from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,0,9]))
pari: [g,chi] = znchar(Mod(19,4176))
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 \(\chi_{464}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4176.dy
\(\chi_{4176}(19,\cdot)\) \(\chi_{4176}(739,\cdot)\) \(\chi_{4176}(955,\cdot)\) \(\chi_{4176}(1099,\cdot)\) \(\chi_{4176}(1171,\cdot)\) \(\chi_{4176}(2251,\cdot)\) \(\chi_{4176}(2323,\cdot)\) \(\chi_{4176}(2467,\cdot)\) \(\chi_{4176}(2683,\cdot)\) \(\chi_{4176}(3403,\cdot)\) \(\chi_{4176}(3691,\cdot)\) \(\chi_{4176}(3907,\cdot)\)
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: | 28.28.461376647922644386311538110285539877693458386764746599325237248.1 |
Values on generators
\((1567,1045,929,4033)\) → \((-1,-i,1,e\left(\frac{9}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
\( \chi_{ 4176 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
sage: chi.jacobi_sum(n)