from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(172, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([0,9]))
pari: [g,chi] = znchar(Mod(45,172))
Basic properties
| Modulus: | \(172\) | |
| Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{43}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 172.l
\(\chi_{172}(45,\cdot)\) \(\chi_{172}(65,\cdot)\) \(\chi_{172}(113,\cdot)\) \(\chi_{172}(125,\cdot)\) \(\chi_{172}(137,\cdot)\) \(\chi_{172}(161,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | Number field defined by a degree 14 polynomial |
Values on generators
\((87,89)\) → \((1,e\left(\frac{9}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 172 }(45, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(-1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
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)