from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1005, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,21]))
pari: [g,chi] = znchar(Mod(779,1005))
Basic properties
Modulus: | \(1005\) | |
Conductor: | \(1005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1005.bc
\(\chi_{1005}(119,\cdot)\) \(\chi_{1005}(179,\cdot)\) \(\chi_{1005}(209,\cdot)\) \(\chi_{1005}(254,\cdot)\) \(\chi_{1005}(539,\cdot)\) \(\chi_{1005}(764,\cdot)\) \(\chi_{1005}(779,\cdot)\) \(\chi_{1005}(809,\cdot)\) \(\chi_{1005}(914,\cdot)\) \(\chi_{1005}(929,\cdot)\)
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: | Number field defined by a degree 22 polynomial |
Values on generators
\((671,202,136)\) → \((-1,-1,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1005 }(779, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)