from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2695, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,26,21]))
pari: [g,chi] = znchar(Mod(1374,2695))
Basic properties
| Modulus: | \(2695\) | |
| Conductor: | \(2695\) | 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 2695.ck
\(\chi_{2695}(109,\cdot)\) \(\chi_{2695}(219,\cdot)\) \(\chi_{2695}(494,\cdot)\) \(\chi_{2695}(604,\cdot)\) \(\chi_{2695}(879,\cdot)\) \(\chi_{2695}(989,\cdot)\) \(\chi_{2695}(1264,\cdot)\) \(\chi_{2695}(1374,\cdot)\) \(\chi_{2695}(1649,\cdot)\) \(\chi_{2695}(1759,\cdot)\) \(\chi_{2695}(2034,\cdot)\) \(\chi_{2695}(2144,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((2157,1816,981)\) → \((-1,e\left(\frac{13}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 2695 }(1374, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) |
sage: chi.jacobi_sum(n)