sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(112, base_ring=CyclotomicField(12))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,3,8]))
pari: [g,chi] = znchar(Mod(53,112))
Basic properties
| Modulus: | \(112\) | |
| Conductor: | \(112\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | 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 112.w
\(\chi_{112}(37,\cdot)\) \(\chi_{112}(53,\cdot)\) \(\chi_{112}(93,\cdot)\) \(\chi_{112}(109,\cdot)\)
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: | 12.12.49519263525896192.1 |
Values on generators
\((15,85,17)\) → \((1,i,e\left(\frac{2}{3}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{112}(53,\cdot)) = \sum_{r\in \Z/112\Z} \chi_{112}(53,r) e\left(\frac{r}{56}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{112}(53,\cdot),\chi_{112}(1,\cdot)) = \sum_{r\in \Z/112\Z} \chi_{112}(53,r) \chi_{112}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{112}(53,·))
= \sum_{r \in \Z/112\Z}
\chi_{112}(53,r) e\left(\frac{1 r + 2 r^{-1}}{112}\right)
= 3.3041034702+-4.3059914769i \)