from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(297, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,9]))
pari: [g,chi] = znchar(Mod(8,297))
Basic properties
| Modulus: | \(297\) | |
| Conductor: | \(99\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{99}(74,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 297.t
\(\chi_{297}(8,\cdot)\) \(\chi_{297}(17,\cdot)\) \(\chi_{297}(35,\cdot)\) \(\chi_{297}(62,\cdot)\) \(\chi_{297}(116,\cdot)\) \(\chi_{297}(206,\cdot)\) \(\chi_{297}(233,\cdot)\) \(\chi_{297}(260,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
| Fixed field: | \(\Q(\zeta_{99})^+\) |
Values on generators
\((56,244)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 297 }(8, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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)