from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3200, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,13,8]))
pari: [g,chi] = znchar(Mod(107,3200))
Basic properties
| Modulus: | \(3200\) | |
| Conductor: | \(640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{640}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3200.cf
\(\chi_{3200}(107,\cdot)\) \(\chi_{3200}(243,\cdot)\) \(\chi_{3200}(507,\cdot)\) \(\chi_{3200}(643,\cdot)\) \(\chi_{3200}(907,\cdot)\) \(\chi_{3200}(1043,\cdot)\) \(\chi_{3200}(1307,\cdot)\) \(\chi_{3200}(1443,\cdot)\) \(\chi_{3200}(1707,\cdot)\) \(\chi_{3200}(1843,\cdot)\) \(\chi_{3200}(2107,\cdot)\) \(\chi_{3200}(2243,\cdot)\) \(\chi_{3200}(2507,\cdot)\) \(\chi_{3200}(2643,\cdot)\) \(\chi_{3200}(2907,\cdot)\) \(\chi_{3200}(3043,\cdot)\)
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: | 32.32.187072209578355573530071658587684226515959365500928000000000000000000000000.2 |
Values on generators
\((1151,901,2177)\) → \((-1,e\left(\frac{13}{32}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 3200 }(107, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) |
sage: chi.jacobi_sum(n)