Properties

Conductor 97
Order 12
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 97.g

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(97)
 
sage: chi = H[6]
 
pari: [g,chi] = znchar(Mod(6,97))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 97
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 12
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 97.g
Orbit index = 7

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 ]
 

\(\chi_{97}(6,\cdot)\) \(\chi_{97}(16,\cdot)\) \(\chi_{97}(81,\cdot)\) \(\chi_{97}(91,\cdot)\)

Values on generators

\(5\) → \(e\left(\frac{1}{12}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{6}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 97 }(6,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{97}(6,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(6,r) e\left(\frac{2r}{97}\right) = 5.9110883145+-7.8777557044i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 97 }(6,·),\chi_{ 97 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{97}(6,\cdot),\chi_{97}(1,\cdot)) = \sum_{r\in \Z/97\Z} \chi_{97}(6,r) \chi_{97}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 97 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{97}(6,·)) = \sum_{r \in \Z/97\Z} \chi_{97}(6,r) e\left(\frac{1 r + 2 r^{-1}}{97}\right) = -10.1403897765+5.8545567671i \)