from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1161, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,10]))
pari: [g,chi] = znchar(Mod(53,1161))
Basic properties
| Modulus: | \(1161\) | |
| Conductor: | \(129\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{129}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1161.by
\(\chi_{1161}(53,\cdot)\) \(\chi_{1161}(296,\cdot)\) \(\chi_{1161}(404,\cdot)\) \(\chi_{1161}(539,\cdot)\) \(\chi_{1161}(701,\cdot)\) \(\chi_{1161}(728,\cdot)\) \(\chi_{1161}(755,\cdot)\) \(\chi_{1161}(917,\cdot)\) \(\chi_{1161}(971,\cdot)\) \(\chi_{1161}(998,\cdot)\) \(\chi_{1161}(1106,\cdot)\) \(\chi_{1161}(1133,\cdot)\)
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: | 42.0.2281836760183646137444154412268560109828024514076489472840222217265158917203.1 |
Values on generators
\((947,433)\) → \((-1,e\left(\frac{5}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1161 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)