sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(140, base_ring=CyclotomicField(12))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,9,4]))
pari: [g,chi] = znchar(Mod(93,140))
Basic properties
| Modulus: | \(140\) | |
| Conductor: | \(35\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{35}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 140.v
\(\chi_{140}(37,\cdot)\) \(\chi_{140}(53,\cdot)\) \(\chi_{140}(93,\cdot)\) \(\chi_{140}(137,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.0.11259376953125.1 |
Values on generators
\((71,57,101)\) → \((1,-i,e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{140}(93,\cdot)) = \sum_{r\in \Z/140\Z} \chi_{140}(93,r) e\left(\frac{r}{70}\right) = -10.4387939523+5.5705996823i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{140}(93,\cdot),\chi_{140}(1,\cdot)) = \sum_{r\in \Z/140\Z} \chi_{140}(93,r) \chi_{140}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{140}(93,·))
= \sum_{r \in \Z/140\Z}
\chi_{140}(93,r) e\left(\frac{1 r + 2 r^{-1}}{140}\right)
= -0.0 \)