from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1157, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,8]))
pari: [g,chi] = znchar(Mod(892,1157))
Basic properties
| Modulus: | \(1157\) | |
| Conductor: | \(1157\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 1157.bi
\(\chi_{1157}(8,\cdot)\) \(\chi_{1157}(242,\cdot)\) \(\chi_{1157}(372,\cdot)\) \(\chi_{1157}(395,\cdot)\) \(\chi_{1157}(434,\cdot)\) \(\chi_{1157}(447,\cdot)\) \(\chi_{1157}(512,\cdot)\) \(\chi_{1157}(538,\cdot)\) \(\chi_{1157}(655,\cdot)\) \(\chi_{1157}(668,\cdot)\) \(\chi_{1157}(720,\cdot)\) \(\chi_{1157}(840,\cdot)\) \(\chi_{1157}(879,\cdot)\) \(\chi_{1157}(892,\cdot)\) \(\chi_{1157}(954,\cdot)\) \(\chi_{1157}(957,\cdot)\) \(\chi_{1157}(983,\cdot)\) \(\chi_{1157}(1084,\cdot)\) \(\chi_{1157}(1100,\cdot)\) \(\chi_{1157}(1113,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((535,92)\) → \((i,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1157 }(892, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{44}\right)\) |
sage: chi.jacobi_sum(n)