from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4600, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([0,0,22,60]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,4600))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4600\) | |
| Conductor: | \(575\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(55\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 575.s | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | 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 55 polynomial |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4600}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) |
| \(\chi_{4600}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) |
| \(\chi_{4600}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) |
| \(\chi_{4600}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) |
| \(\chi_{4600}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) |
| \(\chi_{4600}(561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) |
| \(\chi_{4600}(721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) |
| \(\chi_{4600}(761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) |
| \(\chi_{4600}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) |
| \(\chi_{4600}(961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) |
| \(\chi_{4600}(1041,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) |
| \(\chi_{4600}(1281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) |
| \(\chi_{4600}(1361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) |
| \(\chi_{4600}(1481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) |
| \(\chi_{4600}(1521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) |
| \(\chi_{4600}(1641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) |
| \(\chi_{4600}(1681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) |
| \(\chi_{4600}(1761,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) |
| \(\chi_{4600}(1881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) |
| \(\chi_{4600}(1921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) |
| \(\chi_{4600}(1961,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) |
| \(\chi_{4600}(2281,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) |
| \(\chi_{4600}(2441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) |
| \(\chi_{4600}(2561,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) |
| \(\chi_{4600}(2681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) |
| \(\chi_{4600}(2841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) |
| \(\chi_{4600}(2881,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) |
| \(\chi_{4600}(3121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) |
| \(\chi_{4600}(3321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) |
| \(\chi_{4600}(3361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) |
| \(\chi_{4600}(3481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) |