from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(308, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,10,24]))
pari: [g,chi] = znchar(Mod(289,308))
Basic properties
| Modulus: | \(308\) | |
| Conductor: | \(77\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{77}(58,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 308.y
\(\chi_{308}(9,\cdot)\) \(\chi_{308}(25,\cdot)\) \(\chi_{308}(37,\cdot)\) \(\chi_{308}(53,\cdot)\) \(\chi_{308}(81,\cdot)\) \(\chi_{308}(93,\cdot)\) \(\chi_{308}(137,\cdot)\) \(\chi_{308}(289,\cdot)\)
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: | 15.15.886528337182930278529.1 |
Values on generators
\((155,45,57)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 308 }(289, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |
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)