from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6720, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,45,24,36,8]))
pari: [g,chi] = znchar(Mod(563,6720))
Basic properties
| Modulus: | \(6720\) | |
| Conductor: | \(6720\) | 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 6720.kc
\(\chi_{6720}(563,\cdot)\) \(\chi_{6720}(803,\cdot)\) \(\chi_{6720}(1067,\cdot)\) \(\chi_{6720}(1307,\cdot)\) \(\chi_{6720}(2243,\cdot)\) \(\chi_{6720}(2483,\cdot)\) \(\chi_{6720}(2747,\cdot)\) \(\chi_{6720}(2987,\cdot)\) \(\chi_{6720}(3923,\cdot)\) \(\chi_{6720}(4163,\cdot)\) \(\chi_{6720}(4427,\cdot)\) \(\chi_{6720}(4667,\cdot)\) \(\chi_{6720}(5603,\cdot)\) \(\chi_{6720}(5843,\cdot)\) \(\chi_{6720}(6107,\cdot)\) \(\chi_{6720}(6347,\cdot)\)
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
\((1471,3781,4481,5377,1921)\) → \((-1,e\left(\frac{15}{16}\right),-1,-i,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6720 }(563, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)