from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4160, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,9,24,32]))
pari: [g,chi] = znchar(Mod(3389,4160))
Basic properties
| Modulus: | \(4160\) | |
| Conductor: | \(4160\) | 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 4160.jp
\(\chi_{4160}(29,\cdot)\) \(\chi_{4160}(269,\cdot)\) \(\chi_{4160}(549,\cdot)\) \(\chi_{4160}(789,\cdot)\) \(\chi_{4160}(1069,\cdot)\) \(\chi_{4160}(1309,\cdot)\) \(\chi_{4160}(1589,\cdot)\) \(\chi_{4160}(1829,\cdot)\) \(\chi_{4160}(2109,\cdot)\) \(\chi_{4160}(2349,\cdot)\) \(\chi_{4160}(2629,\cdot)\) \(\chi_{4160}(2869,\cdot)\) \(\chi_{4160}(3149,\cdot)\) \(\chi_{4160}(3389,\cdot)\) \(\chi_{4160}(3669,\cdot)\) \(\chi_{4160}(3909,\cdot)\)
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
\((4031,261,2497,1601)\) → \((1,e\left(\frac{3}{16}\right),-1,e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 4160 }(3389, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{35}{48}\right)\) |
sage: chi.jacobi_sum(n)