from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,44,54]))
pari: [g,chi] = znchar(Mod(133,8712))
Basic properties
| Modulus: | \(8712\) | |
| Conductor: | \(8712\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 8712.dj
\(\chi_{8712}(133,\cdot)\) \(\chi_{8712}(661,\cdot)\) \(\chi_{8712}(925,\cdot)\) \(\chi_{8712}(1717,\cdot)\) \(\chi_{8712}(2245,\cdot)\) \(\chi_{8712}(2509,\cdot)\) \(\chi_{8712}(3037,\cdot)\) \(\chi_{8712}(3301,\cdot)\) \(\chi_{8712}(3829,\cdot)\) \(\chi_{8712}(4093,\cdot)\) \(\chi_{8712}(4621,\cdot)\) \(\chi_{8712}(4885,\cdot)\) \(\chi_{8712}(5413,\cdot)\) \(\chi_{8712}(5677,\cdot)\) \(\chi_{8712}(6205,\cdot)\) \(\chi_{8712}(6469,\cdot)\) \(\chi_{8712}(6997,\cdot)\) \(\chi_{8712}(7789,\cdot)\) \(\chi_{8712}(8053,\cdot)\) \(\chi_{8712}(8581,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((6535,4357,1937,5689)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)