from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1332, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,0,23]))
pari: [g,chi] = znchar(Mod(1189,1332))
Basic properties
| Modulus: | \(1332\) | |
| Conductor: | \(37\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{37}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1332.dn
\(\chi_{1332}(109,\cdot)\) \(\chi_{1332}(217,\cdot)\) \(\chi_{1332}(505,\cdot)\) \(\chi_{1332}(577,\cdot)\) \(\chi_{1332}(649,\cdot)\) \(\chi_{1332}(685,\cdot)\) \(\chi_{1332}(721,\cdot)\) \(\chi_{1332}(757,\cdot)\) \(\chi_{1332}(829,\cdot)\) \(\chi_{1332}(901,\cdot)\) \(\chi_{1332}(1189,\cdot)\) \(\chi_{1332}(1297,\cdot)\)
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
\((667,1037,1297)\) → \((1,1,e\left(\frac{23}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1332 }(1189, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)