from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,34]))
pari: [g,chi] = znchar(Mod(67,380))
Basic properties
| Modulus: | \(380\) | |
| Conductor: | \(380\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 380.bi
\(\chi_{380}(3,\cdot)\) \(\chi_{380}(67,\cdot)\) \(\chi_{380}(127,\cdot)\) \(\chi_{380}(143,\cdot)\) \(\chi_{380}(147,\cdot)\) \(\chi_{380}(167,\cdot)\) \(\chi_{380}(203,\cdot)\) \(\chi_{380}(223,\cdot)\) \(\chi_{380}(243,\cdot)\) \(\chi_{380}(287,\cdot)\) \(\chi_{380}(307,\cdot)\) \(\chi_{380}(363,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.0.15377735736821953341412273192192323957407601152000000000000000000000000000.1 |
Values on generators
\((191,77,21)\) → \((-1,i,e\left(\frac{17}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 380 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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)