from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4620, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,15,15,20,24]))
pari: [g,chi] = znchar(Mod(179,4620))
Basic properties
Modulus: | \(4620\) | |
Conductor: | \(4620\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4620.go
\(\chi_{4620}(179,\cdot)\) \(\chi_{4620}(599,\cdot)\) \(\chi_{4620}(779,\cdot)\) \(\chi_{4620}(1439,\cdot)\) \(\chi_{4620}(2039,\cdot)\) \(\chi_{4620}(2699,\cdot)\) \(\chi_{4620}(4139,\cdot)\) \(\chi_{4620}(4559,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((2311,1541,3697,661,2521)\) → \((-1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{4}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 4620 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(e\left(\frac{7}{30}\right)\) |
sage: chi.jacobi_sum(n)