from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,10]))
pari: [g,chi] = znchar(Mod(615,2368))
Basic properties
| Modulus: | \(2368\) | |
| Conductor: | \(1184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1184}(763,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2368.cp
\(\chi_{2368}(615,\cdot)\) \(\chi_{2368}(695,\cdot)\) \(\chi_{2368}(711,\cdot)\) \(\chi_{2368}(791,\cdot)\) \(\chi_{2368}(1799,\cdot)\) \(\chi_{2368}(1879,\cdot)\) \(\chi_{2368}(1895,\cdot)\) \(\chi_{2368}(1975,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.313492767805246124597661880230428322713253467077415687734427648.1 |
Values on generators
\((1407,1925,705)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{5}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2368 }(615, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) |
sage: chi.jacobi_sum(n)