from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5376, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,11,0,8]))
pari: [g,chi] = znchar(Mod(559,5376))
Basic properties
| Modulus: | \(5376\) | |
| Conductor: | \(448\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{448}(419,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5376.ch
\(\chi_{5376}(559,\cdot)\) \(\chi_{5376}(1231,\cdot)\) \(\chi_{5376}(1903,\cdot)\) \(\chi_{5376}(2575,\cdot)\) \(\chi_{5376}(3247,\cdot)\) \(\chi_{5376}(3919,\cdot)\) \(\chi_{5376}(4591,\cdot)\) \(\chi_{5376}(5263,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.3484608386920116940487669055488.4 |
Values on generators
\((2815,5125,1793,4609)\) → \((-1,e\left(\frac{11}{16}\right),1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5376 }(559, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(-1\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)