from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2652, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,24,16,39]))
pari: [g,chi] = znchar(Mod(1355,2652))
Basic properties
| Modulus: | \(2652\) | |
| Conductor: | \(2652\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2652.fa
\(\chi_{2652}(107,\cdot)\) \(\chi_{2652}(347,\cdot)\) \(\chi_{2652}(419,\cdot)\) \(\chi_{2652}(503,\cdot)\) \(\chi_{2652}(575,\cdot)\) \(\chi_{2652}(887,\cdot)\) \(\chi_{2652}(1043,\cdot)\) \(\chi_{2652}(1127,\cdot)\) \(\chi_{2652}(1355,\cdot)\) \(\chi_{2652}(1439,\cdot)\) \(\chi_{2652}(1595,\cdot)\) \(\chi_{2652}(1907,\cdot)\) \(\chi_{2652}(1979,\cdot)\) \(\chi_{2652}(2063,\cdot)\) \(\chi_{2652}(2135,\cdot)\) \(\chi_{2652}(2375,\cdot)\)
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
\((1327,1769,613,1873)\) → \((-1,-1,e\left(\frac{1}{3}\right),e\left(\frac{13}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
| \( \chi_{ 2652 }(1355, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{48}\right)\) |
sage: chi.jacobi_sum(n)