from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6840, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,15,0,10]))
pari: [g,chi] = znchar(Mod(131,6840))
Basic properties
| Modulus: | \(6840\) | |
| Conductor: | \(1368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1368}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6840.le
\(\chi_{6840}(131,\cdot)\) \(\chi_{6840}(491,\cdot)\) \(\chi_{6840}(731,\cdot)\) \(\chi_{6840}(1811,\cdot)\) \(\chi_{6840}(2531,\cdot)\) \(\chi_{6840}(4451,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
| Fixed field: | Number field defined by a degree 18 polynomial |
Values on generators
\((1711,3421,5321,2737,6481)\) → \((-1,-1,e\left(\frac{5}{6}\right),1,e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6840 }(131, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)