Properties

Conductor 89
Order 44
Real No
Primitive Yes
Parity Even
Orbit Label 89.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(89)
sage: chi = H[79]
pari: [g,chi] = znchar(Mod(79,89))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 89
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 44
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 = 89.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 ]

\(\chi_{89}(5,\cdot)\) \(\chi_{89}(9,\cdot)\) \(\chi_{89}(10,\cdot)\) \(\chi_{89}(17,\cdot)\) \(\chi_{89}(18,\cdot)\) \(\chi_{89}(20,\cdot)\) \(\chi_{89}(21,\cdot)\) \(\chi_{89}(36,\cdot)\) \(\chi_{89}(40,\cdot)\) \(\chi_{89}(42,\cdot)\) \(\chi_{89}(47,\cdot)\) \(\chi_{89}(49,\cdot)\) \(\chi_{89}(53,\cdot)\) \(\chi_{89}(68,\cdot)\) \(\chi_{89}(69,\cdot)\) \(\chi_{89}(71,\cdot)\) \(\chi_{89}(72,\cdot)\) \(\chi_{89}(79,\cdot)\) \(\chi_{89}(80,\cdot)\) \(\chi_{89}(84,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{21}{44}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{21}{44}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{1}{11}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{44})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 89 }(79,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{89}(79,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(79,r) e\left(\frac{2r}{89}\right) = 1.324025837+9.3406078808i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 89 }(79,·),\chi_{ 89 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{89}(79,\cdot),\chi_{89}(1,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(79,r) \chi_{89}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 89 }(79,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{89}(79,·)) = \sum_{r \in \Z/89\Z} \chi_{89}(79,r) e\left(\frac{1 r + 2 r^{-1}}{89}\right) = 1.9309798314+-4.228256038i \)