from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,26,14]))
pari: [g,chi] = znchar(Mod(461,8624))
Basic properties
Modulus: | \(8624\) | |
Conductor: | \(8624\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.dm
\(\chi_{8624}(461,\cdot)\) \(\chi_{8624}(1693,\cdot)\) \(\chi_{8624}(2309,\cdot)\) \(\chi_{8624}(2925,\cdot)\) \(\chi_{8624}(3541,\cdot)\) \(\chi_{8624}(4157,\cdot)\) \(\chi_{8624}(4773,\cdot)\) \(\chi_{8624}(6005,\cdot)\) \(\chi_{8624}(6621,\cdot)\) \(\chi_{8624}(7237,\cdot)\) \(\chi_{8624}(7853,\cdot)\) \(\chi_{8624}(8469,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((5391,6469,7745,3137)\) → \((1,-i,e\left(\frac{13}{14}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 8624 }(461, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(i\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) |
sage: chi.jacobi_sum(n)