from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,14,22]))
pari: [g,chi] = znchar(Mod(1075,1764))
Basic properties
Modulus: | \(1764\) | |
Conductor: | \(1764\) | 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 1764.cc
\(\chi_{1764}(319,\cdot)\) \(\chi_{1764}(331,\cdot)\) \(\chi_{1764}(571,\cdot)\) \(\chi_{1764}(583,\cdot)\) \(\chi_{1764}(823,\cdot)\) \(\chi_{1764}(835,\cdot)\) \(\chi_{1764}(1075,\cdot)\) \(\chi_{1764}(1087,\cdot)\) \(\chi_{1764}(1327,\cdot)\) \(\chi_{1764}(1339,\cdot)\) \(\chi_{1764}(1579,\cdot)\) \(\chi_{1764}(1591,\cdot)\)
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: | 42.0.38859850303551633591313302267241649439710273838343565560676548671526960316517023466871195485682305859584.2 |
Values on generators
\((883,785,1081)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{11}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1764 }(1075, a) \) | \(-1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) |
sage: chi.jacobi_sum(n)