Properties

Conductor 45
Order 12
Real no
Primitive no
Minimal yes
Parity odd
Orbit label 315.by

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(315)
 
sage: chi = H[43]
 
pari: [g,chi] = znchar(Mod(43,315))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 45
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 = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 315.by
Orbit index = 51

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 ]
 

\(\chi_{315}(22,\cdot)\) \(\chi_{315}(43,\cdot)\) \(\chi_{315}(148,\cdot)\) \(\chi_{315}(232,\cdot)\)

Values on generators

\((281,127,136)\) → \((e\left(\frac{2}{3}\right),-i,1)\)

Values

-1124811131617192223
\(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-i\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{12}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 315 }(43,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{315}(43,\cdot)) = \sum_{r\in \Z/315\Z} \chi_{315}(43,r) e\left(\frac{2r}{315}\right) = -5.0072399573+4.4640282269i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 315 }(43,·),\chi_{ 315 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{315}(43,\cdot),\chi_{315}(1,\cdot)) = \sum_{r\in \Z/315\Z} \chi_{315}(43,r) \chi_{315}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 315 }(43,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{315}(43,·)) = \sum_{r \in \Z/315\Z} \chi_{315}(43,r) e\left(\frac{1 r + 2 r^{-1}}{315}\right) = -15.9713192059+4.2795020833i \)