from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(178, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([6]))
pari: [g,chi] = znchar(Mod(39,178))
Basic properties
| Modulus: | \(178\) | |
| Conductor: | \(89\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{89}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 178.e
\(\chi_{178}(39,\cdot)\) \(\chi_{178}(45,\cdot)\) \(\chi_{178}(67,\cdot)\) \(\chi_{178}(91,\cdot)\) \(\chi_{178}(93,\cdot)\) \(\chi_{178}(97,\cdot)\) \(\chi_{178}(105,\cdot)\) \(\chi_{178}(121,\cdot)\) \(\chi_{178}(153,\cdot)\) \(\chi_{178}(167,\cdot)\)
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: | 11.11.31181719929966183601.1 |
Values on generators
\(3\) → \(e\left(\frac{3}{11}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 178 }(39, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)