from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([8,3]))
pari: [g,chi] = znchar(Mod(224,7803))
Basic properties
| Modulus: | \(7803\) | |
| Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{153}(20,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7803.bc
\(\chi_{7803}(224,\cdot)\) \(\chi_{7803}(503,\cdot)\) \(\chi_{7803}(827,\cdot)\) \(\chi_{7803}(1196,\cdot)\) \(\chi_{7803}(1520,\cdot)\) \(\chi_{7803}(1799,\cdot)\) \(\chi_{7803}(2825,\cdot)\) \(\chi_{7803}(3599,\cdot)\) \(\chi_{7803}(3626,\cdot)\) \(\chi_{7803}(3797,\cdot)\) \(\chi_{7803}(4121,\cdot)\) \(\chi_{7803}(5705,\cdot)\) \(\chi_{7803}(6029,\cdot)\) \(\chi_{7803}(6200,\cdot)\) \(\chi_{7803}(6227,\cdot)\) \(\chi_{7803}(7001,\cdot)\)
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
\((2891,2026)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{1}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 7803 }(224, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)