Properties

Conductor 61
Order 60
Real No
Primitive Yes
Parity Odd
Orbit Label 61.l

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[59]
pari: [g,chi] = znchar(Mod(59,61))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 61
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 60
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.l
Orbit index = 12

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 ]

\(\chi_{61}(2,\cdot)\) \(\chi_{61}(6,\cdot)\) \(\chi_{61}(7,\cdot)\) \(\chi_{61}(10,\cdot)\) \(\chi_{61}(17,\cdot)\) \(\chi_{61}(18,\cdot)\) \(\chi_{61}(26,\cdot)\) \(\chi_{61}(30,\cdot)\) \(\chi_{61}(31,\cdot)\) \(\chi_{61}(35,\cdot)\) \(\chi_{61}(43,\cdot)\) \(\chi_{61}(44,\cdot)\) \(\chi_{61}(51,\cdot)\) \(\chi_{61}(54,\cdot)\) \(\chi_{61}(55,\cdot)\) \(\chi_{61}(59,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{31}{60}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{53}{60}\right)\)\(-i\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 61 }(59,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(59,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(59,r) e\left(\frac{2r}{61}\right) = 4.2835578457+6.5307834279i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 61 }(59,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(59,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(59,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 61 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(59,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(59,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = -13.252884864+-0.6945542648i \)