sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(764, base_ring=CyclotomicField(38))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([19,20]))
pari: [g,chi] = znchar(Mod(407,764))
Basic properties
| Modulus: | \(764\) | |
| Conductor: | \(764\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 764.k
\(\chi_{764}(107,\cdot)\) \(\chi_{764}(223,\cdot)\) \(\chi_{764}(227,\cdot)\) \(\chi_{764}(243,\cdot)\) \(\chi_{764}(327,\cdot)\) \(\chi_{764}(351,\cdot)\) \(\chi_{764}(371,\cdot)\) \(\chi_{764}(387,\cdot)\) \(\chi_{764}(407,\cdot)\) \(\chi_{764}(451,\cdot)\) \(\chi_{764}(503,\cdot)\) \(\chi_{764}(507,\cdot)\) \(\chi_{764}(535,\cdot)\) \(\chi_{764}(559,\cdot)\) \(\chi_{764}(579,\cdot)\) \(\chi_{764}(603,\cdot)\) \(\chi_{764}(723,\cdot)\) \(\chi_{764}(727,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.3600319611560698860610362435063272860144872381060102455880973038531255093138465367428016635904.1 |
Values on generators
\((383,401)\) → \((-1,e\left(\frac{10}{19}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \(-1\) | \(1\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(-1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{764}(407,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(407,r) e\left(\frac{r}{382}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{764}(407,\cdot),\chi_{764}(1,\cdot)) = \sum_{r\in \Z/764\Z} \chi_{764}(407,r) \chi_{764}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{764}(407,·))
= \sum_{r \in \Z/764\Z}
\chi_{764}(407,r) e\left(\frac{1 r + 2 r^{-1}}{764}\right)
= -5.7295739295+10.5873253017i \)