from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6840, base_ring=CyclotomicField(6))
M = H._module
chi = DirichletCharacter(H, M([0,3,3,0,5]))
pari: [g,chi] = znchar(Mod(2501,6840))
Basic properties
Modulus: | \(6840\) | |
Conductor: | \(456\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{456}(221,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6840.gp
\(\chi_{6840}(2501,\cdot)\) \(\chi_{6840}(3941,\cdot)\)
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: | \(\mathbb{Q}(\zeta_3)\) |
Fixed field: | 6.6.34229592576.1 |
Values on generators
\((1711,3421,5321,2737,6481)\) → \((1,-1,-1,1,e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 6840 }(2501, a) \) | \(1\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)