Properties

Conductor 37
Order 3
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 37.c

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(37)
 
sage: chi = H[10]
 
pari: [g,chi] = znchar(Mod(10,37))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 37
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 3
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 = 37.c
Orbit index = 3

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 ]
 

\(\chi_{37}(10,\cdot)\) \(\chi_{37}(26,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{2}{3}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{2}{3}\right)\)\(1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 37 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{37}(10,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(10,r) e\left(\frac{2r}{37}\right) = -6.0170685482+-0.8915638432i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 37 }(10,·),\chi_{ 37 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{37}(10,\cdot),\chi_{37}(1,\cdot)) = \sum_{r\in \Z/37\Z} \chi_{37}(10,r) \chi_{37}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 37 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{37}(10,·)) = \sum_{r \in \Z/37\Z} \chi_{37}(10,r) e\left(\frac{1 r + 2 r^{-1}}{37}\right) = 3.728645521+-6.4582034858i \)