from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(705, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,23,27]))
pari: [g,chi] = znchar(Mod(644,705))
Basic properties
Modulus: | \(705\) | |
Conductor: | \(705\) | 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 705.o
\(\chi_{705}(29,\cdot)\) \(\chi_{705}(44,\cdot)\) \(\chi_{705}(104,\cdot)\) \(\chi_{705}(134,\cdot)\) \(\chi_{705}(164,\cdot)\) \(\chi_{705}(179,\cdot)\) \(\chi_{705}(254,\cdot)\) \(\chi_{705}(344,\cdot)\) \(\chi_{705}(359,\cdot)\) \(\chi_{705}(374,\cdot)\) \(\chi_{705}(389,\cdot)\) \(\chi_{705}(419,\cdot)\) \(\chi_{705}(434,\cdot)\) \(\chi_{705}(449,\cdot)\) \(\chi_{705}(464,\cdot)\) \(\chi_{705}(509,\cdot)\) \(\chi_{705}(539,\cdot)\) \(\chi_{705}(569,\cdot)\) \(\chi_{705}(584,\cdot)\) \(\chi_{705}(599,\cdot)\) \(\chi_{705}(644,\cdot)\) \(\chi_{705}(689,\cdot)\)
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
\((236,142,616)\) → \((-1,-1,e\left(\frac{27}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 705 }(644, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) |
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)