from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(916, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,5]))
pari: [g,chi] = znchar(Mod(415,916))
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.o
\(\chi_{916}(11,\cdot)\) \(\chi_{916}(15,\cdot)\) \(\chi_{916}(187,\cdot)\) \(\chi_{916}(255,\cdot)\) \(\chi_{916}(415,\cdot)\) \(\chi_{916}(431,\cdot)\) \(\chi_{916}(583,\cdot)\) \(\chi_{916}(627,\cdot)\) \(\chi_{916}(643,\cdot)\) \(\chi_{916}(671,\cdot)\) \(\chi_{916}(691,\cdot)\) \(\chi_{916}(751,\cdot)\) \(\chi_{916}(755,\cdot)\) \(\chi_{916}(795,\cdot)\) \(\chi_{916}(855,\cdot)\) \(\chi_{916}(859,\cdot)\) \(\chi_{916}(863,\cdot)\) \(\chi_{916}(899,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{19})\) |
| Fixed field: | 38.0.566307980489842712804141765373005041569726061524232428759764421442954360220603558656123885789904896.1 |
Values on generators
\((459,693)\) → \((-1,e\left(\frac{5}{38}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 916 }(415, a) \) | \(-1\) | \(1\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{17}{38}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)