Properties

Label 126.85
Modulus $126$
Conductor $9$
Order $3$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,0]))
 
pari: [g,chi] = znchar(Mod(85,126))
 

Basic properties

Modulus: \(126\)
Conductor: \(9\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{9}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 126.f

\(\chi_{126}(43,\cdot)\) \(\chi_{126}(85,\cdot)\)

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

\((29,73)\) → \((e\left(\frac{1}{3}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{9})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 126 }(85,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{126}(85,\cdot)) = \sum_{r\in \Z/126\Z} \chi_{126}(85,r) e\left(\frac{r}{63}\right) = 2.8190778624+-1.02606043i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 126 }(85,·),\chi_{ 126 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{126}(85,\cdot),\chi_{126}(1,\cdot)) = \sum_{r\in \Z/126\Z} \chi_{126}(85,r) \chi_{126}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 126 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{126}(85,·)) = \sum_{r \in \Z/126\Z} \chi_{126}(85,r) e\left(\frac{1 r + 2 r^{-1}}{126}\right) = 3.0733760093+5.3232433988i \)