from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3380, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,5]))
pari: [g,chi] = znchar(Mod(2859,3380))
Basic properties
| Modulus: | \(3380\) | |
| Conductor: | \(3380\) | 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 3380.bq
\(\chi_{3380}(259,\cdot)\) \(\chi_{3380}(519,\cdot)\) \(\chi_{3380}(779,\cdot)\) \(\chi_{3380}(1039,\cdot)\) \(\chi_{3380}(1299,\cdot)\) \(\chi_{3380}(1559,\cdot)\) \(\chi_{3380}(1819,\cdot)\) \(\chi_{3380}(2079,\cdot)\) \(\chi_{3380}(2339,\cdot)\) \(\chi_{3380}(2599,\cdot)\) \(\chi_{3380}(2859,\cdot)\) \(\chi_{3380}(3119,\cdot)\)
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: | 26.0.313772014972693001778186596260839539421506588386702473052160000000000000.1 |
Values on generators
\((1691,677,1861)\) → \((-1,-1,e\left(\frac{5}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 3380 }(2859, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) |
sage: chi.jacobi_sum(n)