Properties

Label 67.17
Modulus $67$
Conductor $67$
Order $33$
Real no
Primitive yes
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(67)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([32]))
 
pari: [g,chi] = znchar(Mod(17,67))
 

Basic properties

Modulus: \(67\)
Conductor: \(67\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 67.g

\(\chi_{67}(4,\cdot)\) \(\chi_{67}(6,\cdot)\) \(\chi_{67}(10,\cdot)\) \(\chi_{67}(16,\cdot)\) \(\chi_{67}(17,\cdot)\) \(\chi_{67}(19,\cdot)\) \(\chi_{67}(21,\cdot)\) \(\chi_{67}(23,\cdot)\) \(\chi_{67}(26,\cdot)\) \(\chi_{67}(33,\cdot)\) \(\chi_{67}(35,\cdot)\) \(\chi_{67}(36,\cdot)\) \(\chi_{67}(39,\cdot)\) \(\chi_{67}(47,\cdot)\) \(\chi_{67}(49,\cdot)\) \(\chi_{67}(54,\cdot)\) \(\chi_{67}(55,\cdot)\) \(\chi_{67}(56,\cdot)\) \(\chi_{67}(60,\cdot)\) \(\chi_{67}(65,\cdot)\)

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

\(2\) → \(e\left(\frac{32}{33}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{7}{33}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree %d polynomial (not computed)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 67 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{67}(17,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(17,r) e\left(\frac{2r}{67}\right) = 5.9112765792+-5.6618732946i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 67 }(17,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{67}(17,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(17,r) \chi_{67}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 67 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{67}(17,·)) = \sum_{r \in \Z/67\Z} \chi_{67}(17,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right) = 2.38435914+-0.2276786924i \)