from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5586, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,5,35]))
pari: [g,chi] = znchar(Mod(145,5586))
Basic properties
| Modulus: | \(5586\) | |
| Conductor: | \(931\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{931}(145,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5586.dz
\(\chi_{5586}(145,\cdot)\) \(\chi_{5586}(787,\cdot)\) \(\chi_{5586}(943,\cdot)\) \(\chi_{5586}(1585,\cdot)\) \(\chi_{5586}(1741,\cdot)\) \(\chi_{5586}(2539,\cdot)\) \(\chi_{5586}(3181,\cdot)\) \(\chi_{5586}(3337,\cdot)\) \(\chi_{5586}(3979,\cdot)\) \(\chi_{5586}(4777,\cdot)\) \(\chi_{5586}(4933,\cdot)\) \(\chi_{5586}(5575,\cdot)\)
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
\((3725,4903,4999)\) → \((1,e\left(\frac{5}{42}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 5586 }(145, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) |
sage: chi.jacobi_sum(n)