sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(384, base_ring=CyclotomicField(16))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,15,8]))
pari: [g,chi] = znchar(Mod(41,384))
Basic properties
| Modulus: | \(384\) | |
| Conductor: | \(192\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{192}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 384.q
\(\chi_{384}(41,\cdot)\) \(\chi_{384}(89,\cdot)\) \(\chi_{384}(137,\cdot)\) \(\chi_{384}(185,\cdot)\) \(\chi_{384}(233,\cdot)\) \(\chi_{384}(281,\cdot)\) \(\chi_{384}(329,\cdot)\) \(\chi_{384}(377,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.0.3965881151245791007623610368.1 |
Values on generators
\((127,133,257)\) → \((1,e\left(\frac{15}{16}\right),-1)\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(-i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(-1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{384}(41,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(41,r) e\left(\frac{r}{192}\right) = 24.44051901+-13.0637295793i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{384}(41,\cdot),\chi_{384}(1,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(41,r) \chi_{384}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{384}(41,·))
= \sum_{r \in \Z/384\Z}
\chi_{384}(41,r) e\left(\frac{1 r + 2 r^{-1}}{384}\right)
= -0.0 \)