from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,52,45]))
pari: [g,chi] = znchar(Mod(61,2268))
Basic properties
| Modulus: | \(2268\) | |
| Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{567}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2268.cv
\(\chi_{2268}(61,\cdot)\) \(\chi_{2268}(157,\cdot)\) \(\chi_{2268}(313,\cdot)\) \(\chi_{2268}(409,\cdot)\) \(\chi_{2268}(565,\cdot)\) \(\chi_{2268}(661,\cdot)\) \(\chi_{2268}(817,\cdot)\) \(\chi_{2268}(913,\cdot)\) \(\chi_{2268}(1069,\cdot)\) \(\chi_{2268}(1165,\cdot)\) \(\chi_{2268}(1321,\cdot)\) \(\chi_{2268}(1417,\cdot)\) \(\chi_{2268}(1573,\cdot)\) \(\chi_{2268}(1669,\cdot)\) \(\chi_{2268}(1825,\cdot)\) \(\chi_{2268}(1921,\cdot)\) \(\chi_{2268}(2077,\cdot)\) \(\chi_{2268}(2173,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1135,1541,325)\) → \((1,e\left(\frac{26}{27}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)