from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,15]))
pari: [g,chi] = znchar(Mod(1331,3015))
Basic properties
Modulus: | \(3015\) | |
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 \(\chi_{201}(125,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.ck
\(\chi_{3015}(161,\cdot)\) \(\chi_{3015}(206,\cdot)\) \(\chi_{3015}(521,\cdot)\) \(\chi_{3015}(611,\cdot)\) \(\chi_{3015}(656,\cdot)\) \(\chi_{3015}(1331,\cdot)\) \(\chi_{3015}(2186,\cdot)\) \(\chi_{3015}(2321,\cdot)\) \(\chi_{3015}(2591,\cdot)\) \(\chi_{3015}(2951,\cdot)\)
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
\((1676,1207,136)\) → \((-1,1,e\left(\frac{15}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3015 }(1331, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) |
sage: chi.jacobi_sum(n)