from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2340, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,2,9,8]))
pari: [g,chi] = znchar(Mod(893,2340))
Basic properties
Modulus: | \(2340\) | |
Conductor: | \(585\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{585}(308,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2340.gs
\(\chi_{2340}(653,\cdot)\) \(\chi_{2340}(893,\cdot)\) \(\chi_{2340}(2057,\cdot)\) \(\chi_{2340}(2297,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | 12.12.617247646136997205078125.1 |
Values on generators
\((1171,2081,937,1081)\) → \((1,e\left(\frac{1}{6}\right),-i,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 2340 }(893, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(i\) |
sage: chi.jacobi_sum(n)