from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(916, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,18]))
pari: [g,chi] = znchar(Mod(271,916))
Basic properties
Modulus: | \(916\) | |
Conductor: | \(916\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | 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 916.p
\(\chi_{916}(27,\cdot)\) \(\chi_{916}(43,\cdot)\) \(\chi_{916}(203,\cdot)\) \(\chi_{916}(271,\cdot)\) \(\chi_{916}(443,\cdot)\) \(\chi_{916}(447,\cdot)\) \(\chi_{916}(475,\cdot)\) \(\chi_{916}(511,\cdot)\) \(\chi_{916}(515,\cdot)\) \(\chi_{916}(519,\cdot)\) \(\chi_{916}(579,\cdot)\) \(\chi_{916}(619,\cdot)\) \(\chi_{916}(623,\cdot)\) \(\chi_{916}(683,\cdot)\) \(\chi_{916}(703,\cdot)\) \(\chi_{916}(731,\cdot)\) \(\chi_{916}(747,\cdot)\) \(\chi_{916}(791,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | 38.0.2472960613492762938009352687218362626942035203162587025151809700624254848124906369677396881178624.1 |
Values on generators
\((459,693)\) → \((-1,e\left(\frac{9}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 916 }(271, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) |
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)