from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([30,31]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,205))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(205\) | |
Conductor: | \(205\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | 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: | \(\Q(\zeta_{40})\) |
Fixed field: | 40.40.737344664758480175980241689422890987152982395783688920879217945970594882965087890625.1 |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{205}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) |
\(\chi_{205}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) |
\(\chi_{205}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
\(\chi_{205}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |
\(\chi_{205}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
\(\chi_{205}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) |
\(\chi_{205}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) |
\(\chi_{205}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) |
\(\chi_{205}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
\(\chi_{205}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
\(\chi_{205}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) |
\(\chi_{205}(142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) |
\(\chi_{205}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) |
\(\chi_{205}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
\(\chi_{205}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
\(\chi_{205}(198,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |