from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3484, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,2]))
pari: [g,chi] = znchar(Mod(131,3484))
Basic properties
| Modulus: | \(3484\) | |
| Conductor: | \(268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{268}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3484.cn
\(\chi_{3484}(131,\cdot)\) \(\chi_{3484}(911,\cdot)\) \(\chi_{3484}(963,\cdot)\) \(\chi_{3484}(1067,\cdot)\) \(\chi_{3484}(1431,\cdot)\) \(\chi_{3484}(1483,\cdot)\) \(\chi_{3484}(2159,\cdot)\) \(\chi_{3484}(2367,\cdot)\) \(\chi_{3484}(2471,\cdot)\) \(\chi_{3484}(2627,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.0.13936571865431899047915848208912554030792704.1 |
Values on generators
\((1743,1341,3017)\) → \((-1,1,e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3484 }(131, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)