Properties

Conductor 107
Order 53
Real No
Primitive Yes
Parity Even
Orbit Label 107.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(107)
 
sage: chi = H[40]
 
pari: [g,chi] = znchar(Mod(40,107))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 107
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 53
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 = 107.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 ]
 

\(\chi_{107}(3,\cdot)\) \(\chi_{107}(4,\cdot)\) \(\chi_{107}(9,\cdot)\) \(\chi_{107}(10,\cdot)\) \(\chi_{107}(11,\cdot)\) \(\chi_{107}(12,\cdot)\) \(\chi_{107}(13,\cdot)\) \(\chi_{107}(14,\cdot)\) \(\chi_{107}(16,\cdot)\) \(\chi_{107}(19,\cdot)\) \(\chi_{107}(23,\cdot)\) \(\chi_{107}(25,\cdot)\) \(\chi_{107}(27,\cdot)\) \(\chi_{107}(29,\cdot)\) \(\chi_{107}(30,\cdot)\) \(\chi_{107}(33,\cdot)\) \(\chi_{107}(34,\cdot)\) \(\chi_{107}(35,\cdot)\) \(\chi_{107}(36,\cdot)\) \(\chi_{107}(37,\cdot)\) \(\chi_{107}(39,\cdot)\) \(\chi_{107}(40,\cdot)\) \(\chi_{107}(41,\cdot)\) \(\chi_{107}(42,\cdot)\) \(\chi_{107}(44,\cdot)\) \(\chi_{107}(47,\cdot)\) \(\chi_{107}(48,\cdot)\) \(\chi_{107}(49,\cdot)\) \(\chi_{107}(52,\cdot)\) \(\chi_{107}(53,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{25}{53}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{25}{53}\right)\)\(e\left(\frac{1}{53}\right)\)\(e\left(\frac{50}{53}\right)\)\(e\left(\frac{9}{53}\right)\)\(e\left(\frac{26}{53}\right)\)\(e\left(\frac{15}{53}\right)\)\(e\left(\frac{22}{53}\right)\)\(e\left(\frac{2}{53}\right)\)\(e\left(\frac{34}{53}\right)\)\(e\left(\frac{20}{53}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{53})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 107 }(40,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{107}(40,\cdot)) = \sum_{r\in \Z/107\Z} \chi_{107}(40,r) e\left(\frac{2r}{107}\right) = -2.5180083898+-10.0329274765i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 107 }(40,·),\chi_{ 107 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{107}(40,\cdot),\chi_{107}(1,\cdot)) = \sum_{r\in \Z/107\Z} \chi_{107}(40,r) \chi_{107}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 107 }(40,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{107}(40,·)) = \sum_{r \in \Z/107\Z} \chi_{107}(40,r) e\left(\frac{1 r + 2 r^{-1}}{107}\right) = 1.2839710729+14.4026835979i \)