from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([35,27]))
pari: [g,chi] = znchar(Mod(158,261))
Basic properties
| Modulus: | \(261\) | |
| Conductor: | \(261\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 261.v
\(\chi_{261}(5,\cdot)\) \(\chi_{261}(38,\cdot)\) \(\chi_{261}(92,\cdot)\) \(\chi_{261}(122,\cdot)\) \(\chi_{261}(149,\cdot)\) \(\chi_{261}(158,\cdot)\) \(\chi_{261}(167,\cdot)\) \(\chi_{261}(209,\cdot)\) \(\chi_{261}(212,\cdot)\) \(\chi_{261}(236,\cdot)\) \(\chi_{261}(245,\cdot)\) \(\chi_{261}(254,\cdot)\)
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: | 42.0.1236405605609949863710440275882328463150065157474288679507165498832965910232619214817863.1 |
Values on generators
\((146,118)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 261 }(158, a) \) | \(-1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)