from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5586, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,20,7]))
pari: [g,chi] = znchar(Mod(65,5586))
Basic properties
| Modulus: | \(5586\) | |
| Conductor: | \(2793\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2793}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5586.de
\(\chi_{5586}(65,\cdot)\) \(\chi_{5586}(221,\cdot)\) \(\chi_{5586}(1019,\cdot)\) \(\chi_{5586}(1661,\cdot)\) \(\chi_{5586}(1817,\cdot)\) \(\chi_{5586}(2459,\cdot)\) \(\chi_{5586}(3257,\cdot)\) \(\chi_{5586}(3413,\cdot)\) \(\chi_{5586}(4055,\cdot)\) \(\chi_{5586}(4211,\cdot)\) \(\chi_{5586}(4853,\cdot)\) \(\chi_{5586}(5009,\cdot)\)
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
\((3725,4903,4999)\) → \((-1,e\left(\frac{10}{21}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5586 }(65, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)