from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
pari: [g,chi] = znchar(Mod(757,987))
Basic properties
Modulus: | \(987\) | |
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 987.t
\(\chi_{987}(22,\cdot)\) \(\chi_{987}(43,\cdot)\) \(\chi_{987}(85,\cdot)\) \(\chi_{987}(127,\cdot)\) \(\chi_{987}(211,\cdot)\) \(\chi_{987}(232,\cdot)\) \(\chi_{987}(274,\cdot)\) \(\chi_{987}(295,\cdot)\) \(\chi_{987}(358,\cdot)\) \(\chi_{987}(421,\cdot)\) \(\chi_{987}(442,\cdot)\) \(\chi_{987}(463,\cdot)\) \(\chi_{987}(505,\cdot)\) \(\chi_{987}(547,\cdot)\) \(\chi_{987}(631,\cdot)\) \(\chi_{987}(652,\cdot)\) \(\chi_{987}(673,\cdot)\) \(\chi_{987}(715,\cdot)\) \(\chi_{987}(736,\cdot)\) \(\chi_{987}(757,\cdot)\) \(\chi_{987}(778,\cdot)\) \(\chi_{987}(904,\cdot)\)
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
\((659,283,757)\) → \((1,1,e\left(\frac{1}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 987 }(757, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) |
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)