Properties

Conductor 67
Order 66
Real No
Primitive Yes
Parity Odd
Orbit Label 67.h

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(67)
 
sage: chi = H[41]
 
pari: [g,chi] = znchar(Mod(41,67))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 67
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 66
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = Yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 67.h
Orbit index = 8

Galois orbit

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

\(\chi_{67}(2,\cdot)\) \(\chi_{67}(7,\cdot)\) \(\chi_{67}(11,\cdot)\) \(\chi_{67}(12,\cdot)\) \(\chi_{67}(13,\cdot)\) \(\chi_{67}(18,\cdot)\) \(\chi_{67}(20,\cdot)\) \(\chi_{67}(28,\cdot)\) \(\chi_{67}(31,\cdot)\) \(\chi_{67}(32,\cdot)\) \(\chi_{67}(34,\cdot)\) \(\chi_{67}(41,\cdot)\) \(\chi_{67}(44,\cdot)\) \(\chi_{67}(46,\cdot)\) \(\chi_{67}(48,\cdot)\) \(\chi_{67}(50,\cdot)\) \(\chi_{67}(51,\cdot)\) \(\chi_{67}(57,\cdot)\) \(\chi_{67}(61,\cdot)\) \(\chi_{67}(63,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{53}{66}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{25}{66}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{33})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 67 }(41,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{67}(41,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(41,r) e\left(\frac{2r}{67}\right) = -1.8421286904+-7.97537221i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 67 }(41,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{67}(41,\cdot),\chi_{67}(1,\cdot)) = \sum_{r\in \Z/67\Z} \chi_{67}(41,r) \chi_{67}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 67 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{67}(41,·)) = \sum_{r \in \Z/67\Z} \chi_{67}(41,r) e\left(\frac{1 r + 2 r^{-1}}{67}\right) = -5.9671067778+-8.3796289715i \)