Properties

Label 384.367
Modulus $384$
Conductor $32$
Order $8$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(384, base_ring=CyclotomicField(8))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,7,0]))
 
pari: [g,chi] = znchar(Mod(367,384))
 

Basic properties

Modulus: \(384\)
Conductor: \(32\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{32}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 384.m

\(\chi_{384}(79,\cdot)\) \(\chi_{384}(175,\cdot)\) \(\chi_{384}(271,\cdot)\) \(\chi_{384}(367,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.2147483648.1

Values on generators

\((127,133,257)\) → \((-1,e\left(\frac{7}{8}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(-i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(-1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(367,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{384}(367,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(367,r) e\left(\frac{r}{192}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(367,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{384}(367,\cdot),\chi_{384}(1,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(367,r) \chi_{384}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(367,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{384}(367,·)) = \sum_{r \in \Z/384\Z} \chi_{384}(367,r) e\left(\frac{1 r + 2 r^{-1}}{384}\right) = 0.0 \)