from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9680, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,10]))
pari: [g,chi] = znchar(Mod(881,9680))
Basic properties
| Modulus: | \(9680\) | |
| Conductor: | \(121\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{121}(34,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9680.ce
\(\chi_{9680}(881,\cdot)\) \(\chi_{9680}(1761,\cdot)\) \(\chi_{9680}(2641,\cdot)\) \(\chi_{9680}(3521,\cdot)\) \(\chi_{9680}(4401,\cdot)\) \(\chi_{9680}(5281,\cdot)\) \(\chi_{9680}(6161,\cdot)\) \(\chi_{9680}(7041,\cdot)\) \(\chi_{9680}(7921,\cdot)\) \(\chi_{9680}(8801,\cdot)\)
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: | 11.11.672749994932560009201.1 |
Values on generators
\((3631,2421,1937,4721)\) → \((1,1,1,e\left(\frac{5}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 9680 }(881, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)