Properties

Conductor 731
Order 84
Real No
Primitive Yes
Parity Even
Orbit Label 731.bh

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(731)
 
sage: chi = H[13]
 
pari: [g,chi] = znchar(Mod(13,731))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 84
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 = 731.bh
Orbit index = 34

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 ]
 

\(\chi_{731}(13,\cdot)\) \(\chi_{731}(38,\cdot)\) \(\chi_{731}(81,\cdot)\) \(\chi_{731}(225,\cdot)\) \(\chi_{731}(268,\cdot)\) \(\chi_{731}(310,\cdot)\) \(\chi_{731}(353,\cdot)\) \(\chi_{731}(361,\cdot)\) \(\chi_{731}(404,\cdot)\) \(\chi_{731}(412,\cdot)\) \(\chi_{731}(455,\cdot)\) \(\chi_{731}(497,\cdot)\) \(\chi_{731}(531,\cdot)\) \(\chi_{731}(540,\cdot)\) \(\chi_{731}(574,\cdot)\) \(\chi_{731}(582,\cdot)\) \(\chi_{731}(599,\cdot)\) \(\chi_{731}(616,\cdot)\) \(\chi_{731}(625,\cdot)\) \(\chi_{731}(633,\cdot)\) \(\chi_{731}(642,\cdot)\) \(\chi_{731}(659,\cdot)\) \(\chi_{731}(676,\cdot)\) \(\chi_{731}(701,\cdot)\)

Values on generators

\((173,562)\) → \((i,e\left(\frac{16}{21}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{84}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{17}{28}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{84})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{731}(13,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(13,r) e\left(\frac{2r}{731}\right) = -7.4372345375+25.9939905062i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 731 }(13,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{731}(13,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(13,r) \chi_{731}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{731}(13,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(13,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = -0.5836813961+-0.1332214698i \)