from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3822, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,4,21]))
pari: [g,chi] = znchar(Mod(239,3822))
Basic properties
Modulus: | \(3822\) | |
Conductor: | \(1911\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1911}(239,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3822.cu
\(\chi_{3822}(239,\cdot)\) \(\chi_{3822}(281,\cdot)\) \(\chi_{3822}(827,\cdot)\) \(\chi_{3822}(1331,\cdot)\) \(\chi_{3822}(1877,\cdot)\) \(\chi_{3822}(1919,\cdot)\) \(\chi_{3822}(2423,\cdot)\) \(\chi_{3822}(2465,\cdot)\) \(\chi_{3822}(2969,\cdot)\) \(\chi_{3822}(3011,\cdot)\) \(\chi_{3822}(3515,\cdot)\) \(\chi_{3822}(3557,\cdot)\)
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: | Number field defined by a degree 28 polynomial |
Values on generators
\((2549,3433,1471)\) → \((-1,e\left(\frac{1}{7}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3822 }(239, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-i\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(-i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
sage: chi.jacobi_sum(n)