Properties

Conductor 113
Order 56
Real No
Primitive Yes
Parity Even
Orbit Label 113.i

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(113)
 
sage: chi = H[9]
 
pari: [g,chi] = znchar(Mod(9,113))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 113
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 56
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 = 113.i
Orbit index = 9

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 ]
 

\(\chi_{113}(9,\cdot)\) \(\chi_{113}(11,\cdot)\) \(\chi_{113}(13,\cdot)\) \(\chi_{113}(22,\cdot)\) \(\chi_{113}(25,\cdot)\) \(\chi_{113}(26,\cdot)\) \(\chi_{113}(31,\cdot)\) \(\chi_{113}(36,\cdot)\) \(\chi_{113}(41,\cdot)\) \(\chi_{113}(50,\cdot)\) \(\chi_{113}(51,\cdot)\) \(\chi_{113}(52,\cdot)\) \(\chi_{113}(61,\cdot)\) \(\chi_{113}(62,\cdot)\) \(\chi_{113}(63,\cdot)\) \(\chi_{113}(72,\cdot)\) \(\chi_{113}(77,\cdot)\) \(\chi_{113}(82,\cdot)\) \(\chi_{113}(87,\cdot)\) \(\chi_{113}(88,\cdot)\) \(\chi_{113}(91,\cdot)\) \(\chi_{113}(100,\cdot)\) \(\chi_{113}(102,\cdot)\) \(\chi_{113}(104,\cdot)\)

Values on generators

\(3\) → \(e\left(\frac{1}{56}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{56}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{27}{56}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{39}{56}\right)\)\(e\left(\frac{9}{28}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{56})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 113 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{113}(9,\cdot)) = \sum_{r\in \Z/113\Z} \chi_{113}(9,r) e\left(\frac{2r}{113}\right) = 7.8128993324+7.2082316848i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 113 }(9,·),\chi_{ 113 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{113}(9,\cdot),\chi_{113}(1,\cdot)) = \sum_{r\in \Z/113\Z} \chi_{113}(9,r) \chi_{113}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 113 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{113}(9,·)) = \sum_{r \in \Z/113\Z} \chi_{113}(9,r) e\left(\frac{1 r + 2 r^{-1}}{113}\right) = -10.1279615292+-8.0767798032i \)