from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6900, base_ring=CyclotomicField(110))
M = H._module
chi = DirichletCharacter(H, M([55,55,77,70]))
chi.galois_orbit()
[g,chi] = znchar(Mod(59,6900))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6900\) | |
| Conductor: | \(6900\) | 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: | 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 110 polynomial (not computed) |
First 31 of 40 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6900}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{6900}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{6900}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{6900}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{6900}(719,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{51}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{6900}(959,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{26}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{6900}(1139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{42}{55}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{6900}(1319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{6900}(1439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{47}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{6900}(1559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{6}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{31}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{6900}(1619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{21}{55}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{6900}(2279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{23}{55}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{24}{55}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{6900}(2339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{52}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{6900}(2519,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{32}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{6900}(2579,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{49}{55}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{73}{110}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{6900}(2819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{57}{110}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{2}{11}\right)\) |
| \(\chi_{6900}(2879,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{4}{55}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{6900}(2939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{37}{110}\right)\) | \(e\left(\frac{12}{55}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{101}{110}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{27}{110}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{6900}(3479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{53}{55}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{29}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{9}{110}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{6900}(3659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{13}{55}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{59}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{23}{110}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{6900}(3719,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{37}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{17}{110}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{9}{11}\right)\) |
| \(\chi_{6900}(3959,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{41}{55}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{51}{110}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{6900}(4079,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{55}\right)\) | \(e\left(\frac{79}{110}\right)\) | \(e\left(\frac{39}{55}\right)\) | \(e\left(\frac{43}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) |
| \(\chi_{6900}(4259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{46}{55}\right)\) | \(e\left(\frac{93}{110}\right)\) | \(e\left(\frac{48}{55}\right)\) | \(e\left(\frac{91}{110}\right)\) | \(e\left(\frac{19}{110}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{10}{11}\right)\) |
| \(\chi_{6900}(4319,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{17}{55}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{1}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{61}{110}\right)\) | \(e\left(\frac{71}{110}\right)\) | \(e\left(\frac{8}{11}\right)\) |
| \(\chi_{6900}(4379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{38}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{54}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{39}{110}\right)\) | \(e\left(\frac{3}{11}\right)\) |
| \(\chi_{6900}(4859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{31}{55}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{18}{55}\right)\) | \(e\left(\frac{41}{110}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{1}{110}\right)\) | \(e\left(\frac{53}{110}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{1}{11}\right)\) |
| \(\chi_{6900}(5039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{34}{55}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{2}{55}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{49}{110}\right)\) | \(e\left(\frac{67}{110}\right)\) | \(e\left(\frac{87}{110}\right)\) | \(e\left(\frac{5}{11}\right)\) |
| \(\chi_{6900}(5279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{43}{55}\right)\) | \(e\left(\frac{69}{110}\right)\) | \(e\left(\frac{9}{55}\right)\) | \(e\left(\frac{103}{110}\right)\) | \(e\left(\frac{7}{110}\right)\) | \(e\left(\frac{83}{110}\right)\) | \(e\left(\frac{109}{110}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{6}{11}\right)\) |
| \(\chi_{6900}(5339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{55}\right)\) | \(e\left(\frac{97}{110}\right)\) | \(e\left(\frac{27}{55}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{29}{110}\right)\) | \(e\left(\frac{107}{110}\right)\) | \(e\left(\frac{47}{110}\right)\) | \(e\left(\frac{7}{11}\right)\) |
| \(\chi_{6900}(5459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{36}{55}\right)\) | \(e\left(\frac{13}{110}\right)\) | \(e\left(\frac{28}{55}\right)\) | \(e\left(\frac{21}{110}\right)\) | \(e\left(\frac{89}{110}\right)\) | \(e\left(\frac{81}{110}\right)\) | \(e\left(\frac{3}{110}\right)\) | \(e\left(\frac{63}{110}\right)\) | \(e\left(\frac{4}{11}\right)\) |