from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,30,6,40]))
pari: [g,chi] = znchar(Mod(179,8400))
Basic properties
| Modulus: | \(8400\) | |
| Conductor: | \(8400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 8400.lo
\(\chi_{8400}(179,\cdot)\) \(\chi_{8400}(779,\cdot)\) \(\chi_{8400}(1019,\cdot)\) \(\chi_{8400}(1619,\cdot)\) \(\chi_{8400}(1859,\cdot)\) \(\chi_{8400}(2459,\cdot)\) \(\chi_{8400}(3539,\cdot)\) \(\chi_{8400}(4139,\cdot)\) \(\chi_{8400}(4379,\cdot)\) \(\chi_{8400}(4979,\cdot)\) \(\chi_{8400}(5219,\cdot)\) \(\chi_{8400}(5819,\cdot)\) \(\chi_{8400}(6059,\cdot)\) \(\chi_{8400}(6659,\cdot)\) \(\chi_{8400}(7739,\cdot)\) \(\chi_{8400}(8339,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((3151,2101,2801,5377,3601)\) → \((-1,-i,-1,e\left(\frac{1}{10}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8400 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)