from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(141, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(52,141))
Basic properties
Modulus: | \(141\) | |
Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{47}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 141.f
\(\chi_{141}(10,\cdot)\) \(\chi_{141}(13,\cdot)\) \(\chi_{141}(19,\cdot)\) \(\chi_{141}(22,\cdot)\) \(\chi_{141}(31,\cdot)\) \(\chi_{141}(40,\cdot)\) \(\chi_{141}(43,\cdot)\) \(\chi_{141}(52,\cdot)\) \(\chi_{141}(58,\cdot)\) \(\chi_{141}(67,\cdot)\) \(\chi_{141}(70,\cdot)\) \(\chi_{141}(73,\cdot)\) \(\chi_{141}(76,\cdot)\) \(\chi_{141}(82,\cdot)\) \(\chi_{141}(85,\cdot)\) \(\chi_{141}(88,\cdot)\) \(\chi_{141}(91,\cdot)\) \(\chi_{141}(109,\cdot)\) \(\chi_{141}(124,\cdot)\) \(\chi_{141}(127,\cdot)\) \(\chi_{141}(133,\cdot)\) \(\chi_{141}(139,\cdot)\)
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
\((95,52)\) → \((1,e\left(\frac{1}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 141 }(52, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) |
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)