from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,24]))
pari: [g,chi] = znchar(Mod(361,3381))
Basic properties
Modulus: | \(3381\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3381.bh
\(\chi_{3381}(361,\cdot)\) \(\chi_{3381}(508,\cdot)\) \(\chi_{3381}(814,\cdot)\) \(\chi_{3381}(949,\cdot)\) \(\chi_{3381}(961,\cdot)\) \(\chi_{3381}(1108,\cdot)\) \(\chi_{3381}(1255,\cdot)\) \(\chi_{3381}(1549,\cdot)\) \(\chi_{3381}(1843,\cdot)\) \(\chi_{3381}(1990,\cdot)\) \(\chi_{3381}(2125,\cdot)\) \(\chi_{3381}(2272,\cdot)\) \(\chi_{3381}(2419,\cdot)\) \(\chi_{3381}(2431,\cdot)\) \(\chi_{3381}(2566,\cdot)\) \(\chi_{3381}(2578,\cdot)\) \(\chi_{3381}(2860,\cdot)\) \(\chi_{3381}(3019,\cdot)\) \(\chi_{3381}(3154,\cdot)\) \(\chi_{3381}(3301,\cdot)\)
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: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((2255,346,442)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3381 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) |
sage: chi.jacobi_sum(n)