sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(135, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([8,0]))
pari: [g,chi] = znchar(Mod(121,135))
Basic properties
| Modulus: | \(135\) | |
| Conductor: | \(27\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{27}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 135.k
\(\chi_{135}(16,\cdot)\) \(\chi_{135}(31,\cdot)\) \(\chi_{135}(61,\cdot)\) \(\chi_{135}(76,\cdot)\) \(\chi_{135}(106,\cdot)\) \(\chi_{135}(121,\cdot)\)
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
\((56,82)\) → \((e\left(\frac{4}{9}\right),1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | \(\Q(\zeta_{27})^+\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{135}(121,\cdot)) = \sum_{r\in \Z/135\Z} \chi_{135}(121,r) e\left(\frac{2r}{135}\right) = -4.1679543713+3.102927063i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{135}(121,\cdot),\chi_{135}(1,\cdot)) = \sum_{r\in \Z/135\Z} \chi_{135}(121,r) \chi_{135}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{135}(121,·))
= \sum_{r \in \Z/135\Z}
\chi_{135}(121,r) e\left(\frac{1 r + 2 r^{-1}}{135}\right)
= 1.9317685659+10.9556039474i \)