Properties

Conductor 21
Order 2
Real Yes
Primitive Yes
Parity Even
Orbit Label 21.c

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(21)
 
sage: chi = H[20]
 
pari: [g,chi] = znchar(Mod(20,21))
 

Kronecker symbol representation

sage: kronecker_character(21)
 
pari: znchartokronecker(g,chi)
 

\(\displaystyle\left(\frac{21}{\bullet}\right)\)

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 21
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 2
Real = Yes
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 = 21.c
Orbit index = 3

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 ]
 

\(\chi_{21}(20,\cdot)\)

Values on generators

\((8,10)\) → \((-1,-1)\)

Values

-112458101113161719
\(1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(-1\)\(-1\)\(-1\)\(1\)\(1\)\(-1\)
value at  e.g. 2

Related number fields

Field of values \(\Q\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 21 }(20,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{21}(20,\cdot)) = \sum_{r\in \Z/21\Z} \chi_{21}(20,r) e\left(\frac{2r}{21}\right) = -4.582575695 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 21 }(20,·),\chi_{ 21 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{21}(20,\cdot),\chi_{21}(1,\cdot)) = \sum_{r\in \Z/21\Z} \chi_{21}(20,r) \chi_{21}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 21 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{21}(20,·)) = \sum_{r \in \Z/21\Z} \chi_{21}(20,r) e\left(\frac{1 r + 2 r^{-1}}{21}\right) = 0.0 \)