from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,22,11,38]))
pari: [g,chi] = znchar(Mod(467,920))
Basic properties
| Modulus: | \(920\) | |
| Conductor: | \(920\) | 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 920.bs
\(\chi_{920}(43,\cdot)\) \(\chi_{920}(67,\cdot)\) \(\chi_{920}(83,\cdot)\) \(\chi_{920}(107,\cdot)\) \(\chi_{920}(203,\cdot)\) \(\chi_{920}(227,\cdot)\) \(\chi_{920}(267,\cdot)\) \(\chi_{920}(283,\cdot)\) \(\chi_{920}(387,\cdot)\) \(\chi_{920}(467,\cdot)\) \(\chi_{920}(523,\cdot)\) \(\chi_{920}(563,\cdot)\) \(\chi_{920}(603,\cdot)\) \(\chi_{920}(707,\cdot)\) \(\chi_{920}(723,\cdot)\) \(\chi_{920}(747,\cdot)\) \(\chi_{920}(787,\cdot)\) \(\chi_{920}(803,\cdot)\) \(\chi_{920}(843,\cdot)\) \(\chi_{920}(907,\cdot)\)
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: | 44.0.13383169230192059253459701104387771124501004765020501667165784506368000000000000000000000000000000000.1 |
Values on generators
\((231,461,737,281)\) → \((-1,-1,i,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
| \( \chi_{ 920 }(467, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) |
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)