from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,9]))
pari: [g,chi] = znchar(Mod(137,400))
Basic properties
| Modulus: | \(400\) | |
| Conductor: | \(200\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{200}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 400.bj
\(\chi_{400}(73,\cdot)\) \(\chi_{400}(137,\cdot)\) \(\chi_{400}(153,\cdot)\) \(\chi_{400}(217,\cdot)\) \(\chi_{400}(233,\cdot)\) \(\chi_{400}(297,\cdot)\) \(\chi_{400}(313,\cdot)\) \(\chi_{400}(377,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.0.3125000000000000000000000000000000.1 |
Values on generators
\((351,101,177)\) → \((1,-1,e\left(\frac{9}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 400 }(137, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) |
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)