from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(759, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,99,35]))
chi.galois_orbit()
[g,chi] = znchar(Mod(17,759))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(759\) | |
| Conductor: | \(759\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(110\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{759}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) |
| \(\chi_{759}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) |
| \(\chi_{759}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{31}{110}\right)\) |
| \(\chi_{759}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) |
| \(\chi_{759}(134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) |
| \(\chi_{759}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) |
| \(\chi_{759}(182,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{81}{110}\right)\) |
| \(\chi_{759}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) |
| \(\chi_{759}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) |
| \(\chi_{759}(260,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) |
| \(\chi_{759}(272,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) |
| \(\chi_{759}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) |
| \(\chi_{759}(293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) |
| \(\chi_{759}(314,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) |
| \(\chi_{759}(332,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) |
| \(\chi_{759}(359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) |
| \(\chi_{759}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{109}{110}\right)\) |
| \(\chi_{759}(398,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) |
| \(\chi_{759}(425,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) |
| \(\chi_{759}(431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) |
| \(\chi_{759}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) |
| \(\chi_{759}(470,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) |
| \(\chi_{759}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) |
| \(\chi_{759}(497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) |
| \(\chi_{759}(503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) |
| \(\chi_{759}(536,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) |
| \(\chi_{759}(557,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) |
| \(\chi_{759}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{14}{55}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) |
| \(\chi_{759}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) |
| \(\chi_{759}(590,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) |
| \(\chi_{759}(596,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) |