from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,3]))
pari: [g,chi] = znchar(Mod(98,1089))
Basic properties
| Modulus: | \(1089\) | |
| Conductor: | \(363\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{363}(98,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1089.o
\(\chi_{1089}(98,\cdot)\) \(\chi_{1089}(197,\cdot)\) \(\chi_{1089}(296,\cdot)\) \(\chi_{1089}(395,\cdot)\) \(\chi_{1089}(494,\cdot)\) \(\chi_{1089}(593,\cdot)\) \(\chi_{1089}(692,\cdot)\) \(\chi_{1089}(791,\cdot)\) \(\chi_{1089}(890,\cdot)\) \(\chi_{1089}(989,\cdot)\)
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
\((848,244)\) → \((-1,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1089 }(98, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
sage: chi.jacobi_sum(n)