from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5824, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([8,11,0,4]))
pari: [g,chi] = znchar(Mod(99,5824))
Basic properties
| Modulus: | \(5824\) | |
| Conductor: | \(832\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{832}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5824.ii
\(\chi_{5824}(99,\cdot)\) \(\chi_{5824}(603,\cdot)\) \(\chi_{5824}(1555,\cdot)\) \(\chi_{5824}(2059,\cdot)\) \(\chi_{5824}(3011,\cdot)\) \(\chi_{5824}(3515,\cdot)\) \(\chi_{5824}(4467,\cdot)\) \(\chi_{5824}(4971,\cdot)\)
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: | 16.16.14082828326073370534001867210073571328.2 |
Values on generators
\((2367,1093,4161,4929)\) → \((-1,e\left(\frac{11}{16}\right),1,i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 5824 }(99, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)