from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6840, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,0,2,9,10]))
pari: [g,chi] = znchar(Mod(1703,6840))
Basic properties
| Modulus: | \(6840\) | |
| Conductor: | \(3420\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3420}(1703,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6840.hs
\(\chi_{6840}(1703,\cdot)\) \(\chi_{6840}(2687,\cdot)\) \(\chi_{6840}(5423,\cdot)\) \(\chi_{6840}(5807,\cdot)\)
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: | 12.12.19002405501509307877512000000000.2 |
Values on generators
\((1711,3421,5321,2737,6481)\) → \((-1,1,e\left(\frac{1}{6}\right),-i,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6840 }(1703, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{2}{3}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)