from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(277, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([40]))
pari: [g,chi] = znchar(Mod(52,277))
Basic properties
Modulus: | \(277\) | |
Conductor: | \(277\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 277.g
\(\chi_{277}(16,\cdot)\) \(\chi_{277}(19,\cdot)\) \(\chi_{277}(27,\cdot)\) \(\chi_{277}(30,\cdot)\) \(\chi_{277}(52,\cdot)\) \(\chi_{277}(69,\cdot)\) \(\chi_{277}(84,\cdot)\) \(\chi_{277}(131,\cdot)\) \(\chi_{277}(155,\cdot)\) \(\chi_{277}(157,\cdot)\) \(\chi_{277}(164,\cdot)\) \(\chi_{277}(169,\cdot)\) \(\chi_{277}(175,\cdot)\) \(\chi_{277}(201,\cdot)\) \(\chi_{277}(203,\cdot)\) \(\chi_{277}(211,\cdot)\) \(\chi_{277}(213,\cdot)\) \(\chi_{277}(218,\cdot)\) \(\chi_{277}(236,\cdot)\) \(\chi_{277}(256,\cdot)\) \(\chi_{277}(264,\cdot)\) \(\chi_{277}(273,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\(5\) → \(e\left(\frac{20}{23}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 277 }(52, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
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)