Properties

Conductor 81
Order 54
Real No
Primitive Yes
Parity Odd
Orbit Label 81.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(81)
 
sage: chi = H[23]
 
pari: [g,chi] = znchar(Mod(23,81))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 81
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 54
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 = 81.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 ]
 

\(\chi_{81}(2,\cdot)\) \(\chi_{81}(5,\cdot)\) \(\chi_{81}(11,\cdot)\) \(\chi_{81}(14,\cdot)\) \(\chi_{81}(20,\cdot)\) \(\chi_{81}(23,\cdot)\) \(\chi_{81}(29,\cdot)\) \(\chi_{81}(32,\cdot)\) \(\chi_{81}(38,\cdot)\) \(\chi_{81}(41,\cdot)\) \(\chi_{81}(47,\cdot)\) \(\chi_{81}(50,\cdot)\) \(\chi_{81}(56,\cdot)\) \(\chi_{81}(59,\cdot)\) \(\chi_{81}(65,\cdot)\) \(\chi_{81}(68,\cdot)\) \(\chi_{81}(74,\cdot)\) \(\chi_{81}(77,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{11}{54}\right)\)

Values

-11245781011131416
\(-1\)\(1\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{11}{27}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{25}{54}\right)\)\(e\left(\frac{22}{27}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{27})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 81 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{81}(23,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(23,r) e\left(\frac{2r}{81}\right) = 3.2415995232+-8.3959533426i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 81 }(23,·),\chi_{ 81 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{81}(23,\cdot),\chi_{81}(1,\cdot)) = \sum_{r\in \Z/81\Z} \chi_{81}(23,r) \chi_{81}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 81 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{81}(23,·)) = \sum_{r \in \Z/81\Z} \chi_{81}(23,r) e\left(\frac{1 r + 2 r^{-1}}{81}\right) = -13.6085277003+18.2794249959i \)