from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(261, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([7,30]))
pari: [g,chi] = znchar(Mod(110,261))
Basic properties
Modulus: | \(261\) | |
Conductor: | \(261\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 261.t
\(\chi_{261}(20,\cdot)\) \(\chi_{261}(23,\cdot)\) \(\chi_{261}(65,\cdot)\) \(\chi_{261}(74,\cdot)\) \(\chi_{261}(83,\cdot)\) \(\chi_{261}(110,\cdot)\) \(\chi_{261}(140,\cdot)\) \(\chi_{261}(194,\cdot)\) \(\chi_{261}(227,\cdot)\) \(\chi_{261}(239,\cdot)\) \(\chi_{261}(248,\cdot)\) \(\chi_{261}(257,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | 42.0.50695215285987529776146634789549734025587976443244441326301426824919673222871754267.1 |
Values on generators
\((146,118)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 261 }(110, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) |
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)