Properties

Conductor 61
Order 12
Real No
Primitive Yes
Parity Odd
Orbit Label 61.h

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(61)
sage: chi = H[40]
pari: [g,chi] = znchar(Mod(40,61))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 61
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 12
Real = No
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 = 61.h
Orbit index = 8

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 ]

\(\chi_{61}(21,\cdot)\) \(\chi_{61}(29,\cdot)\) \(\chi_{61}(32,\cdot)\) \(\chi_{61}(40,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{5}{12}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(i\)\(1\)\(e\left(\frac{7}{12}\right)\)\(i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{12})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 61 }(40,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(40,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(40,r) e\left(\frac{2r}{61}\right) = -7.7941518334+-0.5011957688i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 61 }(40,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(40,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(40,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 61 }(40,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(40,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(40,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = -7.1927944074+1.9273034528i \)