from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(352, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,5,8]))
pari: [g,chi] = znchar(Mod(59,352))
Basic properties
| Modulus: | \(352\) | |
| Conductor: | \(352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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 352.bd
\(\chi_{352}(3,\cdot)\) \(\chi_{352}(27,\cdot)\) \(\chi_{352}(59,\cdot)\) \(\chi_{352}(75,\cdot)\) \(\chi_{352}(91,\cdot)\) \(\chi_{352}(115,\cdot)\) \(\chi_{352}(147,\cdot)\) \(\chi_{352}(163,\cdot)\) \(\chi_{352}(179,\cdot)\) \(\chi_{352}(203,\cdot)\) \(\chi_{352}(235,\cdot)\) \(\chi_{352}(251,\cdot)\) \(\chi_{352}(267,\cdot)\) \(\chi_{352}(291,\cdot)\) \(\chi_{352}(323,\cdot)\) \(\chi_{352}(339,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.96430685261162182749113906515642066253992366248338958954046471967872161601814528.1 |
Values on generators
\((287,133,321)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 352 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) |
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)