from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(723, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,1]))
pari: [g,chi] = znchar(Mod(41,723))
Basic properties
| Modulus: | \(723\) | |
| Conductor: | \(723\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 723.bc
\(\chi_{723}(5,\cdot)\) \(\chi_{723}(41,\cdot)\) \(\chi_{723}(47,\cdot)\) \(\chi_{723}(116,\cdot)\) \(\chi_{723}(125,\cdot)\) \(\chi_{723}(194,\cdot)\) \(\chi_{723}(200,\cdot)\) \(\chi_{723}(236,\cdot)\) \(\chi_{723}(302,\cdot)\) \(\chi_{723}(320,\cdot)\) \(\chi_{723}(434,\cdot)\) \(\chi_{723}(455,\cdot)\) \(\chi_{723}(509,\cdot)\) \(\chi_{723}(530,\cdot)\) \(\chi_{723}(644,\cdot)\) \(\chi_{723}(662,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.2761148438544725016155574079241161263031982418438872703421427975839546892558978425706480520807960995361.1 |
Values on generators
\((242,7)\) → \((-1,e\left(\frac{1}{40}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 723 }(41, a) \) | \(-1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(-i\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(1\) |
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)