from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(979, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,43]))
pari: [g,chi] = znchar(Mod(10,979))
Basic properties
| Modulus: | \(979\) | |
| Conductor: | \(979\) | 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 979.v
\(\chi_{979}(10,\cdot)\) \(\chi_{979}(21,\cdot)\) \(\chi_{979}(98,\cdot)\) \(\chi_{979}(109,\cdot)\) \(\chi_{979}(131,\cdot)\) \(\chi_{979}(142,\cdot)\) \(\chi_{979}(285,\cdot)\) \(\chi_{979}(307,\cdot)\) \(\chi_{979}(351,\cdot)\) \(\chi_{979}(373,\cdot)\) \(\chi_{979}(428,\cdot)\) \(\chi_{979}(450,\cdot)\) \(\chi_{979}(494,\cdot)\) \(\chi_{979}(516,\cdot)\) \(\chi_{979}(659,\cdot)\) \(\chi_{979}(670,\cdot)\) \(\chi_{979}(692,\cdot)\) \(\chi_{979}(703,\cdot)\) \(\chi_{979}(780,\cdot)\) \(\chi_{979}(791,\cdot)\)
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: | 44.0.54251201284655984639017235295455464252427028689652940739777799052008902639640985288139310463127832243817649.1 |
Values on generators
\((90,804)\) → \((-1,e\left(\frac{43}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 979 }(10, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(i\) |
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)