from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(806, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([25,26]))
pari: [g,chi] = znchar(Mod(49,806))
Basic properties
| Modulus: | \(806\) | |
| Conductor: | \(403\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{403}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 806.bo
\(\chi_{806}(49,\cdot)\) \(\chi_{806}(69,\cdot)\) \(\chi_{806}(121,\cdot)\) \(\chi_{806}(257,\cdot)\) \(\chi_{806}(329,\cdot)\) \(\chi_{806}(355,\cdot)\) \(\chi_{806}(361,\cdot)\) \(\chi_{806}(413,\cdot)\)
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: | 30.30.4043069762060388435860383865333476800658978190634895286990232354626413.2 |
Values on generators
\((249,313)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{13}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 806 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) |
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)