from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(376, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,0,40]))
pari: [g,chi] = znchar(Mod(9,376))
Basic properties
| Modulus: | \(376\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{47}(9,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 376.i
\(\chi_{376}(9,\cdot)\) \(\chi_{376}(17,\cdot)\) \(\chi_{376}(25,\cdot)\) \(\chi_{376}(49,\cdot)\) \(\chi_{376}(65,\cdot)\) \(\chi_{376}(81,\cdot)\) \(\chi_{376}(89,\cdot)\) \(\chi_{376}(97,\cdot)\) \(\chi_{376}(121,\cdot)\) \(\chi_{376}(145,\cdot)\) \(\chi_{376}(153,\cdot)\) \(\chi_{376}(169,\cdot)\) \(\chi_{376}(177,\cdot)\) \(\chi_{376}(209,\cdot)\) \(\chi_{376}(225,\cdot)\) \(\chi_{376}(241,\cdot)\) \(\chi_{376}(249,\cdot)\) \(\chi_{376}(289,\cdot)\) \(\chi_{376}(337,\cdot)\) \(\chi_{376}(345,\cdot)\) \(\chi_{376}(353,\cdot)\) \(\chi_{376}(361,\cdot)\)
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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Values on generators
\((95,189,193)\) → \((1,1,e\left(\frac{20}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 376 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
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)