from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,1]))
pari: [g,chi] = znchar(Mod(409,1110))
Basic properties
| Modulus: | \(1110\) | |
| Conductor: | \(185\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{185}(39,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1110.cf
\(\chi_{1110}(19,\cdot)\) \(\chi_{1110}(79,\cdot)\) \(\chi_{1110}(109,\cdot)\) \(\chi_{1110}(409,\cdot)\) \(\chi_{1110}(439,\cdot)\) \(\chi_{1110}(499,\cdot)\) \(\chi_{1110}(649,\cdot)\) \(\chi_{1110}(679,\cdot)\) \(\chi_{1110}(799,\cdot)\) \(\chi_{1110}(829,\cdot)\) \(\chi_{1110}(949,\cdot)\) \(\chi_{1110}(979,\cdot)\)
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: | 36.0.29411719834995153896864925426307140281034671856927417346954345703125.1 |
Values on generators
\((371,667,631)\) → \((1,-1,e\left(\frac{1}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(41\) | \(43\) |
| \( \chi_{ 1110 }(409, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{1}{18}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)