from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2144, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,40]))
pari: [g,chi] = znchar(Mod(207,2144))
Basic properties
| Modulus: | \(2144\) | |
| Conductor: | \(536\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{536}(475,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2144.bu
\(\chi_{2144}(47,\cdot)\) \(\chi_{2144}(207,\cdot)\) \(\chi_{2144}(303,\cdot)\) \(\chi_{2144}(495,\cdot)\) \(\chi_{2144}(559,\cdot)\) \(\chi_{2144}(591,\cdot)\) \(\chi_{2144}(687,\cdot)\) \(\chi_{2144}(719,\cdot)\) \(\chi_{2144}(1199,\cdot)\) \(\chi_{2144}(1327,\cdot)\) \(\chi_{2144}(1359,\cdot)\) \(\chi_{2144}(1423,\cdot)\) \(\chi_{2144}(1551,\cdot)\) \(\chi_{2144}(1647,\cdot)\) \(\chi_{2144}(1679,\cdot)\) \(\chi_{2144}(1711,\cdot)\) \(\chi_{2144}(1775,\cdot)\) \(\chi_{2144}(1807,\cdot)\) \(\chi_{2144}(1999,\cdot)\) \(\chi_{2144}(2031,\cdot)\)
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
\((671,805,1409)\) → \((-1,-1,e\left(\frac{20}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2144 }(207, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) |
sage: chi.jacobi_sum(n)