from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1332, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,12,17]))
pari: [g,chi] = znchar(Mod(499,1332))
Basic properties
| Modulus: | \(1332\) | |
| Conductor: | \(1332\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1332.dr
\(\chi_{1332}(187,\cdot)\) \(\chi_{1332}(283,\cdot)\) \(\chi_{1332}(331,\cdot)\) \(\chi_{1332}(463,\cdot)\) \(\chi_{1332}(499,\cdot)\) \(\chi_{1332}(679,\cdot)\) \(\chi_{1332}(799,\cdot)\) \(\chi_{1332}(871,\cdot)\) \(\chi_{1332}(967,\cdot)\) \(\chi_{1332}(979,\cdot)\) \(\chi_{1332}(1051,\cdot)\) \(\chi_{1332}(1327,\cdot)\)
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: | 36.36.42263008854516662438364849366996329933779192991109099024104855333205078569792559831318528.2 |
Values on generators
\((667,1037,1297)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{17}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 1332 }(499, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(i\) | \(e\left(\frac{1}{18}\right)\) | \(i\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)