from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3968, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,9,16]))
pari: [g,chi] = znchar(Mod(3099,3968))
Basic properties
| Modulus: | \(3968\) | |
| Conductor: | \(3968\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3968.ch
\(\chi_{3968}(123,\cdot)\) \(\chi_{3968}(371,\cdot)\) \(\chi_{3968}(619,\cdot)\) \(\chi_{3968}(867,\cdot)\) \(\chi_{3968}(1115,\cdot)\) \(\chi_{3968}(1363,\cdot)\) \(\chi_{3968}(1611,\cdot)\) \(\chi_{3968}(1859,\cdot)\) \(\chi_{3968}(2107,\cdot)\) \(\chi_{3968}(2355,\cdot)\) \(\chi_{3968}(2603,\cdot)\) \(\chi_{3968}(2851,\cdot)\) \(\chi_{3968}(3099,\cdot)\) \(\chi_{3968}(3347,\cdot)\) \(\chi_{3968}(3595,\cdot)\) \(\chi_{3968}(3843,\cdot)\)
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: | 32.32.2283054469939826689646603434893544205211768622361793362696380065480141214463295488.1 |
Values on generators
\((2047,3845,2049)\) → \((-1,e\left(\frac{9}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 3968 }(3099, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) |
sage: chi.jacobi_sum(n)