from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4998, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,16,3]))
pari: [g,chi] = znchar(Mod(275,4998))
Basic properties
| Modulus: | \(4998\) | |
| Conductor: | \(357\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{357}(275,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4998.ce
\(\chi_{4998}(275,\cdot)\) \(\chi_{4998}(1145,\cdot)\) \(\chi_{4998}(1439,\cdot)\) \(\chi_{4998}(1451,\cdot)\) \(\chi_{4998}(1745,\cdot)\) \(\chi_{4998}(2615,\cdot)\) \(\chi_{4998}(2921,\cdot)\) \(\chi_{4998}(3203,\cdot)\) \(\chi_{4998}(3497,\cdot)\) \(\chi_{4998}(3509,\cdot)\) \(\chi_{4998}(3803,\cdot)\) \(\chi_{4998}(4085,\cdot)\) \(\chi_{4998}(4379,\cdot)\) \(\chi_{4998}(4391,\cdot)\) \(\chi_{4998}(4685,\cdot)\) \(\chi_{4998}(4967,\cdot)\)
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
\((1667,2551,4117)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{1}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4998 }(275, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(i\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)