from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,6]))
pari: [g,chi] = znchar(Mod(989,3600))
Basic properties
Modulus: | \(3600\) | |
Conductor: | \(1200\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1200}(989,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.eo
\(\chi_{3600}(269,\cdot)\) \(\chi_{3600}(629,\cdot)\) \(\chi_{3600}(989,\cdot)\) \(\chi_{3600}(1709,\cdot)\) \(\chi_{3600}(2069,\cdot)\) \(\chi_{3600}(2429,\cdot)\) \(\chi_{3600}(2789,\cdot)\) \(\chi_{3600}(3509,\cdot)\)
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: | 20.0.1238347284480000000000000000000000000000000000.1 |
Values on generators
\((3151,901,2801,577)\) → \((1,-i,-1,e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3600 }(989, a) \) | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)