from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1183, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,17]))
pari: [g,chi] = znchar(Mod(90,1183))
Basic properties
| Modulus: | \(1183\) | |
| Conductor: | \(1183\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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)
|
Galois orbit 1183.bh
\(\chi_{1183}(90,\cdot)\) \(\chi_{1183}(181,\cdot)\) \(\chi_{1183}(272,\cdot)\) \(\chi_{1183}(363,\cdot)\) \(\chi_{1183}(454,\cdot)\) \(\chi_{1183}(545,\cdot)\) \(\chi_{1183}(636,\cdot)\) \(\chi_{1183}(727,\cdot)\) \(\chi_{1183}(818,\cdot)\) \(\chi_{1183}(909,\cdot)\) \(\chi_{1183}(1000,\cdot)\) \(\chi_{1183}(1091,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.0.371106689747492823121371955941215691181243146994852066064005948411.1 |
Values on generators
\((339,1016)\) → \((-1,e\left(\frac{17}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1183 }(90, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
sage: chi.jacobi_sum(n)