Properties

Conductor 69
Order 22
Real No
Primitive Yes
Parity Even
Orbit Label 69.g

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(69)
sage: chi = H[5]
pari: [g,chi] = znchar(Mod(5,69))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 69
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 22
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 = 69.g
Orbit index = 7

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 ]

\(\chi_{69}(5,\cdot)\) \(\chi_{69}(11,\cdot)\) \(\chi_{69}(14,\cdot)\) \(\chi_{69}(17,\cdot)\) \(\chi_{69}(20,\cdot)\) \(\chi_{69}(38,\cdot)\) \(\chi_{69}(44,\cdot)\) \(\chi_{69}(53,\cdot)\) \(\chi_{69}(56,\cdot)\) \(\chi_{69}(65,\cdot)\)

Values on generators

\((47,28)\) → \((-1,e\left(\frac{1}{22}\right))\)

Values

-11245781011131416
\(1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{4}{11}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{11})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 69 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{69}(5,\cdot)) = \sum_{r\in \Z/69\Z} \chi_{69}(5,r) e\left(\frac{2r}{69}\right) = 4.3685403304+7.0651153835i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 69 }(5,·),\chi_{ 69 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{69}(5,\cdot),\chi_{69}(1,\cdot)) = \sum_{r\in \Z/69\Z} \chi_{69}(5,r) \chi_{69}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 69 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{69}(5,·)) = \sum_{r \in \Z/69\Z} \chi_{69}(5,r) e\left(\frac{1 r + 2 r^{-1}}{69}\right) = -0.0 \)