from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7488, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,9,20,12]))
pari: [g,chi] = znchar(Mod(311,7488))
Basic properties
| Modulus: | \(7488\) | |
| Conductor: | \(3744\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3744}(3587,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7488.kq
\(\chi_{7488}(311,\cdot)\) \(\chi_{7488}(1559,\cdot)\) \(\chi_{7488}(2183,\cdot)\) \(\chi_{7488}(3431,\cdot)\) \(\chi_{7488}(4055,\cdot)\) \(\chi_{7488}(5303,\cdot)\) \(\chi_{7488}(5927,\cdot)\) \(\chi_{7488}(7175,\cdot)\)
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: | 24.24.34631794385751755449277772240000060381154481106891615240192.1 |
Values on generators
\((703,6085,5825,5761)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 7488 }(311, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{8}\right)\) |
sage: chi.jacobi_sum(n)