Properties

Conductor 731
Order 168
Real No
Primitive Yes
Parity Even
Orbit Label 731.bl

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[9]
 
pari: [g,chi] = znchar(Mod(9,731))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 168
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.bl
Orbit index = 38

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 ]
 

\(\chi_{731}(9,\cdot)\) \(\chi_{731}(15,\cdot)\) \(\chi_{731}(25,\cdot)\) \(\chi_{731}(53,\cdot)\) \(\chi_{731}(60,\cdot)\) \(\chi_{731}(66,\cdot)\) \(\chi_{731}(83,\cdot)\) \(\chi_{731}(100,\cdot)\) \(\chi_{731}(110,\cdot)\) \(\chi_{731}(111,\cdot)\) \(\chi_{731}(117,\cdot)\) \(\chi_{731}(138,\cdot)\) \(\chi_{731}(144,\cdot)\) \(\chi_{731}(185,\cdot)\) \(\chi_{731}(189,\cdot)\) \(\chi_{731}(195,\cdot)\) \(\chi_{731}(196,\cdot)\) \(\chi_{731}(212,\cdot)\) \(\chi_{731}(229,\cdot)\) \(\chi_{731}(230,\cdot)\) \(\chi_{731}(240,\cdot)\) \(\chi_{731}(246,\cdot)\) \(\chi_{731}(253,\cdot)\) \(\chi_{731}(281,\cdot)\) \(\chi_{731}(298,\cdot)\) \(\chi_{731}(314,\cdot)\) \(\chi_{731}(315,\cdot)\) \(\chi_{731}(325,\cdot)\) \(\chi_{731}(332,\cdot)\) \(\chi_{731}(359,\cdot)\) ...

Values on generators

\((173,562)\) → \((e\left(\frac{1}{8}\right),e\left(\frac{1}{21}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{29}{168}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{137}{168}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{29}{84}\right)\)\(e\left(\frac{143}{168}\right)\)\(e\left(\frac{17}{56}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{168})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{731}(9,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(9,r) e\left(\frac{2r}{731}\right) = 16.5399583892+21.3876080122i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 731 }(9,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{731}(9,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(9,r) \chi_{731}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{731}(9,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(9,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = 1.1815617874+0.1331300403i \)