Properties

Conductor 77
Order 10
Real no
Primitive no
Minimal yes
Parity even
Orbit label 231.w

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(231)
 
sage: chi = H[13]
 
pari: [g,chi] = znchar(Mod(13,231))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 77
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 10
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 231.w
Orbit index = 23

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 ]
 

\(\chi_{231}(13,\cdot)\) \(\chi_{231}(118,\cdot)\) \(\chi_{231}(139,\cdot)\) \(\chi_{231}(160,\cdot)\)

Values on generators

\((155,199,211)\) → \((1,-1,e\left(\frac{1}{10}\right))\)

Values

-112458101316171920
\(1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{10}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 231 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{231}(13,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(13,r) e\left(\frac{2r}{231}\right) = 8.2838582491+-2.894424383i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 231 }(13,·),\chi_{ 231 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{231}(13,\cdot),\chi_{231}(1,\cdot)) = \sum_{r\in \Z/231\Z} \chi_{231}(13,r) \chi_{231}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 231 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{231}(13,·)) = \sum_{r \in \Z/231\Z} \chi_{231}(13,r) e\left(\frac{1 r + 2 r^{-1}}{231}\right) = 22.6197623985+7.3496063274i \)