from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4928, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,33,40,24]))
pari: [g,chi] = znchar(Mod(285,4928))
Basic properties
| Modulus: | \(4928\) | |
| Conductor: | \(4928\) | 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 4928.es
\(\chi_{4928}(285,\cdot)\) \(\chi_{4928}(549,\cdot)\) \(\chi_{4928}(901,\cdot)\) \(\chi_{4928}(1165,\cdot)\) \(\chi_{4928}(1517,\cdot)\) \(\chi_{4928}(1781,\cdot)\) \(\chi_{4928}(2133,\cdot)\) \(\chi_{4928}(2397,\cdot)\) \(\chi_{4928}(2749,\cdot)\) \(\chi_{4928}(3013,\cdot)\) \(\chi_{4928}(3365,\cdot)\) \(\chi_{4928}(3629,\cdot)\) \(\chi_{4928}(3981,\cdot)\) \(\chi_{4928}(4245,\cdot)\) \(\chi_{4928}(4597,\cdot)\) \(\chi_{4928}(4861,\cdot)\)
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
\((4159,1541,2817,3137)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{5}{6}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 4928 }(285, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)