from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,32]))
pari: [g,chi] = znchar(Mod(269,2368))
Basic properties
| Modulus: | \(2368\) | |
| Conductor: | \(2368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2368.dj
\(\chi_{2368}(269,\cdot)\) \(\chi_{2368}(285,\cdot)\) \(\chi_{2368}(565,\cdot)\) \(\chi_{2368}(581,\cdot)\) \(\chi_{2368}(861,\cdot)\) \(\chi_{2368}(877,\cdot)\) \(\chi_{2368}(1157,\cdot)\) \(\chi_{2368}(1173,\cdot)\) \(\chi_{2368}(1453,\cdot)\) \(\chi_{2368}(1469,\cdot)\) \(\chi_{2368}(1749,\cdot)\) \(\chi_{2368}(1765,\cdot)\) \(\chi_{2368}(2045,\cdot)\) \(\chi_{2368}(2061,\cdot)\) \(\chi_{2368}(2341,\cdot)\) \(\chi_{2368}(2357,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((1407,1925,705)\) → \((1,e\left(\frac{15}{16}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2368 }(269, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) |
sage: chi.jacobi_sum(n)