from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(203, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,27]))
pari: [g,chi] = znchar(Mod(100,203))
Basic properties
Modulus: | \(203\) | |
Conductor: | \(203\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 203.t
\(\chi_{203}(4,\cdot)\) \(\chi_{203}(9,\cdot)\) \(\chi_{203}(51,\cdot)\) \(\chi_{203}(67,\cdot)\) \(\chi_{203}(93,\cdot)\) \(\chi_{203}(100,\cdot)\) \(\chi_{203}(109,\cdot)\) \(\chi_{203}(121,\cdot)\) \(\chi_{203}(149,\cdot)\) \(\chi_{203}(151,\cdot)\) \(\chi_{203}(158,\cdot)\) \(\chi_{203}(179,\cdot)\)
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: | 42.42.496897759422042196258605771077406782550407598249513303021389442457964675897236469.1 |
Values on generators
\((59,176)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 203 }(100, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)