from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8624, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,38,0]))
pari: [g,chi] = znchar(Mod(23,8624))
Basic properties
| Modulus: | \(8624\) | |
| Conductor: | \(392\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{392}(219,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8624.ev
\(\chi_{8624}(23,\cdot)\) \(\chi_{8624}(375,\cdot)\) \(\chi_{8624}(1607,\cdot)\) \(\chi_{8624}(2487,\cdot)\) \(\chi_{8624}(2839,\cdot)\) \(\chi_{8624}(3719,\cdot)\) \(\chi_{8624}(4071,\cdot)\) \(\chi_{8624}(4951,\cdot)\) \(\chi_{8624}(5303,\cdot)\) \(\chi_{8624}(6183,\cdot)\) \(\chi_{8624}(7415,\cdot)\) \(\chi_{8624}(7767,\cdot)\)
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: | 42.0.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1 |
Values on generators
\((5391,6469,7745,3137)\) → \((-1,-1,e\left(\frac{19}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 8624 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) |
sage: chi.jacobi_sum(n)