from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,11,60]))
pari: [g,chi] = znchar(Mod(472,3381))
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 \(\chi_{161}(150,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3381.bu
\(\chi_{3381}(31,\cdot)\) \(\chi_{3381}(325,\cdot)\) \(\chi_{3381}(472,\cdot)\) \(\chi_{3381}(607,\cdot)\) \(\chi_{3381}(754,\cdot)\) \(\chi_{3381}(901,\cdot)\) \(\chi_{3381}(913,\cdot)\) \(\chi_{3381}(1048,\cdot)\) \(\chi_{3381}(1060,\cdot)\) \(\chi_{3381}(1342,\cdot)\) \(\chi_{3381}(1501,\cdot)\) \(\chi_{3381}(1636,\cdot)\) \(\chi_{3381}(1783,\cdot)\) \(\chi_{3381}(2224,\cdot)\) \(\chi_{3381}(2371,\cdot)\) \(\chi_{3381}(2677,\cdot)\) \(\chi_{3381}(2812,\cdot)\) \(\chi_{3381}(2824,\cdot)\) \(\chi_{3381}(2971,\cdot)\) \(\chi_{3381}(3118,\cdot)\)
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
\((2255,346,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{10}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3381 }(472, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) |
sage: chi.jacobi_sum(n)