from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(141, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,18]))
pari: [g,chi] = znchar(Mod(2,141))
Basic properties
Modulus: | \(141\) | |
Conductor: | \(141\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 141.h
\(\chi_{141}(2,\cdot)\) \(\chi_{141}(8,\cdot)\) \(\chi_{141}(14,\cdot)\) \(\chi_{141}(17,\cdot)\) \(\chi_{141}(32,\cdot)\) \(\chi_{141}(50,\cdot)\) \(\chi_{141}(53,\cdot)\) \(\chi_{141}(56,\cdot)\) \(\chi_{141}(59,\cdot)\) \(\chi_{141}(65,\cdot)\) \(\chi_{141}(68,\cdot)\) \(\chi_{141}(71,\cdot)\) \(\chi_{141}(74,\cdot)\) \(\chi_{141}(83,\cdot)\) \(\chi_{141}(89,\cdot)\) \(\chi_{141}(98,\cdot)\) \(\chi_{141}(101,\cdot)\) \(\chi_{141}(110,\cdot)\) \(\chi_{141}(119,\cdot)\) \(\chi_{141}(122,\cdot)\) \(\chi_{141}(128,\cdot)\) \(\chi_{141}(131,\cdot)\)
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: | 46.0.3516370336176915030779886015601767871077707157889593350075735586626118367196692091787.1 |
Values on generators
\((95,52)\) → \((-1,e\left(\frac{9}{23}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 141 }(2, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) |
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)