from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7488, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([4,1,0,2]))
pari: [g,chi] = znchar(Mod(4519,7488))
Basic properties
Modulus: | \(7488\) | |
Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{416}(411,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7488.ed
\(\chi_{7488}(343,\cdot)\) \(\chi_{7488}(775,\cdot)\) \(\chi_{7488}(4087,\cdot)\) \(\chi_{7488}(4519,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
Fixed field: | 8.8.10365493399519232.1 |
Values on generators
\((703,6085,5825,5761)\) → \((-1,e\left(\frac{1}{8}\right),1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 7488 }(4519, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(-i\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)