from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2268, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,44,0]))
pari: [g,chi] = znchar(Mod(715,2268))
Basic properties
| Modulus: | \(2268\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{324}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2268.cx
\(\chi_{2268}(43,\cdot)\) \(\chi_{2268}(211,\cdot)\) \(\chi_{2268}(295,\cdot)\) \(\chi_{2268}(463,\cdot)\) \(\chi_{2268}(547,\cdot)\) \(\chi_{2268}(715,\cdot)\) \(\chi_{2268}(799,\cdot)\) \(\chi_{2268}(967,\cdot)\) \(\chi_{2268}(1051,\cdot)\) \(\chi_{2268}(1219,\cdot)\) \(\chi_{2268}(1303,\cdot)\) \(\chi_{2268}(1471,\cdot)\) \(\chi_{2268}(1555,\cdot)\) \(\chi_{2268}(1723,\cdot)\) \(\chi_{2268}(1807,\cdot)\) \(\chi_{2268}(1975,\cdot)\) \(\chi_{2268}(2059,\cdot)\) \(\chi_{2268}(2227,\cdot)\)
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
\((1135,1541,325)\) → \((-1,e\left(\frac{22}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2268 }(715, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)