from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4598, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([9,11]))
pari: [g,chi] = znchar(Mod(417,4598))
Basic properties
| Modulus: | \(4598\) | |
| Conductor: | \(2299\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2299}(417,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4598.t
\(\chi_{4598}(417,\cdot)\) \(\chi_{4598}(835,\cdot)\) \(\chi_{4598}(1253,\cdot)\) \(\chi_{4598}(1671,\cdot)\) \(\chi_{4598}(2089,\cdot)\) \(\chi_{4598}(2507,\cdot)\) \(\chi_{4598}(2925,\cdot)\) \(\chi_{4598}(3343,\cdot)\) \(\chi_{4598}(3761,\cdot)\) \(\chi_{4598}(4179,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((3269,3631)\) → \((e\left(\frac{9}{22}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 4598 }(417, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)