from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(871, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,32]))
pari: [g,chi] = znchar(Mod(330,871))
Basic properties
| Modulus: | \(871\) | |
| Conductor: | \(871\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 871.bm
\(\chi_{871}(148,\cdot)\) \(\chi_{871}(174,\cdot)\) \(\chi_{871}(216,\cdot)\) \(\chi_{871}(226,\cdot)\) \(\chi_{871}(265,\cdot)\) \(\chi_{871}(330,\cdot)\) \(\chi_{871}(359,\cdot)\) \(\chi_{871}(411,\cdot)\) \(\chi_{871}(424,\cdot)\) \(\chi_{871}(528,\cdot)\) \(\chi_{871}(551,\cdot)\) \(\chi_{871}(684,\cdot)\) \(\chi_{871}(694,\cdot)\) \(\chi_{871}(710,\cdot)\) \(\chi_{871}(746,\cdot)\) \(\chi_{871}(759,\cdot)\) \(\chi_{871}(762,\cdot)\) \(\chi_{871}(801,\cdot)\) \(\chi_{871}(863,\cdot)\) \(\chi_{871}(866,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((470,404)\) → \((-i,e\left(\frac{8}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 871 }(330, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) |
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)