from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(352, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,16]))
pari: [g,chi] = znchar(Mod(5,352))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 352.bf
\(\chi_{352}(5,\cdot)\) \(\chi_{352}(37,\cdot)\) \(\chi_{352}(53,\cdot)\) \(\chi_{352}(69,\cdot)\) \(\chi_{352}(93,\cdot)\) \(\chi_{352}(125,\cdot)\) \(\chi_{352}(141,\cdot)\) \(\chi_{352}(157,\cdot)\) \(\chi_{352}(181,\cdot)\) \(\chi_{352}(213,\cdot)\) \(\chi_{352}(229,\cdot)\) \(\chi_{352}(245,\cdot)\) \(\chi_{352}(269,\cdot)\) \(\chi_{352}(301,\cdot)\) \(\chi_{352}(317,\cdot)\) \(\chi_{352}(333,\cdot)\)
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: | 40.40.96430685261162182749113906515642066253992366248338958954046471967872161601814528.1 |
Values on generators
\((287,133,321)\) → \((1,e\left(\frac{1}{8}\right),e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 352 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) |
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)