Properties

Conductor 731
Order 48
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 731.bd

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[609]
 
pari: [g,chi] = znchar(Mod(609,731))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 48
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 731.bd
Orbit index = 30

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 ]
 

\(\chi_{731}(7,\cdot)\) \(\chi_{731}(37,\cdot)\) \(\chi_{731}(80,\cdot)\) \(\chi_{731}(209,\cdot)\) \(\chi_{731}(252,\cdot)\) \(\chi_{731}(265,\cdot)\) \(\chi_{731}(295,\cdot)\) \(\chi_{731}(351,\cdot)\) \(\chi_{731}(381,\cdot)\) \(\chi_{731}(394,\cdot)\) \(\chi_{731}(437,\cdot)\) \(\chi_{731}(566,\cdot)\) \(\chi_{731}(609,\cdot)\) \(\chi_{731}(639,\cdot)\) \(\chi_{731}(652,\cdot)\) \(\chi_{731}(725,\cdot)\)

Values on generators

\((173,562)\) → \((e\left(\frac{9}{16}\right),e\left(\frac{5}{6}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{19}{48}\right)\)\(-i\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{17}{48}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{15}{16}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{48})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(609,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{731}(609,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(609,r) e\left(\frac{2r}{731}\right) = -22.6586798921+14.7507364408i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 731 }(609,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{731}(609,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(609,r) \chi_{731}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(609,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{731}(609,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(609,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = 12.8568679495+31.0392249732i \)