Properties

Conductor 235
Order 92
Real No
Primitive Yes
Parity Odd
Orbit Label 235.k

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[17]
pari: [g,chi] = znchar(Mod(17,235))

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 = Odd
Orbit label = 235.k
Orbit index = 11

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 ]

\(\chi_{235}(2,\cdot)\) \(\chi_{235}(3,\cdot)\) \(\chi_{235}(7,\cdot)\) \(\chi_{235}(8,\cdot)\) \(\chi_{235}(12,\cdot)\) \(\chi_{235}(17,\cdot)\) \(\chi_{235}(18,\cdot)\) \(\chi_{235}(27,\cdot)\) \(\chi_{235}(28,\cdot)\) \(\chi_{235}(32,\cdot)\) \(\chi_{235}(37,\cdot)\) \(\chi_{235}(42,\cdot)\) \(\chi_{235}(53,\cdot)\) \(\chi_{235}(63,\cdot)\) \(\chi_{235}(68,\cdot)\) \(\chi_{235}(72,\cdot)\) \(\chi_{235}(83,\cdot)\) \(\chi_{235}(97,\cdot)\) \(\chi_{235}(98,\cdot)\) \(\chi_{235}(102,\cdot)\) \(\chi_{235}(103,\cdot)\) \(\chi_{235}(108,\cdot)\) \(\chi_{235}(112,\cdot)\) \(\chi_{235}(118,\cdot)\) \(\chi_{235}(122,\cdot)\) \(\chi_{235}(128,\cdot)\) \(\chi_{235}(143,\cdot)\) \(\chi_{235}(147,\cdot)\) \(\chi_{235}(148,\cdot)\) \(\chi_{235}(153,\cdot)\) ...

Values on generators

\((142,146)\) → \((i,e\left(\frac{8}{23}\right))\)

Values

-112346789111213
\(-1\)\(1\)\(e\left(\frac{47}{92}\right)\)\(e\left(\frac{65}{92}\right)\)\(e\left(\frac{1}{46}\right)\)\(e\left(\frac{5}{23}\right)\)\(e\left(\frac{35}{92}\right)\)\(e\left(\frac{49}{92}\right)\)\(e\left(\frac{19}{46}\right)\)\(e\left(\frac{10}{23}\right)\)\(e\left(\frac{67}{92}\right)\)\(e\left(\frac{53}{92}\right)\)
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)
\( \tau_{ a }( \chi_{ 235 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{235}(17,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(17,r) e\left(\frac{2r}{235}\right) = -9.604463614+-11.9479822015i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 235 }(17,·),\chi_{ 235 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{235}(17,\cdot),\chi_{235}(1,\cdot)) = \sum_{r\in \Z/235\Z} \chi_{235}(17,r) \chi_{235}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 235 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{235}(17,·)) = \sum_{r \in \Z/235\Z} \chi_{235}(17,r) e\left(\frac{1 r + 2 r^{-1}}{235}\right) = 0.2618734558+0.0089458657i \)