Properties

Label 768.19
Modulus $768$
Conductor $256$
Order $64$
Real no
Primitive no
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([32,23,0]))
 
pari: [g,chi] = znchar(Mod(19,768))
 

Basic properties

Modulus: \(768\)
Conductor: \(256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{256}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 768.bb

\(\chi_{768}(19,\cdot)\) \(\chi_{768}(43,\cdot)\) \(\chi_{768}(67,\cdot)\) \(\chi_{768}(91,\cdot)\) \(\chi_{768}(115,\cdot)\) \(\chi_{768}(139,\cdot)\) \(\chi_{768}(163,\cdot)\) \(\chi_{768}(187,\cdot)\) \(\chi_{768}(211,\cdot)\) \(\chi_{768}(235,\cdot)\) \(\chi_{768}(259,\cdot)\) \(\chi_{768}(283,\cdot)\) \(\chi_{768}(307,\cdot)\) \(\chi_{768}(331,\cdot)\) \(\chi_{768}(355,\cdot)\) \(\chi_{768}(379,\cdot)\) \(\chi_{768}(403,\cdot)\) \(\chi_{768}(427,\cdot)\) \(\chi_{768}(451,\cdot)\) \(\chi_{768}(475,\cdot)\) \(\chi_{768}(499,\cdot)\) \(\chi_{768}(523,\cdot)\) \(\chi_{768}(547,\cdot)\) \(\chi_{768}(571,\cdot)\) \(\chi_{768}(595,\cdot)\) \(\chi_{768}(619,\cdot)\) \(\chi_{768}(643,\cdot)\) \(\chi_{768}(667,\cdot)\) \(\chi_{768}(691,\cdot)\) \(\chi_{768}(715,\cdot)\) ...

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

\((511,517,257)\) → \((-1,e\left(\frac{23}{64}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{49}{64}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{23}{32}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{3}{8}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 768 }(19,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{768}(19,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(19,r) e\left(\frac{r}{384}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 768 }(19,·),\chi_{ 768 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{768}(19,\cdot),\chi_{768}(1,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(19,r) \chi_{768}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 768 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{768}(19,·)) = \sum_{r \in \Z/768\Z} \chi_{768}(19,r) e\left(\frac{1 r + 2 r^{-1}}{768}\right) = -31.5288845564+-5.4707804403i \)