Properties

Conductor 11
Order 5
Real No
Primitive No
Parity Even
Orbit Label 44.e

Related objects

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Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(44)
 
sage: chi = H[5]
 
pari: [g,chi] = znchar(Mod(5,44))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 11
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 5
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 44.e
Orbit index = 5

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 ]
 

\(\chi_{44}(5,\cdot)\) \(\chi_{44}(9,\cdot)\) \(\chi_{44}(25,\cdot)\) \(\chi_{44}(37,\cdot)\)

Inducing primitive character

\(\chi_{11}(5,\cdot)\)

Values on generators

\((23,13)\) → \((1,e\left(\frac{2}{5}\right))\)

Values

-113579131517192123
\(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(1\)\(1\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{5})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 44 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{44}(5,\cdot)) = \sum_{r\in \Z/44\Z} \chi_{44}(5,r) e\left(\frac{r}{22}\right) = 6.3957562102+1.7590629608i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 44 }(5,·),\chi_{ 44 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{44}(5,\cdot),\chi_{44}(1,\cdot)) = \sum_{r\in \Z/44\Z} \chi_{44}(5,r) \chi_{44}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 44 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{44}(5,·)) = \sum_{r \in \Z/44\Z} \chi_{44}(5,r) e\left(\frac{1 r + 2 r^{-1}}{44}\right) = 0.0 \)