from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4760, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,24,36,16,45]))
pari: [g,chi] = znchar(Mod(1213,4760))
Basic properties
| Modulus: | \(4760\) | |
| Conductor: | \(4760\) | 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 4760.kd
\(\chi_{4760}(37,\cdot)\) \(\chi_{4760}(277,\cdot)\) \(\chi_{4760}(333,\cdot)\) \(\chi_{4760}(653,\cdot)\) \(\chi_{4760}(1117,\cdot)\) \(\chi_{4760}(1213,\cdot)\) \(\chi_{4760}(1397,\cdot)\) \(\chi_{4760}(2013,\cdot)\) \(\chi_{4760}(2573,\cdot)\) \(\chi_{4760}(3173,\cdot)\) \(\chi_{4760}(3397,\cdot)\) \(\chi_{4760}(3677,\cdot)\) \(\chi_{4760}(3733,\cdot)\) \(\chi_{4760}(4517,\cdot)\) \(\chi_{4760}(4533,\cdot)\) \(\chi_{4760}(4757,\cdot)\)
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
\((1191,2381,2857,1361,3641)\) → \((1,-1,-i,e\left(\frac{1}{3}\right),e\left(\frac{15}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 4760 }(1213, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(-1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)