Properties

Conductor 400
Order 20
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 400.bf

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(400)
 
sage: chi = H[139]
 
pari: [g,chi] = znchar(Mod(139,400))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 400
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 20
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 = odd
Orbit label = 400.bf
Orbit index = 32

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 ]
 

\(\chi_{400}(19,\cdot)\) \(\chi_{400}(59,\cdot)\) \(\chi_{400}(139,\cdot)\) \(\chi_{400}(179,\cdot)\) \(\chi_{400}(219,\cdot)\) \(\chi_{400}(259,\cdot)\) \(\chi_{400}(339,\cdot)\) \(\chi_{400}(379,\cdot)\)

Values on generators

\((351,101,177)\) → \((-1,i,e\left(\frac{3}{10}\right))\)

Values

-1137911131719212327
\(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(-1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{20}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{20})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 400 }(139,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{400}(139,\cdot)) = \sum_{r\in \Z/400\Z} \chi_{400}(139,r) e\left(\frac{r}{200}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 400 }(139,·),\chi_{ 400 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{400}(139,\cdot),\chi_{400}(1,\cdot)) = \sum_{r\in \Z/400\Z} \chi_{400}(139,r) \chi_{400}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 400 }(139,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{400}(139,·)) = \sum_{r \in \Z/400\Z} \chi_{400}(139,r) e\left(\frac{1 r + 2 r^{-1}}{400}\right) = 0.0 \)