Properties

Label 768.41
Modulus $768$
Conductor $384$
Order $32$
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(768, base_ring=CyclotomicField(32))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,31,16]))
 
pari: [g,chi] = znchar(Mod(41,768))
 

Basic properties

Modulus: \(768\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{384}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 768.x

\(\chi_{768}(41,\cdot)\) \(\chi_{768}(89,\cdot)\) \(\chi_{768}(137,\cdot)\) \(\chi_{768}(185,\cdot)\) \(\chi_{768}(233,\cdot)\) \(\chi_{768}(281,\cdot)\) \(\chi_{768}(329,\cdot)\) \(\chi_{768}(377,\cdot)\) \(\chi_{768}(425,\cdot)\) \(\chi_{768}(473,\cdot)\) \(\chi_{768}(521,\cdot)\) \(\chi_{768}(569,\cdot)\) \(\chi_{768}(617,\cdot)\) \(\chi_{768}(665,\cdot)\) \(\chi_{768}(713,\cdot)\) \(\chi_{768}(761,\cdot)\)

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: \(\Q(\zeta_{32})\)
Fixed field: 32.0.135104323545903136978453058557785670637514001130337144105502507008.1

Values on generators

\((511,517,257)\) → \((1,e\left(\frac{31}{32}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{21}{32}\right)\)\(-i\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 768 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{768}(41,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(41,r) e\left(\frac{r}{384}\right) = -38.7676434394+-5.7506366741i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 768 }(41,·),\chi_{ 768 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{768}(41,\cdot),\chi_{768}(1,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(41,r) \chi_{768}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 768 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{768}(41,·)) = \sum_{r \in \Z/768\Z} \chi_{768}(41,r) e\left(\frac{1 r + 2 r^{-1}}{768}\right) = 0.0 \)