Properties

Conductor 11
Order 5
Real No
Primitive No
Parity Even
Orbit Label 187.g

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(187)
 
sage: chi = H[69]
 
pari: [g,chi] = znchar(Mod(69,187))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 11
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 5
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 187.g
Orbit index = 7

Galois orbit

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

\(\chi_{187}(69,\cdot)\) \(\chi_{187}(86,\cdot)\) \(\chi_{187}(103,\cdot)\) \(\chi_{187}(137,\cdot)\)

Inducing primitive character

\(\chi_{11}(3,\cdot)\)

Values on generators

\((35,122)\) → \((e\left(\frac{4}{5}\right),1)\)

Values

-11234567891012
\(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 187 }(69,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{187}(69,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(69,r) e\left(\frac{2r}{187}\right) = 0.9496208436+-3.177769698i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 187 }(69,·),\chi_{ 187 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{187}(69,\cdot),\chi_{187}(1,\cdot)) = \sum_{r\in \Z/187\Z} \chi_{187}(69,r) \chi_{187}(1,1-r) = -15 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 187 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{187}(69,·)) = \sum_{r \in \Z/187\Z} \chi_{187}(69,r) e\left(\frac{1 r + 2 r^{-1}}{187}\right) = 17.9956698435+-13.0746194612i \)