from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,44,15]))
pari: [g,chi] = znchar(Mod(43,8712))
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.dl
\(\chi_{8712}(43,\cdot)\) \(\chi_{8712}(571,\cdot)\) \(\chi_{8712}(835,\cdot)\) \(\chi_{8712}(1363,\cdot)\) \(\chi_{8712}(1627,\cdot)\) \(\chi_{8712}(2155,\cdot)\) \(\chi_{8712}(2947,\cdot)\) \(\chi_{8712}(3211,\cdot)\) \(\chi_{8712}(3739,\cdot)\) \(\chi_{8712}(4003,\cdot)\) \(\chi_{8712}(4531,\cdot)\) \(\chi_{8712}(4795,\cdot)\) \(\chi_{8712}(5587,\cdot)\) \(\chi_{8712}(6115,\cdot)\) \(\chi_{8712}(6379,\cdot)\) \(\chi_{8712}(6907,\cdot)\) \(\chi_{8712}(7171,\cdot)\) \(\chi_{8712}(7699,\cdot)\) \(\chi_{8712}(7963,\cdot)\) \(\chi_{8712}(8491,\cdot)\)
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
\((6535,4357,1937,5689)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) |
sage: chi.jacobi_sum(n)