from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(485, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,43]))
pari: [g,chi] = znchar(Mod(11,485))
Basic properties
Modulus: | \(485\) | |
Conductor: | \(97\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{97}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 485.bk
\(\chi_{485}(11,\cdot)\) \(\chi_{485}(31,\cdot)\) \(\chi_{485}(66,\cdot)\) \(\chi_{485}(86,\cdot)\) \(\chi_{485}(141,\cdot)\) \(\chi_{485}(146,\cdot)\) \(\chi_{485}(191,\cdot)\) \(\chi_{485}(196,\cdot)\) \(\chi_{485}(226,\cdot)\) \(\chi_{485}(266,\cdot)\) \(\chi_{485}(316,\cdot)\) \(\chi_{485}(356,\cdot)\) \(\chi_{485}(386,\cdot)\) \(\chi_{485}(391,\cdot)\) \(\chi_{485}(436,\cdot)\) \(\chi_{485}(441,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((292,296)\) → \((1,e\left(\frac{43}{48}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 485 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{48}\right)\) |
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)