\) \(e\left(\frac{13}{60}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{13}{30}\right)\) \(e\left(\frac{11}{15}\right)\) \(i\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{1}{30}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{47}{60}\right)\)
\(\chi_{475}(28,\cdot)\) 475.bj 180 yes \(-1\) \(1\) \(e\left(\frac{143}{180}\right)\) \(e\left(\frac{41}{180}\right)\) \(e\left(\frac{53}{90}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{23}{60}\right)\) \(e\left(\frac{41}{90}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{49}{60}\right)\) \(e\left(\frac{157}{180}\right)\)
\(\chi_{475}(29,\cdot)\) 475.bh 90 yes \(-1\) \(1\) \(e\left(\frac{2}{45}\right)\) \(e\left(\frac{44}{45}\right)\) \(e\left(\frac{4}{45}\right)\) \(e\left(\frac{1}{45}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{2}{15}\right)\) \(e\left(\frac{43}{45}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{28}{45}\right)\)
\(\chi_{475}(31,\cdot)\) 475.y 30 yes \(-1\) \(1\) \(e\left(\frac{7}{30}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{13}{15}\right)\) \(1\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{4}{15}\right)\) \(e\left(\frac{2}{5}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{23}{30}\right)\)
\(\chi_{475}(32,\cdot)\) 475.bb 36 no \(1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{36}\right)\)
\(\chi_{475}(33,\cdot)\) 475.bi 180 yes \(1\) \(1\) \(e\left(\frac{97}{180}\right)\) \(e\left(\frac{19}{180}\right)\) \(e\left(\frac{7}{90}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{37}{60}\right)\) \(e\left(\frac{19}{90}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{11}{60}\right)\) \(e\left(\frac{143}{180}\right)\)
\(\chi_{475}(34,\cdot)\) 475.bh 90 yes \(-1\) \(1\) \(e\left(\frac{14}{45}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{28}{45}\right)\) \(e\left(\frac{7}{45}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{14}{15}\right)\) \(e\left(\frac{31}{45}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{16}{45}\right)\)
\(\chi_{475}(36,\cdot)\) 475.bc 45 yes \(1\) \(1\) \(e\left(\frac{16}{45}\right)\) \(e\left(\frac{37}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{8}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{15}\right)\) \(e\left(\frac{29}{45}\right)\) \(e\left(\frac{7}{15}\right)\) \(e\left(\frac{8}{15}\right)\) \(e\left(\frac{44}{45}\right)\)
\(\chi_{475}(37,\cdot)\) 475.v 20 yes \(1\) \(1\) \(e\left(\frac{19}{20}\right)\) \(e\left(\frac{13}{20}\right)\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(i\) \(e\left(\frac{17}{20}\right)\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{11}{20}\right)\) \(e\left(\frac{1}{20}\right)\)
\(\chi_{475}(39,\cdot)\) 475.n 10 no \(1\) \(1\) \(e\left(\frac{3}{10}\right)\) \(e\left(\frac{1}{10}\right)\) \(e\left(\frac{3}{5}\right)\) \(e\left(\frac{2}{5}\right)\) \(-1\) \(e\left(\frac{9}{10}\right)\) \(e\left(\frac{1}{5}\right)\) \(e\left(\frac{4}{5}\right)\) \(e\left(\frac{7}{10}\right)\) \(e\left(\frac{7}{10}\right)\)
\(\chi_{475}(41,\cdot)\) 475.bf 90 yes \(-1\) \(1\) \(e\left(\frac{83}{90}\right)\) \(e\left(\frac{71}{90}\right)\) \(e\left(\frac{38}{45}\right)\) \(e\left(\frac{32}{45}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{30}\right)\) \(e\left(\frac{26}{45}\right)\) \(e\left(\frac{13}{15}\right)\) \(e\left(\frac{19}{30}\right)\) \(e\left(\frac{37}{90}\right)\)
\(\chi_{475}(42,\cdot)\) 475.bj 180 yes \(-1\) \(1\) \(e\left(\frac{137}{180}\right)\) \(e\left(\frac{179}{180}\right)\) \(e\left(\frac{47}{90}\right)\) \(e\left(\frac{34}{45}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{17}{60}\right)\) \(e\left(\frac{89}{90}\right)\) \(e\left(\frac{11}{15}\right)\) \(e\left(\frac{31}{60}\right)\) \(e\left(\frac{163}{180}\right)\)