sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(916, base_ring=CyclotomicField(38))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([19,2]))
pari: [g,chi] = znchar(Mod(623,916))
Basic properties
| Modulus: | \(916\) | |
| Conductor: | \(916\) | 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 916.p
\(\chi_{916}(27,\cdot)\) \(\chi_{916}(43,\cdot)\) \(\chi_{916}(203,\cdot)\) \(\chi_{916}(271,\cdot)\) \(\chi_{916}(443,\cdot)\) \(\chi_{916}(447,\cdot)\) \(\chi_{916}(475,\cdot)\) \(\chi_{916}(511,\cdot)\) \(\chi_{916}(515,\cdot)\) \(\chi_{916}(519,\cdot)\) \(\chi_{916}(579,\cdot)\) \(\chi_{916}(619,\cdot)\) \(\chi_{916}(623,\cdot)\) \(\chi_{916}(683,\cdot)\) \(\chi_{916}(703,\cdot)\) \(\chi_{916}(731,\cdot)\) \(\chi_{916}(747,\cdot)\) \(\chi_{916}(791,\cdot)\)
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: | 38.0.2472960613492762938009352687218362626942035203162587025151809700624254848124906369677396881178624.1 |
Values on generators
\((459,693)\) → \((-1,e\left(\frac{1}{19}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \(-1\) | \(1\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{916}(623,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(623,r) e\left(\frac{r}{458}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{916}(623,\cdot),\chi_{916}(1,\cdot)) = \sum_{r\in \Z/916\Z} \chi_{916}(623,r) \chi_{916}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{916}(623,·))
= \sum_{r \in \Z/916\Z}
\chi_{916}(623,r) e\left(\frac{1 r + 2 r^{-1}}{916}\right)
= -3.4375474551+10.0132336533i \)