from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4100, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,32,7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(111,4100))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(4100\) | |
Conductor: | \(4100\) | 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: | Number field defined by a degree 40 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{4100}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) |
\(\chi_{4100}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) |
\(\chi_{4100}(591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) |
\(\chi_{4100}(891,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) |
\(\chi_{4100}(1031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) |
\(\chi_{4100}(1611,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) |
\(\chi_{4100}(2311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) |
\(\chi_{4100}(2331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) |
\(\chi_{4100}(2371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) |
\(\chi_{4100}(2971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) |
\(\chi_{4100}(3291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) |
\(\chi_{4100}(3431,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) |
\(\chi_{4100}(3511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) |
\(\chi_{4100}(3591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) |
\(\chi_{4100}(3671,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) |
\(\chi_{4100}(4031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) |