Properties

Conductor 731
Order 42
Real No
Primitive Yes
Parity Even
Orbit Label 731.z

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[67]
 
pari: [g,chi] = znchar(Mod(67,731))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 731
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 42
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.z
Orbit index = 26

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 ]
 

\(\chi_{731}(67,\cdot)\) \(\chi_{731}(101,\cdot)\) \(\chi_{731}(152,\cdot)\) \(\chi_{731}(169,\cdot)\) \(\chi_{731}(186,\cdot)\) \(\chi_{731}(203,\cdot)\) \(\chi_{731}(271,\cdot)\) \(\chi_{731}(339,\cdot)\) \(\chi_{731}(526,\cdot)\) \(\chi_{731}(611,\cdot)\) \(\chi_{731}(662,\cdot)\) \(\chi_{731}(713,\cdot)\)

Values on generators

\((173,562)\) → \((-1,e\left(\frac{20}{21}\right))\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{14}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{21})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 731 }(67,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{731}(67,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(67,r) e\left(\frac{2r}{731}\right) = 2.1766577639+26.9492515848i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 731 }(67,·),\chi_{ 731 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{731}(67,\cdot),\chi_{731}(1,\cdot)) = \sum_{r\in \Z/731\Z} \chi_{731}(67,r) \chi_{731}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 731 }(67,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{731}(67,·)) = \sum_{r \in \Z/731\Z} \chi_{731}(67,r) e\left(\frac{1 r + 2 r^{-1}}{731}\right) = 1.9803137342+-2.4832348788i \)