from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2541, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,22,63]))
pari: [g,chi] = znchar(Mod(2144,2541))
Basic properties
| Modulus: | \(2541\) | |
| Conductor: | \(2541\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2541.bv
\(\chi_{2541}(32,\cdot)\) \(\chi_{2541}(65,\cdot)\) \(\chi_{2541}(263,\cdot)\) \(\chi_{2541}(296,\cdot)\) \(\chi_{2541}(494,\cdot)\) \(\chi_{2541}(527,\cdot)\) \(\chi_{2541}(758,\cdot)\) \(\chi_{2541}(956,\cdot)\) \(\chi_{2541}(989,\cdot)\) \(\chi_{2541}(1187,\cdot)\) \(\chi_{2541}(1220,\cdot)\) \(\chi_{2541}(1418,\cdot)\) \(\chi_{2541}(1649,\cdot)\) \(\chi_{2541}(1682,\cdot)\) \(\chi_{2541}(1880,\cdot)\) \(\chi_{2541}(1913,\cdot)\) \(\chi_{2541}(2111,\cdot)\) \(\chi_{2541}(2144,\cdot)\) \(\chi_{2541}(2342,\cdot)\) \(\chi_{2541}(2375,\cdot)\)
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
\((848,1816,2059)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{21}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 2541 }(2144, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)