from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8820, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,0,4]))
pari: [g,chi] = znchar(Mod(5471,8820))
Basic properties
| Modulus: | \(8820\) | |
| Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{588}(179,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8820.gy
\(\chi_{8820}(431,\cdot)\) \(\chi_{8820}(611,\cdot)\) \(\chi_{8820}(1691,\cdot)\) \(\chi_{8820}(1871,\cdot)\) \(\chi_{8820}(2951,\cdot)\) \(\chi_{8820}(3131,\cdot)\) \(\chi_{8820}(4211,\cdot)\) \(\chi_{8820}(5471,\cdot)\) \(\chi_{8820}(5651,\cdot)\) \(\chi_{8820}(6911,\cdot)\) \(\chi_{8820}(7991,\cdot)\) \(\chi_{8820}(8171,\cdot)\)
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
\((4411,7841,7057,1081)\) → \((-1,-1,1,e\left(\frac{2}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 8820 }(5471, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) |
sage: chi.jacobi_sum(n)