Properties

Label 768.5
Modulus $768$
Conductor $768$
Order $64$
Real no
Primitive yes
Minimal yes
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(64))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1,32]))
 
pari: [g,chi] = znchar(Mod(5,768))
 

Basic properties

Modulus: \(768\)
Conductor: \(768\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 768.y

\(\chi_{768}(5,\cdot)\) \(\chi_{768}(29,\cdot)\) \(\chi_{768}(53,\cdot)\) \(\chi_{768}(77,\cdot)\) \(\chi_{768}(101,\cdot)\) \(\chi_{768}(125,\cdot)\) \(\chi_{768}(149,\cdot)\) \(\chi_{768}(173,\cdot)\) \(\chi_{768}(197,\cdot)\) \(\chi_{768}(221,\cdot)\) \(\chi_{768}(245,\cdot)\) \(\chi_{768}(269,\cdot)\) \(\chi_{768}(293,\cdot)\) \(\chi_{768}(317,\cdot)\) \(\chi_{768}(341,\cdot)\) \(\chi_{768}(365,\cdot)\) \(\chi_{768}(389,\cdot)\) \(\chi_{768}(413,\cdot)\) \(\chi_{768}(437,\cdot)\) \(\chi_{768}(461,\cdot)\) \(\chi_{768}(485,\cdot)\) \(\chi_{768}(509,\cdot)\) \(\chi_{768}(533,\cdot)\) \(\chi_{768}(557,\cdot)\) \(\chi_{768}(581,\cdot)\) \(\chi_{768}(605,\cdot)\) \(\chi_{768}(629,\cdot)\) \(\chi_{768}(653,\cdot)\) \(\chi_{768}(677,\cdot)\) \(\chi_{768}(701,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((511,517,257)\) → \((1,e\left(\frac{1}{64}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{33}{64}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{53}{64}\right)\)\(e\left(\frac{47}{64}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{27}{64}\right)\)\(e\left(\frac{1}{8}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 768 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{768}(5,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(5,r) e\left(\frac{r}{384}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 768 }(5,·),\chi_{ 768 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{768}(5,\cdot),\chi_{768}(1,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(5,r) \chi_{768}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 768 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{768}(5,·)) = \sum_{r \in \Z/768\Z} \chi_{768}(5,r) e\left(\frac{1 r + 2 r^{-1}}{768}\right) = 0.0 \)