from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(384, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,7,16]))
pari: [g,chi] = znchar(Mod(83,384))
Basic properties
Modulus: | \(384\) | |
Conductor: | \(384\) | 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 384.w
\(\chi_{384}(11,\cdot)\) \(\chi_{384}(35,\cdot)\) \(\chi_{384}(59,\cdot)\) \(\chi_{384}(83,\cdot)\) \(\chi_{384}(107,\cdot)\) \(\chi_{384}(131,\cdot)\) \(\chi_{384}(155,\cdot)\) \(\chi_{384}(179,\cdot)\) \(\chi_{384}(203,\cdot)\) \(\chi_{384}(227,\cdot)\) \(\chi_{384}(251,\cdot)\) \(\chi_{384}(275,\cdot)\) \(\chi_{384}(299,\cdot)\) \(\chi_{384}(323,\cdot)\) \(\chi_{384}(347,\cdot)\) \(\chi_{384}(371,\cdot)\)
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: | 32.32.135104323545903136978453058557785670637514001130337144105502507008.1 |
Values on generators
\((127,133,257)\) → \((-1,e\left(\frac{7}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 384 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)