from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,7,2]))
pari: [g,chi] = znchar(Mod(27,980))
Basic properties
Modulus: | \(980\) | |
Conductor: | \(980\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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 980.bj
\(\chi_{980}(27,\cdot)\) \(\chi_{980}(83,\cdot)\) \(\chi_{980}(167,\cdot)\) \(\chi_{980}(223,\cdot)\) \(\chi_{980}(307,\cdot)\) \(\chi_{980}(363,\cdot)\) \(\chi_{980}(447,\cdot)\) \(\chi_{980}(503,\cdot)\) \(\chi_{980}(643,\cdot)\) \(\chi_{980}(727,\cdot)\) \(\chi_{980}(867,\cdot)\) \(\chi_{980}(923,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.0.230203525458868754767395883592915116704159872000000000000000000000.1 |
Values on generators
\((491,197,101)\) → \((-1,i,e\left(\frac{1}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 980 }(27, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) |
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)