Properties

Conductor 11
Order 2
Real Yes
Primitive Yes
Parity Odd

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(11)
sage: chi = H[10]
pari: [g,chi] = znchar(Mod(10,11))

Kronecker symbol representation

sage: kronecker_character(-11)

\(\displaystyle\left(\frac{-11}{\bullet}\right)\)

Values on generators

sage: chi(k) for k in H.gens()
pari: [ chareval(g,chi,x) | x <- g.gen ] \\ value in Q/Z

\(2\) → \(-1\)

Values

12345678910
1\(-1\)111\(-1\)\(-1\)\(-1\)1\(-1\)
value at  e.g. 2

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 11
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 2
sage: chi.is_odd()
pari: zncharisodd(g,chi)
Parity = Odd
Real = Yes
sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
Primitive = Yes

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 ]

\(\chi_{11}(10,\cdot)\)

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 11 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{11}(10,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(10,r) e\left(\frac{2r}{11}\right) = -3.3166247904i. \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 11 }(10,·),\chi_{ 11 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{11}(10,\cdot),\chi_{11}(1,\cdot)) = \sum_{r\in \Z/11\Z} \chi_{11}(10,r) \chi_{11}(1,1-r) = -1.\)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 11 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{11}(10,·)) = \sum_{r \in \Z/11\Z} \chi_{11}(10,r) e\left(\frac{1 r + 2 r^{-1}}{11}\right) = -0.0. \)