Properties

Label 324.163
Modulus $324$
Conductor $4$
Order $2$
Real yes
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(324)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([1,0]))
 
pari: [g,chi] = znchar(Mod(163,324))
 

Basic properties

Modulus: \(324\)
Conductor: \(4\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{4}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 324.d

\(\chi_{324}(163,\cdot)\)

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

Values on generators

\((163,245)\) → \((-1,1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{-1}) \)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 324 }(163,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{324}(163,\cdot)) = \sum_{r\in \Z/324\Z} \chi_{324}(163,r) e\left(\frac{r}{162}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 324 }(163,·),\chi_{ 324 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{324}(163,\cdot),\chi_{324}(1,\cdot)) = \sum_{r\in \Z/324\Z} \chi_{324}(163,r) \chi_{324}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 324 }(163,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{324}(163,·)) = \sum_{r \in \Z/324\Z} \chi_{324}(163,r) e\left(\frac{1 r + 2 r^{-1}}{324}\right) = 0.0 \)