from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3328, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,27,0]))
pari: [g,chi] = znchar(Mod(183,3328))
Basic properties
Modulus: | \(3328\) | |
Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{128}(99,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3328.cq
\(\chi_{3328}(183,\cdot)\) \(\chi_{3328}(391,\cdot)\) \(\chi_{3328}(599,\cdot)\) \(\chi_{3328}(807,\cdot)\) \(\chi_{3328}(1015,\cdot)\) \(\chi_{3328}(1223,\cdot)\) \(\chi_{3328}(1431,\cdot)\) \(\chi_{3328}(1639,\cdot)\) \(\chi_{3328}(1847,\cdot)\) \(\chi_{3328}(2055,\cdot)\) \(\chi_{3328}(2263,\cdot)\) \(\chi_{3328}(2471,\cdot)\) \(\chi_{3328}(2679,\cdot)\) \(\chi_{3328}(2887,\cdot)\) \(\chi_{3328}(3095,\cdot)\) \(\chi_{3328}(3303,\cdot)\)
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: | 32.0.3138550867693340381917894711603833208051177722232017256448.1 |
Values on generators
\((1535,261,769)\) → \((-1,e\left(\frac{27}{32}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 3328 }(183, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)