from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,0,17]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,8470))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(8470\) | |
Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(110\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 605.v | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
Field of values: | $\Q(\zeta_{55})$ |
Fixed field: | Number field defined by a degree 110 polynomial (not computed) |
First 31 of 40 characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{8470}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) |
\(\chi_{8470}(519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) |
\(\chi_{8470}(589,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) |
\(\chi_{8470}(799,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) |
\(\chi_{8470}(1009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{31}{110}\right)\) |
\(\chi_{8470}(1289,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) |
\(\chi_{8470}(1359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) |
\(\chi_{8470}(1569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) |
\(\chi_{8470}(1779,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) |
\(\chi_{8470}(2059,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) |
\(\chi_{8470}(2129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) |
\(\chi_{8470}(2549,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{71}{110}\right)\) |
\(\chi_{8470}(2829,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) |
\(\chi_{8470}(2899,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) |
\(\chi_{8470}(3109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) |
\(\chi_{8470}(3319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) |
\(\chi_{8470}(3599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) |
\(\chi_{8470}(3669,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) |
\(\chi_{8470}(3879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) |
\(\chi_{8470}(4089,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) |
\(\chi_{8470}(4369,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) |
\(\chi_{8470}(4439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) |
\(\chi_{8470}(4649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) |
\(\chi_{8470}(4859,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) |
\(\chi_{8470}(5139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) |
\(\chi_{8470}(5209,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) |
\(\chi_{8470}(5419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) |
\(\chi_{8470}(5629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) |
\(\chi_{8470}(5909,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) |
\(\chi_{8470}(5979,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) |
\(\chi_{8470}(6189,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) |