from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(980, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,20]))
pari: [g,chi] = znchar(Mod(261,980))
Basic properties
| Modulus: | \(980\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 980.bg
\(\chi_{980}(81,\cdot)\) \(\chi_{980}(121,\cdot)\) \(\chi_{980}(221,\cdot)\) \(\chi_{980}(261,\cdot)\) \(\chi_{980}(401,\cdot)\) \(\chi_{980}(501,\cdot)\) \(\chi_{980}(541,\cdot)\) \(\chi_{980}(641,\cdot)\) \(\chi_{980}(681,\cdot)\) \(\chi_{980}(781,\cdot)\) \(\chi_{980}(821,\cdot)\) \(\chi_{980}(921,\cdot)\)
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: | Number field defined by a degree 21 polynomial |
Values on generators
\((491,197,101)\) → \((1,1,e\left(\frac{10}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
| \( \chi_{ 980 }(261, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
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)