from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(564, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,23,6]))
pari: [g,chi] = znchar(Mod(491,564))
Basic properties
| Modulus: | \(564\) | |
| Conductor: | \(564\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(46\) | 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 564.o
\(\chi_{564}(59,\cdot)\) \(\chi_{564}(71,\cdot)\) \(\chi_{564}(83,\cdot)\) \(\chi_{564}(119,\cdot)\) \(\chi_{564}(131,\cdot)\) \(\chi_{564}(143,\cdot)\) \(\chi_{564}(155,\cdot)\) \(\chi_{564}(191,\cdot)\) \(\chi_{564}(215,\cdot)\) \(\chi_{564}(239,\cdot)\) \(\chi_{564}(251,\cdot)\) \(\chi_{564}(263,\cdot)\) \(\chi_{564}(299,\cdot)\) \(\chi_{564}(335,\cdot)\) \(\chi_{564}(347,\cdot)\) \(\chi_{564}(371,\cdot)\) \(\chi_{564}(383,\cdot)\) \(\chi_{564}(431,\cdot)\) \(\chi_{564}(455,\cdot)\) \(\chi_{564}(479,\cdot)\) \(\chi_{564}(491,\cdot)\) \(\chi_{564}(551,\cdot)\)
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 46 polynomial |
Values on generators
\((283,377,193)\) → \((-1,-1,e\left(\frac{3}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 564 }(491, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) |
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)