Properties

Conductor 61
Order 30
Real No
Primitive Yes
Parity Even
Orbit Label 61.k

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[4]
pari: [g,chi] = znchar(Mod(4,61))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 61
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 30
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 = Even
Orbit label = 61.k
Orbit index = 11

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 ]

\(\chi_{61}(4,\cdot)\) \(\chi_{61}(5,\cdot)\) \(\chi_{61}(19,\cdot)\) \(\chi_{61}(36,\cdot)\) \(\chi_{61}(39,\cdot)\) \(\chi_{61}(45,\cdot)\) \(\chi_{61}(46,\cdot)\) \(\chi_{61}(49,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{1}{30}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{15})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 61 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{61}(4,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(4,r) e\left(\frac{2r}{61}\right) = 7.0074516095+3.4490030357i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 61 }(4,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{61}(4,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(4,r) \chi_{61}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 61 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{61}(4,·)) = \sum_{r \in \Z/61\Z} \chi_{61}(4,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right) = -6.1318220849+-0.644480471i \)