from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,2,0]))
pari: [g,chi] = znchar(Mod(205,4998))
Basic properties
| Modulus: | \(4998\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4998.bo
\(\chi_{4998}(205,\cdot)\) \(\chi_{4998}(613,\cdot)\) \(\chi_{4998}(919,\cdot)\) \(\chi_{4998}(1327,\cdot)\) \(\chi_{4998}(1633,\cdot)\) \(\chi_{4998}(2041,\cdot)\) \(\chi_{4998}(2347,\cdot)\) \(\chi_{4998}(2755,\cdot)\) \(\chi_{4998}(3061,\cdot)\) \(\chi_{4998}(3469,\cdot)\) \(\chi_{4998}(3775,\cdot)\) \(\chi_{4998}(4897,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((1667,2551,4117)\) → \((1,e\left(\frac{1}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(205, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)