from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,22,6]))
pari: [g,chi] = znchar(Mod(1871,8470))
Basic properties
Modulus: | \(8470\) | |
Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{847}(177,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.ca
\(\chi_{8470}(221,\cdot)\) \(\chi_{8470}(331,\cdot)\) \(\chi_{8470}(991,\cdot)\) \(\chi_{8470}(1101,\cdot)\) \(\chi_{8470}(1761,\cdot)\) \(\chi_{8470}(1871,\cdot)\) \(\chi_{8470}(2531,\cdot)\) \(\chi_{8470}(2641,\cdot)\) \(\chi_{8470}(3301,\cdot)\) \(\chi_{8470}(3411,\cdot)\) \(\chi_{8470}(4071,\cdot)\) \(\chi_{8470}(4181,\cdot)\) \(\chi_{8470}(4951,\cdot)\) \(\chi_{8470}(5611,\cdot)\) \(\chi_{8470}(5721,\cdot)\) \(\chi_{8470}(6381,\cdot)\) \(\chi_{8470}(6491,\cdot)\) \(\chi_{8470}(7151,\cdot)\) \(\chi_{8470}(7921,\cdot)\) \(\chi_{8470}(8031,\cdot)\)
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 33 polynomial |
Values on generators
\((6777,6051,7141)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 8470 }(1871, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) |
sage: chi.jacobi_sum(n)