from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(141, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,1]))
pari: [g,chi] = znchar(Mod(5,141))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 141.g
\(\chi_{141}(5,\cdot)\) \(\chi_{141}(11,\cdot)\) \(\chi_{141}(20,\cdot)\) \(\chi_{141}(23,\cdot)\) \(\chi_{141}(26,\cdot)\) \(\chi_{141}(29,\cdot)\) \(\chi_{141}(35,\cdot)\) \(\chi_{141}(38,\cdot)\) \(\chi_{141}(41,\cdot)\) \(\chi_{141}(44,\cdot)\) \(\chi_{141}(62,\cdot)\) \(\chi_{141}(77,\cdot)\) \(\chi_{141}(80,\cdot)\) \(\chi_{141}(86,\cdot)\) \(\chi_{141}(92,\cdot)\) \(\chi_{141}(104,\cdot)\) \(\chi_{141}(107,\cdot)\) \(\chi_{141}(113,\cdot)\) \(\chi_{141}(116,\cdot)\) \(\chi_{141}(125,\cdot)\) \(\chi_{141}(134,\cdot)\) \(\chi_{141}(137,\cdot)\)
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: | \(\Q(\zeta_{141})^+\) |
Values on generators
\((95,52)\) → \((-1,e\left(\frac{1}{46}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 141 }(5, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) |
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)