from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,12,16,15]))
pari: [g,chi] = znchar(Mod(1249,9792))
Basic properties
| Modulus: | \(9792\) | |
| Conductor: | \(1224\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1224}(637,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.jc
\(\chi_{9792}(1249,\cdot)\) \(\chi_{9792}(2977,\cdot)\) \(\chi_{9792}(4129,\cdot)\) \(\chi_{9792}(4513,\cdot)\) \(\chi_{9792}(5857,\cdot)\) \(\chi_{9792}(6241,\cdot)\) \(\chi_{9792}(7393,\cdot)\) \(\chi_{9792}(9121,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((7039,5509,8705,9217)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9792 }(1249, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)