from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,15,4,9]))
pari: [g,chi] = znchar(Mod(3143,9792))
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 \(\chi_{4896}(3755,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.is
\(\chi_{9792}(3143,\cdot)\) \(\chi_{9792}(3623,\cdot)\) \(\chi_{9792}(4055,\cdot)\) \(\chi_{9792}(4343,\cdot)\) \(\chi_{9792}(6887,\cdot)\) \(\chi_{9792}(7319,\cdot)\) \(\chi_{9792}(7607,\cdot)\) \(\chi_{9792}(9671,\cdot)\)
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: | 24.24.102702760180080482238862265758511365777986854716768014313536599353196544.4 |
Values on generators
\((7039,5509,8705,9217)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{6}\right),e\left(\frac{3}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 9792 }(3143, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)