from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,1]))
pari: [g,chi] = znchar(Mod(395,1764))
Basic properties
Modulus: | \(1764\) | |
Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{588}(395,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1764.cx
\(\chi_{1764}(143,\cdot)\) \(\chi_{1764}(395,\cdot)\) \(\chi_{1764}(467,\cdot)\) \(\chi_{1764}(647,\cdot)\) \(\chi_{1764}(719,\cdot)\) \(\chi_{1764}(899,\cdot)\) \(\chi_{1764}(971,\cdot)\) \(\chi_{1764}(1151,\cdot)\) \(\chi_{1764}(1223,\cdot)\) \(\chi_{1764}(1475,\cdot)\) \(\chi_{1764}(1655,\cdot)\) \(\chi_{1764}(1727,\cdot)\)
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: | 42.0.5436948860695888782893198886016377149049148530413040928765325951335574011833955525853184.1 |
Values on generators
\((883,785,1081)\) → \((-1,-1,e\left(\frac{1}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1764 }(395, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{16}{21}\right)\) |
sage: chi.jacobi_sum(n)