from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9792, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,12,40,21]))
pari: [g,chi] = znchar(Mod(113,9792))
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 \(\chi_{2448}(725,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9792.nq
\(\chi_{9792}(113,\cdot)\) \(\chi_{9792}(209,\cdot)\) \(\chi_{9792}(785,\cdot)\) \(\chi_{9792}(1265,\cdot)\) \(\chi_{9792}(2513,\cdot)\) \(\chi_{9792}(3089,\cdot)\) \(\chi_{9792}(3377,\cdot)\) \(\chi_{9792}(4529,\cdot)\) \(\chi_{9792}(5297,\cdot)\) \(\chi_{9792}(6449,\cdot)\) \(\chi_{9792}(6737,\cdot)\) \(\chi_{9792}(7313,\cdot)\) \(\chi_{9792}(8561,\cdot)\) \(\chi_{9792}(9041,\cdot)\) \(\chi_{9792}(9617,\cdot)\) \(\chi_{9792}(9713,\cdot)\)
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
\((7039,5509,8705,9217)\) → \((1,i,e\left(\frac{5}{6}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 9792 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)