Properties

Modulus 315
Conductor 315
Order 12
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 315.bx

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(315)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,3,4]))
 
pari: [g,chi] = znchar(Mod(2,315))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 315
Conductor = 315
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 = 315.bx
Orbit index = 50

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{315}(2,\cdot)\) \(\chi_{315}(32,\cdot)\) \(\chi_{315}(128,\cdot)\) \(\chi_{315}(158,\cdot)\)

Values on generators

\((281,127,136)\) → \((e\left(\frac{1}{6}\right),i,e\left(\frac{1}{3}\right))\)

Values

-1124811131617192223
\(1\)\(1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(i\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)
value at  e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 315 }(2,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{315}(2,\cdot)) = \sum_{r\in \Z/315\Z} \chi_{315}(2,r) e\left(\frac{2r}{315}\right) = 10.0913270452+14.6001752889i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 315 }(2,·),\chi_{ 315 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{315}(2,\cdot),\chi_{315}(1,\cdot)) = \sum_{r\in \Z/315\Z} \chi_{315}(2,r) \chi_{315}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 315 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{315}(2,·)) = \sum_{r \in \Z/315\Z} \chi_{315}(2,r) e\left(\frac{1 r + 2 r^{-1}}{315}\right) = 13.1893144274+3.5340661495i \)