from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([3]))
pari: [g,chi] = znchar(Mod(94,361))
Basic properties
Modulus: | \(361\) | |
Conductor: | \(361\) | 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 361.h
\(\chi_{361}(18,\cdot)\) \(\chi_{361}(37,\cdot)\) \(\chi_{361}(56,\cdot)\) \(\chi_{361}(75,\cdot)\) \(\chi_{361}(94,\cdot)\) \(\chi_{361}(113,\cdot)\) \(\chi_{361}(132,\cdot)\) \(\chi_{361}(151,\cdot)\) \(\chi_{361}(170,\cdot)\) \(\chi_{361}(189,\cdot)\) \(\chi_{361}(208,\cdot)\) \(\chi_{361}(227,\cdot)\) \(\chi_{361}(246,\cdot)\) \(\chi_{361}(265,\cdot)\) \(\chi_{361}(284,\cdot)\) \(\chi_{361}(303,\cdot)\) \(\chi_{361}(322,\cdot)\) \(\chi_{361}(341,\cdot)\)
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: | 38.0.2233638411813024816853081773648251688534529753590642239923912316757382599022775822751448518259.1 |
Values on generators
\(2\) → \(e\left(\frac{3}{38}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 361 }(94, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) |
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)