Properties

Label 384.41
Modulus $384$
Conductor $192$
Order $16$
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(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,8]))
 
pari: [g,chi] = znchar(Mod(41,384))
 

Basic properties

Modulus: \(384\)
Conductor: \(192\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{192}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 384.q

\(\chi_{384}(41,\cdot)\) \(\chi_{384}(89,\cdot)\) \(\chi_{384}(137,\cdot)\) \(\chi_{384}(185,\cdot)\) \(\chi_{384}(233,\cdot)\) \(\chi_{384}(281,\cdot)\) \(\chi_{384}(329,\cdot)\) \(\chi_{384}(377,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.0.3965881151245791007623610368.1

Values on generators

\((127,133,257)\) → \((1,e\left(\frac{15}{16}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(-i\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(-1\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 384 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{384}(41,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(41,r) e\left(\frac{r}{192}\right) = 24.44051901+-13.0637295793i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 384 }(41,·),\chi_{ 384 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{384}(41,\cdot),\chi_{384}(1,\cdot)) = \sum_{r\in \Z/384\Z} \chi_{384}(41,r) \chi_{384}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 384 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{384}(41,·)) = \sum_{r \in \Z/384\Z} \chi_{384}(41,r) e\left(\frac{1 r + 2 r^{-1}}{384}\right) = -0.0 \)