from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1340, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,2]))
pari: [g,chi] = znchar(Mod(339,1340))
Basic properties
Modulus: | \(1340\) | |
Conductor: | \(1340\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 1340.bl
\(\chi_{1340}(19,\cdot)\) \(\chi_{1340}(39,\cdot)\) \(\chi_{1340}(199,\cdot)\) \(\chi_{1340}(339,\cdot)\) \(\chi_{1340}(419,\cdot)\) \(\chi_{1340}(479,\cdot)\) \(\chi_{1340}(559,\cdot)\) \(\chi_{1340}(619,\cdot)\) \(\chi_{1340}(639,\cdot)\) \(\chi_{1340}(659,\cdot)\) \(\chi_{1340}(719,\cdot)\) \(\chi_{1340}(839,\cdot)\) \(\chi_{1340}(859,\cdot)\) \(\chi_{1340}(959,\cdot)\) \(\chi_{1340}(1059,\cdot)\) \(\chi_{1340}(1119,\cdot)\) \(\chi_{1340}(1199,\cdot)\) \(\chi_{1340}(1239,\cdot)\) \(\chi_{1340}(1279,\cdot)\) \(\chi_{1340}(1299,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((671,537,1141)\) → \((-1,-1,e\left(\frac{1}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 1340 }(339, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)