from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,8]))
pari: [g,chi] = znchar(Mod(2501,8800))
Basic properties
| Modulus: | \(8800\) | |
| 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 \(\chi_{352}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8800.nn
\(\chi_{8800}(301,\cdot)\) \(\chi_{8800}(1301,\cdot)\) \(\chi_{8800}(1501,\cdot)\) \(\chi_{8800}(1901,\cdot)\) \(\chi_{8800}(2501,\cdot)\) \(\chi_{8800}(3501,\cdot)\) \(\chi_{8800}(3701,\cdot)\) \(\chi_{8800}(4101,\cdot)\) \(\chi_{8800}(4701,\cdot)\) \(\chi_{8800}(5701,\cdot)\) \(\chi_{8800}(5901,\cdot)\) \(\chi_{8800}(6301,\cdot)\) \(\chi_{8800}(6901,\cdot)\) \(\chi_{8800}(7901,\cdot)\) \(\chi_{8800}(8101,\cdot)\) \(\chi_{8800}(8501,\cdot)\)
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
\((2751,3301,4577,5601)\) → \((1,e\left(\frac{1}{8}\right),1,e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8800 }(2501, a) \) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |
sage: chi.jacobi_sum(n)