from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1694, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,63]))
pari: [g,chi] = znchar(Mod(87,1694))
Basic properties
| Modulus: | \(1694\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{847}(87,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1694.x
\(\chi_{1694}(87,\cdot)\) \(\chi_{1694}(131,\cdot)\) \(\chi_{1694}(285,\cdot)\) \(\chi_{1694}(395,\cdot)\) \(\chi_{1694}(439,\cdot)\) \(\chi_{1694}(549,\cdot)\) \(\chi_{1694}(593,\cdot)\) \(\chi_{1694}(703,\cdot)\) \(\chi_{1694}(747,\cdot)\) \(\chi_{1694}(857,\cdot)\) \(\chi_{1694}(901,\cdot)\) \(\chi_{1694}(1011,\cdot)\) \(\chi_{1694}(1055,\cdot)\) \(\chi_{1694}(1165,\cdot)\) \(\chi_{1694}(1319,\cdot)\) \(\chi_{1694}(1363,\cdot)\) \(\chi_{1694}(1473,\cdot)\) \(\chi_{1694}(1517,\cdot)\) \(\chi_{1694}(1627,\cdot)\) \(\chi_{1694}(1671,\cdot)\)
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
\((969,365)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 1694 }(87, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)