from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,9]))
pari: [g,chi] = znchar(Mod(1942,3381))
Basic properties
| Modulus: | \(3381\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(10,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3381.bv
\(\chi_{3381}(19,\cdot)\) \(\chi_{3381}(166,\cdot)\) \(\chi_{3381}(178,\cdot)\) \(\chi_{3381}(313,\cdot)\) \(\chi_{3381}(619,\cdot)\) \(\chi_{3381}(766,\cdot)\) \(\chi_{3381}(1207,\cdot)\) \(\chi_{3381}(1354,\cdot)\) \(\chi_{3381}(1489,\cdot)\) \(\chi_{3381}(1648,\cdot)\) \(\chi_{3381}(1930,\cdot)\) \(\chi_{3381}(1942,\cdot)\) \(\chi_{3381}(2077,\cdot)\) \(\chi_{3381}(2089,\cdot)\) \(\chi_{3381}(2236,\cdot)\) \(\chi_{3381}(2383,\cdot)\) \(\chi_{3381}(2518,\cdot)\) \(\chi_{3381}(2665,\cdot)\) \(\chi_{3381}(2959,\cdot)\) \(\chi_{3381}(3253,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((2255,346,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3381 }(1942, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) |
sage: chi.jacobi_sum(n)