from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,25]))
pari: [g,chi] = znchar(Mod(2351,2368))
Basic properties
| Modulus: | \(2368\) | |
| Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{592}(131,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2368.db
\(\chi_{2368}(79,\cdot)\) \(\chi_{2368}(207,\cdot)\) \(\chi_{2368}(431,\cdot)\) \(\chi_{2368}(463,\cdot)\) \(\chi_{2368}(943,\cdot)\) \(\chi_{2368}(975,\cdot)\) \(\chi_{2368}(1199,\cdot)\) \(\chi_{2368}(1327,\cdot)\) \(\chi_{2368}(1423,\cdot)\) \(\chi_{2368}(1519,\cdot)\) \(\chi_{2368}(2255,\cdot)\) \(\chi_{2368}(2351,\cdot)\)
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: | 36.36.4886860176107258124616704873602845327686728999915307588219200292503475176863258640384.2 |
Values on generators
\((1407,1925,705)\) → \((-1,-i,e\left(\frac{25}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2368 }(2351, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) |
sage: chi.jacobi_sum(n)