from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1005, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,3]))
pari: [g,chi] = znchar(Mod(311,1005))
Basic properties
| Modulus: | \(1005\) | |
| Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{201}(110,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1005.ba
\(\chi_{1005}(161,\cdot)\) \(\chi_{1005}(176,\cdot)\) \(\chi_{1005}(206,\cdot)\) \(\chi_{1005}(311,\cdot)\) \(\chi_{1005}(326,\cdot)\) \(\chi_{1005}(521,\cdot)\) \(\chi_{1005}(581,\cdot)\) \(\chi_{1005}(611,\cdot)\) \(\chi_{1005}(656,\cdot)\) \(\chi_{1005}(941,\cdot)\)
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: | 22.22.39437071573367006679286233687044038294749249.1 |
Values on generators
\((671,202,136)\) → \((-1,1,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1005 }(311, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)