from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1191, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,65]))
pari: [g,chi] = znchar(Mod(137,1191))
Basic properties
Modulus: | \(1191\) | |
Conductor: | \(1191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1191.z
\(\chi_{1191}(65,\cdot)\) \(\chi_{1191}(83,\cdot)\) \(\chi_{1191}(137,\cdot)\) \(\chi_{1191}(191,\cdot)\) \(\chi_{1191}(257,\cdot)\) \(\chi_{1191}(287,\cdot)\) \(\chi_{1191}(440,\cdot)\) \(\chi_{1191}(533,\cdot)\) \(\chi_{1191}(539,\cdot)\) \(\chi_{1191}(560,\cdot)\) \(\chi_{1191}(623,\cdot)\) \(\chi_{1191}(647,\cdot)\) \(\chi_{1191}(674,\cdot)\) \(\chi_{1191}(686,\cdot)\) \(\chi_{1191}(821,\cdot)\) \(\chi_{1191}(824,\cdot)\) \(\chi_{1191}(1001,\cdot)\) \(\chi_{1191}(1019,\cdot)\) \(\chi_{1191}(1040,\cdot)\) \(\chi_{1191}(1085,\cdot)\)
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
\((398,799)\) → \((-1,e\left(\frac{65}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 1191 }(137, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)