from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,0,19,10]))
pari: [g,chi] = znchar(Mod(13,8400))
Basic properties
Modulus: | \(8400\) | |
Conductor: | \(2800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2800}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8400.hz
\(\chi_{8400}(13,\cdot)\) \(\chi_{8400}(517,\cdot)\) \(\chi_{8400}(2197,\cdot)\) \(\chi_{8400}(3373,\cdot)\) \(\chi_{8400}(3877,\cdot)\) \(\chi_{8400}(5053,\cdot)\) \(\chi_{8400}(6733,\cdot)\) \(\chi_{8400}(7237,\cdot)\)
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: | 20.20.29619676669542400000000000000000000000000000000000.2 |
Values on generators
\((3151,2101,2801,5377,3601)\) → \((1,-i,1,e\left(\frac{19}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 8400 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)