from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2688, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,29,16,16]))
pari: [g,chi] = znchar(Mod(2603,2688))
Basic properties
Modulus: | \(2688\) | |
Conductor: | \(2688\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2688.cz
\(\chi_{2688}(83,\cdot)\) \(\chi_{2688}(251,\cdot)\) \(\chi_{2688}(419,\cdot)\) \(\chi_{2688}(587,\cdot)\) \(\chi_{2688}(755,\cdot)\) \(\chi_{2688}(923,\cdot)\) \(\chi_{2688}(1091,\cdot)\) \(\chi_{2688}(1259,\cdot)\) \(\chi_{2688}(1427,\cdot)\) \(\chi_{2688}(1595,\cdot)\) \(\chi_{2688}(1763,\cdot)\) \(\chi_{2688}(1931,\cdot)\) \(\chi_{2688}(2099,\cdot)\) \(\chi_{2688}(2267,\cdot)\) \(\chi_{2688}(2435,\cdot)\) \(\chi_{2688}(2603,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
Fixed field: | 32.0.4489912604053908534055314729400632754872833954383027744299245049859706934263808.1 |
Values on generators
\((127,2437,1793,1921)\) → \((-1,e\left(\frac{29}{32}\right),-1,-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 2688 }(2603, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(i\) | \(e\left(\frac{21}{32}\right)\) |
sage: chi.jacobi_sum(n)