from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,0,12]))
pari: [g,chi] = znchar(Mod(181,4176))
Basic properties
Modulus: | \(4176\) | |
Conductor: | \(464\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{464}(181,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.ek
\(\chi_{4176}(181,\cdot)\) \(\chi_{4176}(397,\cdot)\) \(\chi_{4176}(973,\cdot)\) \(\chi_{4176}(1765,\cdot)\) \(\chi_{4176}(1909,\cdot)\) \(\chi_{4176}(2053,\cdot)\) \(\chi_{4176}(2269,\cdot)\) \(\chi_{4176}(2485,\cdot)\) \(\chi_{4176}(3061,\cdot)\) \(\chi_{4176}(3853,\cdot)\) \(\chi_{4176}(3997,\cdot)\) \(\chi_{4176}(4141,\cdot)\)
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
\((1567,1045,929,4033)\) → \((1,i,1,e\left(\frac{3}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
\( \chi_{ 4176 }(181, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) |
sage: chi.jacobi_sum(n)