Properties

Conductor 193
Order 64
Real No
Primitive Yes
Parity Odd
Orbit Label 193.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(193)
sage: chi = H[105]
pari: [g,chi] = znchar(Mod(105,193))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 193
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 64
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 = 193.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 ]

\(\chi_{193}(11,\cdot)\) \(\chi_{193}(13,\cdot)\) \(\chi_{193}(20,\cdot)\) \(\chi_{193}(29,\cdot)\) \(\chi_{193}(33,\cdot)\) \(\chi_{193}(35,\cdot)\) \(\chi_{193}(39,\cdot)\) \(\chi_{193}(60,\cdot)\) \(\chi_{193}(68,\cdot)\) \(\chi_{193}(71,\cdot)\) \(\chi_{193}(74,\cdot)\) \(\chi_{193}(76,\cdot)\) \(\chi_{193}(87,\cdot)\) \(\chi_{193}(88,\cdot)\) \(\chi_{193}(89,\cdot)\) \(\chi_{193}(94,\cdot)\) \(\chi_{193}(99,\cdot)\) \(\chi_{193}(104,\cdot)\) \(\chi_{193}(105,\cdot)\) \(\chi_{193}(106,\cdot)\) \(\chi_{193}(117,\cdot)\) \(\chi_{193}(119,\cdot)\) \(\chi_{193}(122,\cdot)\) \(\chi_{193}(125,\cdot)\) \(\chi_{193}(133,\cdot)\) \(\chi_{193}(154,\cdot)\) \(\chi_{193}(158,\cdot)\) \(\chi_{193}(160,\cdot)\) \(\chi_{193}(164,\cdot)\) \(\chi_{193}(173,\cdot)\) ...

Values on generators

\(5\) → \(e\left(\frac{63}{64}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{63}{64}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{29}{64}\right)\)\(e\left(\frac{9}{64}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{64})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 193 }(105,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{193}(105,\cdot)) = \sum_{r\in \Z/193\Z} \chi_{193}(105,r) e\left(\frac{2r}{193}\right) = -10.7771686406+8.7665635281i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 193 }(105,·),\chi_{ 193 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{193}(105,\cdot),\chi_{193}(1,\cdot)) = \sum_{r\in \Z/193\Z} \chi_{193}(105,r) \chi_{193}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 193 }(105,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{193}(105,·)) = \sum_{r \in \Z/193\Z} \chi_{193}(105,r) e\left(\frac{1 r + 2 r^{-1}}{193}\right) = 12.7477672773+-1.2555454888i \)