Properties

Label 170.9
Modulus $170$
Conductor $85$
Order $8$
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(170)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4,1]))
 
pari: [g,chi] = znchar(Mod(9,170))
 

Basic properties

Modulus: \(170\)
Conductor: \(85\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{85}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 170.n

\(\chi_{170}(9,\cdot)\) \(\chi_{170}(19,\cdot)\) \(\chi_{170}(49,\cdot)\) \(\chi_{170}(59,\cdot)\)

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

\((137,71)\) → \((-1,e\left(\frac{1}{8}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(i\)\(e\left(\frac{7}{8}\right)\)\(1\)\(-i\)\(-1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.8.256461670625.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 170 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{170}(9,\cdot)) = \sum_{r\in \Z/170\Z} \chi_{170}(9,r) e\left(\frac{r}{85}\right) = 9.2095657883+0.4288332895i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 170 }(9,·),\chi_{ 170 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{170}(9,\cdot),\chi_{170}(1,\cdot)) = \sum_{r\in \Z/170\Z} \chi_{170}(9,r) \chi_{170}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 170 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{170}(9,·)) = \sum_{r \in \Z/170\Z} \chi_{170}(9,r) e\left(\frac{1 r + 2 r^{-1}}{170}\right) = 0.0 \)