from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,25]))
pari: [g,chi] = znchar(Mod(409,980))
Basic properties
Modulus: | \(980\) | |
Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{245}(164,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 980.br
\(\chi_{980}(89,\cdot)\) \(\chi_{980}(229,\cdot)\) \(\chi_{980}(269,\cdot)\) \(\chi_{980}(369,\cdot)\) \(\chi_{980}(409,\cdot)\) \(\chi_{980}(549,\cdot)\) \(\chi_{980}(649,\cdot)\) \(\chi_{980}(689,\cdot)\) \(\chi_{980}(789,\cdot)\) \(\chi_{980}(829,\cdot)\) \(\chi_{980}(929,\cdot)\) \(\chi_{980}(969,\cdot)\)
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: | 42.0.56353276529596271503862578540802938668269419115433656434196014026165008544921875.1 |
Values on generators
\((491,197,101)\) → \((1,-1,e\left(\frac{25}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 980 }(409, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)