from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,35,33]))
pari: [g,chi] = znchar(Mod(1679,1764))
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.cv
\(\chi_{1764}(83,\cdot)\) \(\chi_{1764}(167,\cdot)\) \(\chi_{1764}(335,\cdot)\) \(\chi_{1764}(419,\cdot)\) \(\chi_{1764}(671,\cdot)\) \(\chi_{1764}(839,\cdot)\) \(\chi_{1764}(923,\cdot)\) \(\chi_{1764}(1091,\cdot)\) \(\chi_{1764}(1343,\cdot)\) \(\chi_{1764}(1427,\cdot)\) \(\chi_{1764}(1595,\cdot)\) \(\chi_{1764}(1679,\cdot)\)
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: | 42.0.12140927516266774666314598865493926760663766983493911125885658849232780316031818617435329218169600416415744.1 |
Values on generators
\((883,785,1081)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{11}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1764 }(1679, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)