from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,23,20]))
pari: [g,chi] = znchar(Mod(755,987))
Basic properties
Modulus: | \(987\) | |
Conductor: | \(987\) | 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 987.v
\(\chi_{987}(83,\cdot)\) \(\chi_{987}(209,\cdot)\) \(\chi_{987}(230,\cdot)\) \(\chi_{987}(251,\cdot)\) \(\chi_{987}(272,\cdot)\) \(\chi_{987}(314,\cdot)\) \(\chi_{987}(335,\cdot)\) \(\chi_{987}(356,\cdot)\) \(\chi_{987}(440,\cdot)\) \(\chi_{987}(482,\cdot)\) \(\chi_{987}(524,\cdot)\) \(\chi_{987}(545,\cdot)\) \(\chi_{987}(566,\cdot)\) \(\chi_{987}(629,\cdot)\) \(\chi_{987}(692,\cdot)\) \(\chi_{987}(713,\cdot)\) \(\chi_{987}(755,\cdot)\) \(\chi_{987}(776,\cdot)\) \(\chi_{987}(860,\cdot)\) \(\chi_{987}(902,\cdot)\) \(\chi_{987}(944,\cdot)\) \(\chi_{987}(965,\cdot)\)
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
\((659,283,757)\) → \((-1,-1,e\left(\frac{10}{23}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 987 }(755, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) |
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)