Properties

Conductor 41
Order 20
Real No
Primitive Yes
Parity Even
Orbit Label 41.g

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(41)
 
sage: chi = H[39]
 
pari: [g,chi] = znchar(Mod(39,41))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 41
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
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 41.g
Orbit index = 7

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 ]
 

\(\chi_{41}(2,\cdot)\) \(\chi_{41}(5,\cdot)\) \(\chi_{41}(8,\cdot)\) \(\chi_{41}(20,\cdot)\) \(\chi_{41}(21,\cdot)\) \(\chi_{41}(33,\cdot)\) \(\chi_{41}(36,\cdot)\) \(\chi_{41}(39,\cdot)\)

Values on generators

\(6\) → \(e\left(\frac{3}{20}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(i\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(-1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{9}{20}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 41 }(39,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{41}(39,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(39,r) e\left(\frac{2r}{41}\right) = 5.3782614942+3.4748098222i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 41 }(39,·),\chi_{ 41 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{41}(39,\cdot),\chi_{41}(1,\cdot)) = \sum_{r\in \Z/41\Z} \chi_{41}(39,r) \chi_{41}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 41 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{41}(39,·)) = \sum_{r \in \Z/41\Z} \chi_{41}(39,r) e\left(\frac{1 r + 2 r^{-1}}{41}\right) = -6.6397851297+2.1573969674i \)