from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(301, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,37]))
pari: [g,chi] = znchar(Mod(20,301))
Basic properties
Modulus: | \(301\) | |
Conductor: | \(301\) | 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 301.bn
\(\chi_{301}(20,\cdot)\) \(\chi_{301}(34,\cdot)\) \(\chi_{301}(48,\cdot)\) \(\chi_{301}(55,\cdot)\) \(\chi_{301}(62,\cdot)\) \(\chi_{301}(69,\cdot)\) \(\chi_{301}(76,\cdot)\) \(\chi_{301}(104,\cdot)\) \(\chi_{301}(132,\cdot)\) \(\chi_{301}(202,\cdot)\) \(\chi_{301}(244,\cdot)\) \(\chi_{301}(286,\cdot)\)
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: | 42.42.5239206534209069133889646882729090965733830111939046184442828436267483950448112600301.1 |
Values on generators
\((87,218)\) → \((-1,e\left(\frac{37}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 301 }(20, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) |
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)