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[38]
pari: [g,chi] = znchar(Mod(38,235))

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

\((142,146)\) → \((-i,e\left(\frac{17}{46}\right))\)

Values

-112346789111213
\(1\)\(1\)\(e\left(\frac{37}{92}\right)\)\(e\left(\frac{59}{92}\right)\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{1}{23}\right)\)\(e\left(\frac{53}{92}\right)\)\(e\left(\frac{19}{92}\right)\)\(e\left(\frac{13}{46}\right)\)\(e\left(\frac{27}{46}\right)\)\(e\left(\frac{41}{92}\right)\)\(e\left(\frac{29}{92}\right)\)
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)
\( \tau_{ a }( \chi_{ 235 }(38,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{235}(38,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(38,r) e\left(\frac{2r}{235}\right) = 10.7796533312+-10.8994987985i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 235 }(38,·),\chi_{ 235 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{235}(38,\cdot),\chi_{235}(1,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(38,r) \chi_{235}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 235 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{235}(38,·)) = \sum_{r \in \Z/235\Z} \chi_{235}(38,r) e\left(\frac{1 r + 2 r^{-1}}{235}\right) = 4.9232920047+15.5120142025i \)