from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(309, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,25]))
pari: [g,chi] = znchar(Mod(80,309))
Basic properties
| Modulus: | \(309\) | |
| Conductor: | \(309\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 309.k
\(\chi_{309}(80,\cdot)\) \(\chi_{309}(89,\cdot)\) \(\chi_{309}(95,\cdot)\) \(\chi_{309}(113,\cdot)\) \(\chi_{309}(125,\cdot)\) \(\chi_{309}(134,\cdot)\) \(\chi_{309}(140,\cdot)\) \(\chi_{309}(176,\cdot)\) \(\chi_{309}(197,\cdot)\) \(\chi_{309}(209,\cdot)\) \(\chi_{309}(230,\cdot)\) \(\chi_{309}(233,\cdot)\) \(\chi_{309}(245,\cdot)\) \(\chi_{309}(248,\cdot)\) \(\chi_{309}(275,\cdot)\) \(\chi_{309}(296,\cdot)\)
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: | 34.34.342523005011894297428856269332610453116457630461733441736562419892654124149.1 |
Values on generators
\((104,211)\) → \((-1,e\left(\frac{25}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 309 }(80, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) |
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)