from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,55,21]))
pari: [g,chi] = znchar(Mod(7369,8470))
Basic properties
Modulus: | \(8470\) | |
Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4235}(3134,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8470.cm
\(\chi_{8470}(439,\cdot)\) \(\chi_{8470}(549,\cdot)\) \(\chi_{8470}(1319,\cdot)\) \(\chi_{8470}(1979,\cdot)\) \(\chi_{8470}(2089,\cdot)\) \(\chi_{8470}(2749,\cdot)\) \(\chi_{8470}(2859,\cdot)\) \(\chi_{8470}(3519,\cdot)\) \(\chi_{8470}(4289,\cdot)\) \(\chi_{8470}(4399,\cdot)\) \(\chi_{8470}(5059,\cdot)\) \(\chi_{8470}(5169,\cdot)\) \(\chi_{8470}(5829,\cdot)\) \(\chi_{8470}(5939,\cdot)\) \(\chi_{8470}(6599,\cdot)\) \(\chi_{8470}(6709,\cdot)\) \(\chi_{8470}(7369,\cdot)\) \(\chi_{8470}(7479,\cdot)\) \(\chi_{8470}(8139,\cdot)\) \(\chi_{8470}(8249,\cdot)\)
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
\((6777,6051,7141)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{7}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 8470 }(7369, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(1\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) |
sage: chi.jacobi_sum(n)