Properties

Conductor 399
Order 2
Real Yes
Primitive Yes
Parity Odd
Orbit Label 399.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(399)
 
sage: chi = H[398]
 
pari: [g,chi] = znchar(Mod(398,399))
 

Kronecker symbol representation

sage: kronecker_character(-399)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{-399}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 399
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2
Real = Yes
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 = 399.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 ]
 

\(\chi_{399}(398,\cdot)\)

Values on generators

\((134,115,211)\) → \((-1,-1,-1)\)

Values

-112458101113161720
\(-1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 399 }(398,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{399}(398,\cdot)) = \sum_{r\in \Z/399\Z} \chi_{399}(398,r) e\left(\frac{2r}{399}\right) = 19.9749843554i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 399 }(398,·),\chi_{ 399 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{399}(398,\cdot),\chi_{399}(1,\cdot)) = \sum_{r\in \Z/399\Z} \chi_{399}(398,r) \chi_{399}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 399 }(398,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{399}(398,·)) = \sum_{r \in \Z/399\Z} \chi_{399}(398,r) e\left(\frac{1 r + 2 r^{-1}}{399}\right) = -0.0 \)