from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,16,21]))
pari: [g,chi] = znchar(Mod(79,3332))
Basic properties
| Modulus: | \(3332\) | |
| Conductor: | \(476\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{476}(79,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3332.cg
\(\chi_{3332}(79,\cdot)\) \(\chi_{3332}(275,\cdot)\) \(\chi_{3332}(471,\cdot)\) \(\chi_{3332}(1047,\cdot)\) \(\chi_{3332}(1059,\cdot)\) \(\chi_{3332}(1255,\cdot)\) \(\chi_{3332}(1439,\cdot)\) \(\chi_{3332}(1451,\cdot)\) \(\chi_{3332}(1635,\cdot)\) \(\chi_{3332}(1831,\cdot)\) \(\chi_{3332}(1843,\cdot)\) \(\chi_{3332}(2419,\cdot)\) \(\chi_{3332}(2615,\cdot)\) \(\chi_{3332}(2811,\cdot)\) \(\chi_{3332}(3019,\cdot)\) \(\chi_{3332}(3203,\cdot)\)
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
\((1667,885,785)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(79, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)