from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1344, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,45,0,16]))
pari: [g,chi] = znchar(Mod(205,1344))
Basic properties
| Modulus: | \(1344\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(205,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1344.cz
\(\chi_{1344}(37,\cdot)\) \(\chi_{1344}(109,\cdot)\) \(\chi_{1344}(205,\cdot)\) \(\chi_{1344}(277,\cdot)\) \(\chi_{1344}(373,\cdot)\) \(\chi_{1344}(445,\cdot)\) \(\chi_{1344}(541,\cdot)\) \(\chi_{1344}(613,\cdot)\) \(\chi_{1344}(709,\cdot)\) \(\chi_{1344}(781,\cdot)\) \(\chi_{1344}(877,\cdot)\) \(\chi_{1344}(949,\cdot)\) \(\chi_{1344}(1045,\cdot)\) \(\chi_{1344}(1117,\cdot)\) \(\chi_{1344}(1213,\cdot)\) \(\chi_{1344}(1285,\cdot)\)
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
\((127,1093,449,577)\) → \((1,e\left(\frac{15}{16}\right),1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1344 }(205, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{48}\right)\) |
sage: chi.jacobi_sum(n)