from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,20]))
pari: [g,chi] = znchar(Mod(309,4235))
Basic properties
Modulus: | \(4235\) | |
Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{605}(309,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4235.bq
\(\chi_{4235}(309,\cdot)\) \(\chi_{4235}(694,\cdot)\) \(\chi_{4235}(1079,\cdot)\) \(\chi_{4235}(1464,\cdot)\) \(\chi_{4235}(1849,\cdot)\) \(\chi_{4235}(2234,\cdot)\) \(\chi_{4235}(2619,\cdot)\) \(\chi_{4235}(3004,\cdot)\) \(\chi_{4235}(3774,\cdot)\) \(\chi_{4235}(4159,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((2542,1816,2906)\) → \((-1,1,e\left(\frac{10}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 4235 }(309, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(-1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)