from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2299, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([51,55]))
pari: [g,chi] = znchar(Mod(2254,2299))
Basic properties
| Modulus: | \(2299\) | |
| Conductor: | \(2299\) | 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 2299.bd
\(\chi_{2299}(65,\cdot)\) \(\chi_{2299}(164,\cdot)\) \(\chi_{2299}(274,\cdot)\) \(\chi_{2299}(373,\cdot)\) \(\chi_{2299}(582,\cdot)\) \(\chi_{2299}(692,\cdot)\) \(\chi_{2299}(791,\cdot)\) \(\chi_{2299}(901,\cdot)\) \(\chi_{2299}(1000,\cdot)\) \(\chi_{2299}(1110,\cdot)\) \(\chi_{2299}(1319,\cdot)\) \(\chi_{2299}(1418,\cdot)\) \(\chi_{2299}(1528,\cdot)\) \(\chi_{2299}(1627,\cdot)\) \(\chi_{2299}(1737,\cdot)\) \(\chi_{2299}(1836,\cdot)\) \(\chi_{2299}(1946,\cdot)\) \(\chi_{2299}(2045,\cdot)\) \(\chi_{2299}(2155,\cdot)\) \(\chi_{2299}(2254,\cdot)\)
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
\((970,1332)\) → \((e\left(\frac{17}{22}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 2299 }(2254, a) \) | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)