from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,35]))
pari: [g,chi] = znchar(Mod(167,1110))
Basic properties
Modulus: | \(1110\) | |
Conductor: | \(555\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{555}(167,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1110.cb
\(\chi_{1110}(17,\cdot)\) \(\chi_{1110}(113,\cdot)\) \(\chi_{1110}(143,\cdot)\) \(\chi_{1110}(167,\cdot)\) \(\chi_{1110}(203,\cdot)\) \(\chi_{1110}(227,\cdot)\) \(\chi_{1110}(257,\cdot)\) \(\chi_{1110}(353,\cdot)\) \(\chi_{1110}(533,\cdot)\) \(\chi_{1110}(653,\cdot)\) \(\chi_{1110}(827,\cdot)\) \(\chi_{1110}(947,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((371,667,631)\) → \((-1,i,e\left(\frac{35}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
\( \chi_{ 1110 }(167, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(e\left(\frac{4}{9}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)