sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(63, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([5,3]))
pari: [g,chi] = znchar(Mod(41,63))
Basic properties
| Modulus: | \(63\) | |
| Conductor: | \(63\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 63.o
\(\chi_{63}(20,\cdot)\) \(\chi_{63}(41,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\((29,10)\) → \((e\left(\frac{5}{6}\right),-1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(-1\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | 6.6.6751269.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{63}(41,\cdot)) = \sum_{r\in \Z/63\Z} \chi_{63}(41,r) e\left(\frac{2r}{63}\right) = 5.1019684832+-6.0802892691i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{63}(41,\cdot),\chi_{63}(1,\cdot)) = \sum_{r\in \Z/63\Z} \chi_{63}(41,r) \chi_{63}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{63}(41,·))
= \sum_{r \in \Z/63\Z}
\chi_{63}(41,r) e\left(\frac{1 r + 2 r^{-1}}{63}\right)
= 12.3862741079+-7.1512186904i \)