from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(940, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,23,17]))
pari: [g,chi] = znchar(Mod(179,940))
Basic properties
| Modulus: | \(940\) | |
| Conductor: | \(940\) | 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 940.n
\(\chi_{940}(19,\cdot)\) \(\chi_{940}(39,\cdot)\) \(\chi_{940}(99,\cdot)\) \(\chi_{940}(139,\cdot)\) \(\chi_{940}(179,\cdot)\) \(\chi_{940}(199,\cdot)\) \(\chi_{940}(219,\cdot)\) \(\chi_{940}(279,\cdot)\) \(\chi_{940}(339,\cdot)\) \(\chi_{940}(359,\cdot)\) \(\chi_{940}(399,\cdot)\) \(\chi_{940}(419,\cdot)\) \(\chi_{940}(499,\cdot)\) \(\chi_{940}(539,\cdot)\) \(\chi_{940}(579,\cdot)\) \(\chi_{940}(599,\cdot)\) \(\chi_{940}(699,\cdot)\) \(\chi_{940}(819,\cdot)\) \(\chi_{940}(839,\cdot)\) \(\chi_{940}(859,\cdot)\) \(\chi_{940}(879,\cdot)\) \(\chi_{940}(919,\cdot)\)
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: | 46.46.1472629538247713057115908841446795301275975338441612336549649339098854081833834905600000000000000000000000.1 |
Values on generators
\((471,377,381)\) → \((-1,-1,e\left(\frac{17}{46}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 940 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) |
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)