from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,10]))
pari: [g,chi] = znchar(Mod(188,735))
Basic properties
Modulus: | \(735\) | |
Conductor: | \(735\) | 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 735.bk
\(\chi_{735}(62,\cdot)\) \(\chi_{735}(83,\cdot)\) \(\chi_{735}(167,\cdot)\) \(\chi_{735}(188,\cdot)\) \(\chi_{735}(272,\cdot)\) \(\chi_{735}(377,\cdot)\) \(\chi_{735}(398,\cdot)\) \(\chi_{735}(482,\cdot)\) \(\chi_{735}(503,\cdot)\) \(\chi_{735}(608,\cdot)\) \(\chi_{735}(692,\cdot)\) \(\chi_{735}(713,\cdot)\)
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: | 28.0.4101754449160695184473159618498838032884071911945819854736328125.1 |
Values on generators
\((491,442,346)\) → \((-1,-i,e\left(\frac{5}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
\( \chi_{ 735 }(188, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) |
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)