Properties

Conductor 235
Order 92
Real No
Primitive Yes
Parity Even
Orbit Label 235.l

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(235)
 
sage: chi = H[33]
 
pari: [g,chi] = znchar(Mod(33,235))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 235
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 92
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 = 235.l
Orbit index = 12

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 ]
 

\(\chi_{235}(13,\cdot)\) \(\chi_{235}(22,\cdot)\) \(\chi_{235}(23,\cdot)\) \(\chi_{235}(33,\cdot)\) \(\chi_{235}(38,\cdot)\) \(\chi_{235}(43,\cdot)\) \(\chi_{235}(52,\cdot)\) \(\chi_{235}(57,\cdot)\) \(\chi_{235}(58,\cdot)\) \(\chi_{235}(62,\cdot)\) \(\chi_{235}(67,\cdot)\) \(\chi_{235}(73,\cdot)\) \(\chi_{235}(77,\cdot)\) \(\chi_{235}(78,\cdot)\) \(\chi_{235}(82,\cdot)\) \(\chi_{235}(87,\cdot)\) \(\chi_{235}(88,\cdot)\) \(\chi_{235}(92,\cdot)\) \(\chi_{235}(107,\cdot)\) \(\chi_{235}(113,\cdot)\) \(\chi_{235}(117,\cdot)\) \(\chi_{235}(123,\cdot)\) \(\chi_{235}(127,\cdot)\) \(\chi_{235}(132,\cdot)\) \(\chi_{235}(133,\cdot)\) \(\chi_{235}(137,\cdot)\) \(\chi_{235}(138,\cdot)\) \(\chi_{235}(152,\cdot)\) \(\chi_{235}(163,\cdot)\) \(\chi_{235}(167,\cdot)\) ...

Values on generators

\((142,146)\) → \((-i,e\left(\frac{27}{46}\right))\)

Values

-112346789111213
\(1\)\(1\)\(e\left(\frac{29}{92}\right)\)\(e\left(\frac{91}{92}\right)\)\(e\left(\frac{29}{46}\right)\)\(e\left(\frac{7}{23}\right)\)\(e\left(\frac{49}{92}\right)\)\(e\left(\frac{87}{92}\right)\)\(e\left(\frac{45}{46}\right)\)\(e\left(\frac{5}{46}\right)\)\(e\left(\frac{57}{92}\right)\)\(e\left(\frac{65}{92}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{92})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 235 }(33,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{235}(33,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(33,r) e\left(\frac{2r}{235}\right) = 5.6741749733+-14.2409177503i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 235 }(33,·),\chi_{ 235 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{235}(33,\cdot),\chi_{235}(1,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(33,r) \chi_{235}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 235 }(33,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{235}(33,·)) = \sum_{r \in \Z/235\Z} \chi_{235}(33,r) e\left(\frac{1 r + 2 r^{-1}}{235}\right) = -17.7041991965+-26.9922224143i \)