from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,14,6]))
pari: [g,chi] = znchar(Mod(35,4176))
Basic properties
Modulus: | \(4176\) | |
Conductor: | \(1392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1392}(35,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.ec
\(\chi_{4176}(35,\cdot)\) \(\chi_{4176}(179,\cdot)\) \(\chi_{4176}(323,\cdot)\) \(\chi_{4176}(1115,\cdot)\) \(\chi_{4176}(1691,\cdot)\) \(\chi_{4176}(1907,\cdot)\) \(\chi_{4176}(2123,\cdot)\) \(\chi_{4176}(2267,\cdot)\) \(\chi_{4176}(2411,\cdot)\) \(\chi_{4176}(3203,\cdot)\) \(\chi_{4176}(3779,\cdot)\) \(\chi_{4176}(3995,\cdot)\)
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
\((1567,1045,929,4033)\) → \((-1,-i,-1,e\left(\frac{3}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
\( \chi_{ 4176 }(35, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) |
sage: chi.jacobi_sum(n)