from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,38,35]))
pari: [g,chi] = znchar(Mod(23,1911))
Basic properties
| Modulus: | \(1911\) | |
| Conductor: | \(1911\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1911.da
\(\chi_{1911}(23,\cdot)\) \(\chi_{1911}(95,\cdot)\) \(\chi_{1911}(296,\cdot)\) \(\chi_{1911}(368,\cdot)\) \(\chi_{1911}(641,\cdot)\) \(\chi_{1911}(842,\cdot)\) \(\chi_{1911}(914,\cdot)\) \(\chi_{1911}(1115,\cdot)\) \(\chi_{1911}(1187,\cdot)\) \(\chi_{1911}(1388,\cdot)\) \(\chi_{1911}(1460,\cdot)\) \(\chi_{1911}(1661,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | 42.0.171796661872303139426286971553175729564906991908179786155620948893904232254867038476961967376575716647979404317071.2 |
Values on generators
\((638,1522,1471)\) → \((-1,e\left(\frac{19}{21}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(16\) | \(17\) | \(19\) | \(20\) |
| \( \chi_{ 1911 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)