from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,36,40,9]))
pari: [g,chi] = znchar(Mod(401,9792))
Basic properties
| Modulus: | \(9792\) | |
| Conductor: | \(2448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2448}(2237,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.la
\(\chi_{9792}(401,\cdot)\) \(\chi_{9792}(1553,\cdot)\) \(\chi_{9792}(1841,\cdot)\) \(\chi_{9792}(2417,\cdot)\) \(\chi_{9792}(3665,\cdot)\) \(\chi_{9792}(4145,\cdot)\) \(\chi_{9792}(4721,\cdot)\) \(\chi_{9792}(4817,\cdot)\) \(\chi_{9792}(5009,\cdot)\) \(\chi_{9792}(5105,\cdot)\) \(\chi_{9792}(5681,\cdot)\) \(\chi_{9792}(6161,\cdot)\) \(\chi_{9792}(7409,\cdot)\) \(\chi_{9792}(7985,\cdot)\) \(\chi_{9792}(8273,\cdot)\) \(\chi_{9792}(9425,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((1,-i,e\left(\frac{5}{6}\right),e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9792 }(401, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)