from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1161, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,20]))
pari: [g,chi] = znchar(Mod(143,1161))
Basic properties
Modulus: | \(1161\) | |
Conductor: | \(387\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{387}(272,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1161.bq
\(\chi_{1161}(143,\cdot)\) \(\chi_{1161}(152,\cdot)\) \(\chi_{1161}(197,\cdot)\) \(\chi_{1161}(224,\cdot)\) \(\chi_{1161}(341,\cdot)\) \(\chi_{1161}(359,\cdot)\) \(\chi_{1161}(368,\cdot)\) \(\chi_{1161}(719,\cdot)\) \(\chi_{1161}(791,\cdot)\) \(\chi_{1161}(827,\cdot)\) \(\chi_{1161}(1070,\cdot)\) \(\chi_{1161}(1088,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((947,433)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{10}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 1161 }(143, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) |
sage: chi.jacobi_sum(n)