from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(705, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([23,0,3]))
pari: [g,chi] = znchar(Mod(266,705))
Basic properties
Modulus: | \(705\) | |
Conductor: | \(141\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{141}(125,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 705.q
\(\chi_{705}(11,\cdot)\) \(\chi_{705}(26,\cdot)\) \(\chi_{705}(41,\cdot)\) \(\chi_{705}(86,\cdot)\) \(\chi_{705}(116,\cdot)\) \(\chi_{705}(146,\cdot)\) \(\chi_{705}(161,\cdot)\) \(\chi_{705}(176,\cdot)\) \(\chi_{705}(221,\cdot)\) \(\chi_{705}(266,\cdot)\) \(\chi_{705}(311,\cdot)\) \(\chi_{705}(326,\cdot)\) \(\chi_{705}(386,\cdot)\) \(\chi_{705}(416,\cdot)\) \(\chi_{705}(446,\cdot)\) \(\chi_{705}(461,\cdot)\) \(\chi_{705}(536,\cdot)\) \(\chi_{705}(626,\cdot)\) \(\chi_{705}(641,\cdot)\) \(\chi_{705}(656,\cdot)\) \(\chi_{705}(671,\cdot)\) \(\chi_{705}(701,\cdot)\)
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
\((236,142,616)\) → \((-1,1,e\left(\frac{3}{46}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 705 }(266, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{43}{46}\right)\) |
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)