sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(23, base_ring=CyclotomicField(22))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([4]))
pari: [g,chi] = znchar(Mod(4,23))
Basic properties
| Modulus: | \(23\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | 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 23.c
\(\chi_{23}(2,\cdot)\) \(\chi_{23}(3,\cdot)\) \(\chi_{23}(4,\cdot)\) \(\chi_{23}(6,\cdot)\) \(\chi_{23}(8,\cdot)\) \(\chi_{23}(9,\cdot)\) \(\chi_{23}(12,\cdot)\) \(\chi_{23}(13,\cdot)\) \(\chi_{23}(16,\cdot)\) \(\chi_{23}(18,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | \(\Q(\zeta_{23})^+\) |
Values on generators
\(5\) → \(e\left(\frac{2}{11}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{23}(4,\cdot)) = \sum_{r\in \Z/23\Z} \chi_{23}(4,r) e\left(\frac{2r}{23}\right) = -0.8012485418+-4.7284247667i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{23}(4,\cdot),\chi_{23}(1,\cdot)) = \sum_{r\in \Z/23\Z} \chi_{23}(4,r) \chi_{23}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{23}(4,·))
= \sum_{r \in \Z/23\Z}
\chi_{23}(4,r) e\left(\frac{1 r + 2 r^{-1}}{23}\right)
= -1.997928302+-4.3748527401i \)