from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(94, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([9]))
pari: [g,chi] = znchar(Mod(87,94))
Basic properties
Modulus: | \(94\) | |
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 \(\chi_{47}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 94.d
\(\chi_{94}(5,\cdot)\) \(\chi_{94}(11,\cdot)\) \(\chi_{94}(13,\cdot)\) \(\chi_{94}(15,\cdot)\) \(\chi_{94}(19,\cdot)\) \(\chi_{94}(23,\cdot)\) \(\chi_{94}(29,\cdot)\) \(\chi_{94}(31,\cdot)\) \(\chi_{94}(33,\cdot)\) \(\chi_{94}(35,\cdot)\) \(\chi_{94}(39,\cdot)\) \(\chi_{94}(41,\cdot)\) \(\chi_{94}(43,\cdot)\) \(\chi_{94}(45,\cdot)\) \(\chi_{94}(57,\cdot)\) \(\chi_{94}(67,\cdot)\) \(\chi_{94}(69,\cdot)\) \(\chi_{94}(73,\cdot)\) \(\chi_{94}(77,\cdot)\) \(\chi_{94}(85,\cdot)\) \(\chi_{94}(87,\cdot)\) \(\chi_{94}(91,\cdot)\)
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
\(5\) → \(e\left(\frac{9}{46}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 94 }(87, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) |
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)