from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,20]))
pari: [g,chi] = znchar(Mod(89,804))
Basic properties
| Modulus: | \(804\) | |
| Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{201}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 804.v
\(\chi_{804}(89,\cdot)\) \(\chi_{804}(149,\cdot)\) \(\chi_{804}(293,\cdot)\) \(\chi_{804}(461,\cdot)\) \(\chi_{804}(509,\cdot)\) \(\chi_{804}(533,\cdot)\) \(\chi_{804}(545,\cdot)\) \(\chi_{804}(617,\cdot)\) \(\chi_{804}(665,\cdot)\) \(\chi_{804}(761,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.0.588613008557716517601287069955881168578347.1 |
Values on generators
\((403,269,337)\) → \((1,-1,e\left(\frac{10}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 804 }(89, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(-1\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)