from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(611, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,8]))
pari: [g,chi] = znchar(Mod(337,611))
Basic properties
| Modulus: | \(611\) | |
| Conductor: | \(611\) | 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 611.p
\(\chi_{611}(12,\cdot)\) \(\chi_{611}(25,\cdot)\) \(\chi_{611}(51,\cdot)\) \(\chi_{611}(64,\cdot)\) \(\chi_{611}(103,\cdot)\) \(\chi_{611}(155,\cdot)\) \(\chi_{611}(168,\cdot)\) \(\chi_{611}(194,\cdot)\) \(\chi_{611}(220,\cdot)\) \(\chi_{611}(259,\cdot)\) \(\chi_{611}(272,\cdot)\) \(\chi_{611}(285,\cdot)\) \(\chi_{611}(298,\cdot)\) \(\chi_{611}(324,\cdot)\) \(\chi_{611}(337,\cdot)\) \(\chi_{611}(350,\cdot)\) \(\chi_{611}(363,\cdot)\) \(\chi_{611}(441,\cdot)\) \(\chi_{611}(506,\cdot)\) \(\chi_{611}(519,\cdot)\) \(\chi_{611}(545,\cdot)\) \(\chi_{611}(571,\cdot)\)
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
\((236,287)\) → \((-1,e\left(\frac{4}{23}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 611 }(337, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) |
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)