from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,20,21]))
pari: [g,chi] = znchar(Mod(1143,8624))
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(4312\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4312}(3299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.es
\(\chi_{8624}(1143,\cdot)\) \(\chi_{8624}(1495,\cdot)\) \(\chi_{8624}(2375,\cdot)\) \(\chi_{8624}(2727,\cdot)\) \(\chi_{8624}(3959,\cdot)\) \(\chi_{8624}(4839,\cdot)\) \(\chi_{8624}(5191,\cdot)\) \(\chi_{8624}(6071,\cdot)\) \(\chi_{8624}(6423,\cdot)\) \(\chi_{8624}(7303,\cdot)\) \(\chi_{8624}(7655,\cdot)\) \(\chi_{8624}(8535,\cdot)\)
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
\((5391,6469,7745,3137)\) → \((-1,-1,e\left(\frac{10}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8624 }(1143, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) |
sage: chi.jacobi_sum(n)