from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1617, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,31,0]))
pari: [g,chi] = znchar(Mod(584,1617))
Basic properties
Modulus: | \(1617\) | |
Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{147}(143,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1617.bt
\(\chi_{1617}(89,\cdot)\) \(\chi_{1617}(122,\cdot)\) \(\chi_{1617}(320,\cdot)\) \(\chi_{1617}(353,\cdot)\) \(\chi_{1617}(551,\cdot)\) \(\chi_{1617}(584,\cdot)\) \(\chi_{1617}(782,\cdot)\) \(\chi_{1617}(1013,\cdot)\) \(\chi_{1617}(1046,\cdot)\) \(\chi_{1617}(1277,\cdot)\) \(\chi_{1617}(1475,\cdot)\) \(\chi_{1617}(1508,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((1079,199,442)\) → \((-1,e\left(\frac{31}{42}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
\( \chi_{ 1617 }(584, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)