from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,6]))
pari: [g,chi] = znchar(Mod(529,9680))
Basic properties
| Modulus: | \(9680\) | |
| 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 \(\chi_{605}(529,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9680.dk
\(\chi_{9680}(529,\cdot)\) \(\chi_{9680}(1409,\cdot)\) \(\chi_{9680}(2289,\cdot)\) \(\chi_{9680}(3169,\cdot)\) \(\chi_{9680}(4049,\cdot)\) \(\chi_{9680}(4929,\cdot)\) \(\chi_{9680}(6689,\cdot)\) \(\chi_{9680}(7569,\cdot)\) \(\chi_{9680}(8449,\cdot)\) \(\chi_{9680}(9329,\cdot)\)
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
\((3631,2421,1937,4721)\) → \((1,1,-1,e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 9680 }(529, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{9}{22}\right)\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(-1\) | \(e\left(\frac{7}{11}\right)\) |
sage: chi.jacobi_sum(n)