Properties

Label 171.25
Modulus $171$
Conductor $171$
Order $9$
Real no
Primitive yes
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(171, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([12,14]))
 
pari: [g,chi] = znchar(Mod(25,171))
 

Basic properties

Modulus: \(171\)
Conductor: \(171\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 171.v

\(\chi_{171}(25,\cdot)\) \(\chi_{171}(61,\cdot)\) \(\chi_{171}(112,\cdot)\) \(\chi_{171}(130,\cdot)\) \(\chi_{171}(142,\cdot)\) \(\chi_{171}(157,\cdot)\)

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

\((20,154)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{9}\right)\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{7}{9}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.9025761726072081.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 171 }(25,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{171}(25,\cdot)) = \sum_{r\in \Z/171\Z} \chi_{171}(25,r) e\left(\frac{2r}{171}\right) = -12.3544335541+-4.2857871574i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 171 }(25,·),\chi_{ 171 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{171}(25,\cdot),\chi_{171}(1,\cdot)) = \sum_{r\in \Z/171\Z} \chi_{171}(25,r) \chi_{171}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 171 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{171}(25,·)) = \sum_{r \in \Z/171\Z} \chi_{171}(25,r) e\left(\frac{1 r + 2 r^{-1}}{171}\right) = 3.096786115+17.5627467931i \)