from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,5,0]))
pari: [g,chi] = znchar(Mod(341,4998))
Basic properties
| Modulus: | \(4998\) | |
| Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{147}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4998.cc
\(\chi_{4998}(341,\cdot)\) \(\chi_{4998}(647,\cdot)\) \(\chi_{4998}(1055,\cdot)\) \(\chi_{4998}(1361,\cdot)\) \(\chi_{4998}(1769,\cdot)\) \(\chi_{4998}(2075,\cdot)\) \(\chi_{4998}(2483,\cdot)\) \(\chi_{4998}(2789,\cdot)\) \(\chi_{4998}(3197,\cdot)\) \(\chi_{4998}(3503,\cdot)\) \(\chi_{4998}(3911,\cdot)\) \(\chi_{4998}(4217,\cdot)\)
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: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((1667,2551,4117)\) → \((-1,e\left(\frac{5}{42}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(341, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)