Properties

Conductor 731
Order 336
Real No
Primitive Yes
Parity Even
Orbit Label 731.bm

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[3]
 
pari: [g,chi] = znchar(Mod(3,731))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 336
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.bm
Orbit index = 39

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 ]
 

\(\chi_{731}(3,\cdot)\) \(\chi_{731}(5,\cdot)\) \(\chi_{731}(12,\cdot)\) \(\chi_{731}(20,\cdot)\) \(\chi_{731}(28,\cdot)\) \(\chi_{731}(29,\cdot)\) \(\chi_{731}(46,\cdot)\) \(\chi_{731}(48,\cdot)\) \(\chi_{731}(61,\cdot)\) \(\chi_{731}(62,\cdot)\) \(\chi_{731}(63,\cdot)\) \(\chi_{731}(71,\cdot)\) \(\chi_{731}(73,\cdot)\) \(\chi_{731}(91,\cdot)\) \(\chi_{731}(105,\cdot)\) \(\chi_{731}(112,\cdot)\) \(\chi_{731}(114,\cdot)\) \(\chi_{731}(116,\cdot)\) \(\chi_{731}(141,\cdot)\) \(\chi_{731}(147,\cdot)\) \(\chi_{731}(148,\cdot)\) \(\chi_{731}(158,\cdot)\) \(\chi_{731}(159,\cdot)\) \(\chi_{731}(163,\cdot)\) \(\chi_{731}(175,\cdot)\) \(\chi_{731}(177,\cdot)\) \(\chi_{731}(184,\cdot)\) \(\chi_{731}(190,\cdot)\) \(\chi_{731}(192,\cdot)\) \(\chi_{731}(198,\cdot)\) ...

Values on generators

\((173,562)\) → \((e\left(\frac{1}{16}\right),e\left(\frac{1}{42}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{29}{56}\right)\)\(e\left(\frac{29}{336}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{305}{336}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{31}{56}\right)\)\(e\left(\frac{29}{168}\right)\)\(e\left(\frac{143}{336}\right)\)\(e\left(\frac{17}{112}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{336})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{731}(3,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(3,r) e\left(\frac{2r}{731}\right) = -19.2217542195+-19.0137888051i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 731 }(3,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{731}(3,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(3,r) \chi_{731}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{731}(3,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(3,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = -0.0599737357+1.0679313062i \)