from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(920, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,22,11,12]))
pari: [g,chi] = znchar(Mod(77,920))
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.bu
\(\chi_{920}(13,\cdot)\) \(\chi_{920}(77,\cdot)\) \(\chi_{920}(117,\cdot)\) \(\chi_{920}(133,\cdot)\) \(\chi_{920}(173,\cdot)\) \(\chi_{920}(197,\cdot)\) \(\chi_{920}(213,\cdot)\) \(\chi_{920}(317,\cdot)\) \(\chi_{920}(357,\cdot)\) \(\chi_{920}(397,\cdot)\) \(\chi_{920}(453,\cdot)\) \(\chi_{920}(533,\cdot)\) \(\chi_{920}(637,\cdot)\) \(\chi_{920}(653,\cdot)\) \(\chi_{920}(693,\cdot)\) \(\chi_{920}(717,\cdot)\) \(\chi_{920}(813,\cdot)\) \(\chi_{920}(837,\cdot)\) \(\chi_{920}(853,\cdot)\) \(\chi_{920}(877,\cdot)\)
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
\((231,461,737,281)\) → \((1,-1,i,e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(27\) | \(29\) |
\( \chi_{ 920 }(77, a) \) | \(-1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) |
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)