from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3549, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,18]))
pari: [g,chi] = znchar(Mod(1366,3549))
Basic properties
| Modulus: | \(3549\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(14,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3549.ci
\(\chi_{3549}(274,\cdot)\) \(\chi_{3549}(547,\cdot)\) \(\chi_{3549}(820,\cdot)\) \(\chi_{3549}(1093,\cdot)\) \(\chi_{3549}(1366,\cdot)\) \(\chi_{3549}(1639,\cdot)\) \(\chi_{3549}(1912,\cdot)\) \(\chi_{3549}(2185,\cdot)\) \(\chi_{3549}(2458,\cdot)\) \(\chi_{3549}(2731,\cdot)\) \(\chi_{3549}(3004,\cdot)\) \(\chi_{3549}(3277,\cdot)\)
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: | 13.13.542800770374370512771595361.1 |
Values on generators
\((1184,1522,3382)\) → \((1,1,e\left(\frac{9}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 3549 }(1366, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(1\) | \(e\left(\frac{8}{13}\right)\) |
sage: chi.jacobi_sum(n)