from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,10,14]))
pari: [g,chi] = znchar(Mod(293,8085))
Basic properties
Modulus: | \(8085\) | |
Conductor: | \(1155\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1155}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.di
\(\chi_{8085}(293,\cdot)\) \(\chi_{8085}(1322,\cdot)\) \(\chi_{8085}(3527,\cdot)\) \(\chi_{8085}(3968,\cdot)\) \(\chi_{8085}(4703,\cdot)\) \(\chi_{8085}(6173,\cdot)\) \(\chi_{8085}(7202,\cdot)\) \(\chi_{8085}(7937,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.2830162324148238464682513639699737548828125.1 |
Values on generators
\((2696,4852,1816,3676)\) → \((-1,-i,-1,e\left(\frac{7}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 8085 }(293, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)