from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,35,0,24]))
pari: [g,chi] = znchar(Mod(361,6336))
Basic properties
| Modulus: | \(6336\) | |
| Conductor: | \(352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{352}(141,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6336.en
\(\chi_{6336}(361,\cdot)\) \(\chi_{6336}(1081,\cdot)\) \(\chi_{6336}(1225,\cdot)\) \(\chi_{6336}(1369,\cdot)\) \(\chi_{6336}(1945,\cdot)\) \(\chi_{6336}(2665,\cdot)\) \(\chi_{6336}(2809,\cdot)\) \(\chi_{6336}(2953,\cdot)\) \(\chi_{6336}(3529,\cdot)\) \(\chi_{6336}(4249,\cdot)\) \(\chi_{6336}(4393,\cdot)\) \(\chi_{6336}(4537,\cdot)\) \(\chi_{6336}(5113,\cdot)\) \(\chi_{6336}(5833,\cdot)\) \(\chi_{6336}(5977,\cdot)\) \(\chi_{6336}(6121,\cdot)\)
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
\((4159,4357,3521,1729)\) → \((1,e\left(\frac{7}{8}\right),1,e\left(\frac{3}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6336 }(361, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) |
sage: chi.jacobi_sum(n)