from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(115, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,4]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,115))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(115\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | 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: | \(\Q(\zeta_{44})\) |
Fixed field: | 44.0.342865339180420288801608222738062084913425127327306009945459663867950439453125.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{115}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) |
\(\chi_{115}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) |
\(\chi_{115}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) |
\(\chi_{115}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) |
\(\chi_{115}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) |
\(\chi_{115}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) |
\(\chi_{115}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) |
\(\chi_{115}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) |
\(\chi_{115}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) |
\(\chi_{115}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) |
\(\chi_{115}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) |
\(\chi_{115}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) |
\(\chi_{115}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) |
\(\chi_{115}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) |
\(\chi_{115}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) |
\(\chi_{115}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) |
\(\chi_{115}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) |
\(\chi_{115}(87,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) |
\(\chi_{115}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) |
\(\chi_{115}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) |