from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4730, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,13]))
pari: [g,chi] = znchar(Mod(3409,4730))
Basic properties
Modulus: | \(4730\) | |
Conductor: | \(2365\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2365}(1044,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4730.cm
\(\chi_{4730}(329,\cdot)\) \(\chi_{4730}(549,\cdot)\) \(\chi_{4730}(879,\cdot)\) \(\chi_{4730}(1209,\cdot)\) \(\chi_{4730}(1319,\cdot)\) \(\chi_{4730}(1539,\cdot)\) \(\chi_{4730}(1869,\cdot)\) \(\chi_{4730}(2309,\cdot)\) \(\chi_{4730}(3079,\cdot)\) \(\chi_{4730}(3409,\cdot)\) \(\chi_{4730}(3959,\cdot)\) \(\chi_{4730}(4619,\cdot)\)
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 42 polynomial |
Values on generators
\((947,431,1981)\) → \((-1,-1,e\left(\frac{13}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4730 }(3409, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)