from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7360, base_ring=CyclotomicField(8))
M = H._module
chi = DirichletCharacter(H, M([0,1,0,0]))
pari: [g,chi] = znchar(Mod(921,7360))
Basic properties
| Modulus: | \(7360\) | |
| Conductor: | \(32\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{32}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7360.bu
\(\chi_{7360}(921,\cdot)\) \(\chi_{7360}(2761,\cdot)\) \(\chi_{7360}(4601,\cdot)\) \(\chi_{7360}(6441,\cdot)\)
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: | \(\Q(\zeta_{32})^+\) |
Values on generators
\((1151,5061,4417,6721)\) → \((1,e\left(\frac{1}{8}\right),1,1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 7360 }(921, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) |
sage: chi.jacobi_sum(n)