from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4003, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([8]))
pari: [g,chi] = znchar(Mod(1358,4003))
Basic properties
Modulus: | \(4003\) | |
Conductor: | \(4003\) | 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 4003.e
\(\chi_{4003}(551,\cdot)\) \(\chi_{4003}(703,\cdot)\) \(\chi_{4003}(835,\cdot)\) \(\chi_{4003}(848,\cdot)\) \(\chi_{4003}(1081,\cdot)\) \(\chi_{4003}(1173,\cdot)\) \(\chi_{4003}(1358,\cdot)\) \(\chi_{4003}(1840,\cdot)\) \(\chi_{4003}(1960,\cdot)\) \(\chi_{4003}(2567,\cdot)\) \(\chi_{4003}(2723,\cdot)\) \(\chi_{4003}(2784,\cdot)\) \(\chi_{4003}(2900,\cdot)\) \(\chi_{4003}(3065,\cdot)\) \(\chi_{4003}(3153,\cdot)\) \(\chi_{4003}(3187,\cdot)\) \(\chi_{4003}(3251,\cdot)\) \(\chi_{4003}(3376,\cdot)\) \(\chi_{4003}(3552,\cdot)\) \(\chi_{4003}(3688,\cdot)\) \(\chi_{4003}(3700,\cdot)\) \(\chi_{4003}(3743,\cdot)\)
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
\(2\) → \(e\left(\frac{4}{23}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4003 }(1358, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(1\) | \(e\left(\frac{5}{23}\right)\) |
sage: chi.jacobi_sum(n)