Properties

Conductor 29
Order 7
Real No
Primitive Yes
Parity Even
Orbit Label 29.d

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(29)
 
sage: chi = H[23]
 
pari: [g,chi] = znchar(Mod(23,29))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 29
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 7
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 = 29.d
Orbit index = 4

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 ]
 

\(\chi_{29}(7,\cdot)\) \(\chi_{29}(16,\cdot)\) \(\chi_{29}(20,\cdot)\) \(\chi_{29}(23,\cdot)\) \(\chi_{29}(24,\cdot)\) \(\chi_{29}(25,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{5}{7}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{6}{7}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{7})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 29 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{29}(23,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(23,r) e\left(\frac{2r}{29}\right) = 3.9013142098+-3.7121082199i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 29 }(23,·),\chi_{ 29 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{29}(23,\cdot),\chi_{29}(1,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(23,r) \chi_{29}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 29 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{29}(23,·)) = \sum_{r \in \Z/29\Z} \chi_{29}(23,r) e\left(\frac{1 r + 2 r^{-1}}{29}\right) = 4.868853117+-6.1053486987i \)