from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(412, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,6]))
pari: [g,chi] = znchar(Mod(79,412))
Basic properties
| Modulus: | \(412\) | |
| Conductor: | \(412\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 412.k
\(\chi_{412}(23,\cdot)\) \(\chi_{412}(79,\cdot)\) \(\chi_{412}(111,\cdot)\) \(\chi_{412}(167,\cdot)\) \(\chi_{412}(175,\cdot)\) \(\chi_{412}(179,\cdot)\) \(\chi_{412}(203,\cdot)\) \(\chi_{412}(215,\cdot)\) \(\chi_{412}(219,\cdot)\) \(\chi_{412}(267,\cdot)\) \(\chi_{412}(287,\cdot)\) \(\chi_{412}(299,\cdot)\) \(\chi_{412}(323,\cdot)\) \(\chi_{412}(339,\cdot)\) \(\chi_{412}(343,\cdot)\) \(\chi_{412}(375,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | 34.0.442395848806444333196713449710663325979115564010458272524910178228286521344.1 |
Values on generators
\((207,5)\) → \((-1,e\left(\frac{3}{17}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 412 }(79, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) |
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)