from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1161, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,18]))
pari: [g,chi] = znchar(Mod(35,1161))
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 \(\chi_{387}(164,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1161.bp
\(\chi_{1161}(35,\cdot)\) \(\chi_{1161}(170,\cdot)\) \(\chi_{1161}(305,\cdot)\) \(\chi_{1161}(422,\cdot)\) \(\chi_{1161}(494,\cdot)\) \(\chi_{1161}(557,\cdot)\) \(\chi_{1161}(575,\cdot)\) \(\chi_{1161}(656,\cdot)\) \(\chi_{1161}(692,\cdot)\) \(\chi_{1161}(881,\cdot)\) \(\chi_{1161}(962,\cdot)\) \(\chi_{1161}(1043,\cdot)\)
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
\((947,433)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1161 }(35, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)