from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1081, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,25]))
pari: [g,chi] = znchar(Mod(22,1081))
Basic properties
| Modulus: | \(1081\) | |
| Conductor: | \(1081\) | 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 1081.j
\(\chi_{1081}(22,\cdot)\) \(\chi_{1081}(45,\cdot)\) \(\chi_{1081}(91,\cdot)\) \(\chi_{1081}(114,\cdot)\) \(\chi_{1081}(137,\cdot)\) \(\chi_{1081}(160,\cdot)\) \(\chi_{1081}(229,\cdot)\) \(\chi_{1081}(275,\cdot)\) \(\chi_{1081}(321,\cdot)\) \(\chi_{1081}(344,\cdot)\) \(\chi_{1081}(367,\cdot)\) \(\chi_{1081}(436,\cdot)\) \(\chi_{1081}(505,\cdot)\) \(\chi_{1081}(528,\cdot)\) \(\chi_{1081}(574,\cdot)\) \(\chi_{1081}(597,\cdot)\) \(\chi_{1081}(689,\cdot)\) \(\chi_{1081}(735,\cdot)\) \(\chi_{1081}(781,\cdot)\) \(\chi_{1081}(804,\cdot)\) \(\chi_{1081}(919,\cdot)\) \(\chi_{1081}(1057,\cdot)\)
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: | 46.46.36655895649209236301982551721202737575299278801903892310770876621683915109917518654573451777500377652400169.1 |
Values on generators
\((189,898)\) → \((-1,e\left(\frac{25}{46}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1081 }(22, a) \) | \(1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) |
sage: chi.jacobi_sum(n)