from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(599, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([22]))
pari: [g,chi] = znchar(Mod(102,599))
Basic properties
Modulus: | \(599\) | |
Conductor: | \(599\) | 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 599.d
\(\chi_{599}(18,\cdot)\) \(\chi_{599}(39,\cdot)\) \(\chi_{599}(57,\cdot)\) \(\chi_{599}(102,\cdot)\) \(\chi_{599}(103,\cdot)\) \(\chi_{599}(151,\cdot)\) \(\chi_{599}(221,\cdot)\) \(\chi_{599}(233,\cdot)\) \(\chi_{599}(254,\cdot)\) \(\chi_{599}(322,\cdot)\) \(\chi_{599}(323,\cdot)\) \(\chi_{599}(324,\cdot)\) \(\chi_{599}(379,\cdot)\) \(\chi_{599}(384,\cdot)\) \(\chi_{599}(405,\cdot)\) \(\chi_{599}(423,\cdot)\) \(\chi_{599}(426,\cdot)\) \(\chi_{599}(427,\cdot)\) \(\chi_{599}(441,\cdot)\) \(\chi_{599}(480,\cdot)\) \(\chi_{599}(498,\cdot)\) \(\chi_{599}(578,\cdot)\)
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
\(7\) → \(e\left(\frac{11}{23}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 599 }(102, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
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)