from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(485, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([8,25]))
pari: [g,chi] = znchar(Mod(257,485))
Basic properties
| Modulus: | \(485\) | |
| Conductor: | \(485\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | 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 485.bj
\(\chi_{485}(42,\cdot)\) \(\chi_{485}(78,\cdot)\) \(\chi_{485}(152,\cdot)\) \(\chi_{485}(213,\cdot)\) \(\chi_{485}(222,\cdot)\) \(\chi_{485}(257,\cdot)\) \(\chi_{485}(337,\cdot)\) \(\chi_{485}(342,\cdot)\) \(\chi_{485}(343,\cdot)\) \(\chi_{485}(358,\cdot)\) \(\chi_{485}(368,\cdot)\) \(\chi_{485}(408,\cdot)\) \(\chi_{485}(418,\cdot)\) \(\chi_{485}(422,\cdot)\) \(\chi_{485}(433,\cdot)\) \(\chi_{485}(457,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.32.2318482735674757180308401186646582923236113207344277714859125196933746337890625.1 |
Values on generators
\((292,296)\) → \((i,e\left(\frac{25}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 485 }(257, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) |
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)