from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,15,0,19]))
pari: [g,chi] = znchar(Mod(163,3600))
Basic properties
Modulus: | \(3600\) | |
Conductor: | \(400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{400}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3600.ei
\(\chi_{3600}(163,\cdot)\) \(\chi_{3600}(667,\cdot)\) \(\chi_{3600}(883,\cdot)\) \(\chi_{3600}(1387,\cdot)\) \(\chi_{3600}(1603,\cdot)\) \(\chi_{3600}(2323,\cdot)\) \(\chi_{3600}(2827,\cdot)\) \(\chi_{3600}(3547,\cdot)\)
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: | 20.20.104857600000000000000000000000000000000000.2 |
Values on generators
\((3151,901,2801,577)\) → \((-1,-i,1,e\left(\frac{19}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 3600 }(163, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)