from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,21,4,12]))
pari: [g,chi] = znchar(Mod(407,9792))
Basic properties
| Modulus: | \(9792\) | |
| Conductor: | \(4896\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4896}(4691,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.jn
\(\chi_{9792}(407,\cdot)\) \(\chi_{9792}(2039,\cdot)\) \(\chi_{9792}(2855,\cdot)\) \(\chi_{9792}(4487,\cdot)\) \(\chi_{9792}(5303,\cdot)\) \(\chi_{9792}(6935,\cdot)\) \(\chi_{9792}(7751,\cdot)\) \(\chi_{9792}(9383,\cdot)\)
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: | 24.24.866047721013695910154048112067960927811965142155184048177152.1 |
Values on generators
\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{1}{6}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9792 }(407, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{8}\right)\) |
sage: chi.jacobi_sum(n)