Properties

Conductor 167
Order 2
Real Yes
Primitive Yes
Parity Odd
Orbit Label 167.b

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
sage: H = DirichletGroup_conrey(167)
sage: chi = H[166]
pari: [g,chi] = znchar(Mod(166,167))

Kronecker symbol representation

sage: kronecker_character(-167)
pari: znchartokronecker(g,chi)

\(\displaystyle\left(\frac{-167}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 167
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 2
Real = Yes
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Orbit label = 167.b
Orbit index = 2

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 ]

\(\chi_{167}(166,\cdot)\)

Values on generators

\(5\) → \(-1\)

Values

-11234567891011
\(-1\)\(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 167 }(166,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{167}(166,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(166,r) e\left(\frac{2r}{167}\right) = 12.9228479833i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 167 }(166,·),\chi_{ 167 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{167}(166,\cdot),\chi_{167}(1,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(166,r) \chi_{167}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 167 }(166,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{167}(166,·)) = \sum_{r \in \Z/167\Z} \chi_{167}(166,r) e\left(\frac{1 r + 2 r^{-1}}{167}\right) = 14.4348098641i \)