from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8712, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,39]))
pari: [g,chi] = znchar(Mod(65,8712))
Basic properties
| Modulus: | \(8712\) | |
| Conductor: | \(1089\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1089}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8712.df
\(\chi_{8712}(65,\cdot)\) \(\chi_{8712}(329,\cdot)\) \(\chi_{8712}(857,\cdot)\) \(\chi_{8712}(1121,\cdot)\) \(\chi_{8712}(1649,\cdot)\) \(\chi_{8712}(1913,\cdot)\) \(\chi_{8712}(2441,\cdot)\) \(\chi_{8712}(2705,\cdot)\) \(\chi_{8712}(3233,\cdot)\) \(\chi_{8712}(3497,\cdot)\) \(\chi_{8712}(4025,\cdot)\) \(\chi_{8712}(4289,\cdot)\) \(\chi_{8712}(4817,\cdot)\) \(\chi_{8712}(5609,\cdot)\) \(\chi_{8712}(5873,\cdot)\) \(\chi_{8712}(6401,\cdot)\) \(\chi_{8712}(6665,\cdot)\) \(\chi_{8712}(7193,\cdot)\) \(\chi_{8712}(7457,\cdot)\) \(\chi_{8712}(8249,\cdot)\)
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
\((6535,4357,1937,5689)\) → \((1,1,e\left(\frac{1}{6}\right),e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8712 }(65, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)