from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1740, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,14,21,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,1740))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(1740\) | |
Conductor: | \(1740\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(31\) | \(37\) | \(41\) | \(43\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{1740}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(-1\) | \(e\left(\frac{1}{28}\right)\) |
\(\chi_{1740}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-1\) | \(e\left(\frac{5}{28}\right)\) |
\(\chi_{1740}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(-1\) | \(e\left(\frac{11}{28}\right)\) |
\(\chi_{1740}(227,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(-i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(-1\) | \(e\left(\frac{27}{28}\right)\) |
\(\chi_{1740}(587,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(-1\) | \(e\left(\frac{23}{28}\right)\) |
\(\chi_{1740}(683,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(i\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-1\) | \(e\left(\frac{17}{28}\right)\) |
\(\chi_{1740}(803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-1\) | \(e\left(\frac{25}{28}\right)\) |
\(\chi_{1740}(923,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(i\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-1\) | \(e\left(\frac{13}{28}\right)\) |
\(\chi_{1740}(1067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-1\) | \(e\left(\frac{15}{28}\right)\) |
\(\chi_{1740}(1127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(-1\) | \(e\left(\frac{19}{28}\right)\) |
\(\chi_{1740}(1283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(i\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-1\) | \(e\left(\frac{9}{28}\right)\) |
\(\chi_{1740}(1727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(-1\) | \(e\left(\frac{3}{28}\right)\) |