sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(837, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([20,6]))
pari: [g,chi] = znchar(Mod(667,837))
Basic properties
| Modulus: | \(837\) | |
| Conductor: | \(279\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{279}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 837.bc
\(\chi_{837}(64,\cdot)\) \(\chi_{837}(343,\cdot)\) \(\chi_{837}(388,\cdot)\) \(\chi_{837}(442,\cdot)\) \(\chi_{837}(469,\cdot)\) \(\chi_{837}(667,\cdot)\) \(\chi_{837}(721,\cdot)\) \(\chi_{837}(748,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | 15.15.2746410307762150989067078161.1 |
Values on generators
\((218,406)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{5}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{837}(667,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(667,r) e\left(\frac{2r}{837}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{837}(667,\cdot),\chi_{837}(1,\cdot)) = \sum_{r\in \Z/837\Z} \chi_{837}(667,r) \chi_{837}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{837}(667,·))
= \sum_{r \in \Z/837\Z}
\chi_{837}(667,r) e\left(\frac{1 r + 2 r^{-1}}{837}\right)
= -0.0 \)