Properties

Label 966.59
Modulus $966$
Conductor $483$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(966, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,11,42]))
 
pari: [g,chi] = znchar(Mod(59,966))
 

Basic properties

Modulus: \(966\)
Conductor: \(483\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{483}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 966.bd

\(\chi_{966}(59,\cdot)\) \(\chi_{966}(101,\cdot)\) \(\chi_{966}(131,\cdot)\) \(\chi_{966}(173,\cdot)\) \(\chi_{966}(215,\cdot)\) \(\chi_{966}(257,\cdot)\) \(\chi_{966}(269,\cdot)\) \(\chi_{966}(311,\cdot)\) \(\chi_{966}(353,\cdot)\) \(\chi_{966}(395,\cdot)\) \(\chi_{966}(509,\cdot)\) \(\chi_{966}(593,\cdot)\) \(\chi_{966}(647,\cdot)\) \(\chi_{966}(719,\cdot)\) \(\chi_{966}(731,\cdot)\) \(\chi_{966}(761,\cdot)\) \(\chi_{966}(857,\cdot)\) \(\chi_{966}(887,\cdot)\) \(\chi_{966}(899,\cdot)\) \(\chi_{966}(929,\cdot)\)

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

Values on generators

\((323,829,925)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{7}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 966 }(59,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{966}(59,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(59,r) e\left(\frac{r}{483}\right) = 5.2725460752+21.3354226085i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 966 }(59,·),\chi_{ 966 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{966}(59,\cdot),\chi_{966}(1,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(59,r) \chi_{966}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 966 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{966}(59,·)) = \sum_{r \in \Z/966\Z} \chi_{966}(59,r) e\left(\frac{1 r + 2 r^{-1}}{966}\right) = -0.0 \)