Properties

Conductor 293
Order 73
Real No
Primitive Yes
Parity Even
Orbit Label 293.d

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(293)
sage: chi = H[22]
pari: [g,chi] = znchar(Mod(22,293))

Basic properties

sage: chi.conductor()
pari: znconreyconductor(g,chi)
Conductor = 293
sage: chi.multiplicative_order()
pari: charorder(g,chi)
Order = 73
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 = 293.d
Orbit index = 4

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 ]

\(\chi_{293}(16,\cdot)\) \(\chi_{293}(17,\cdot)\) \(\chi_{293}(22,\cdot)\) \(\chi_{293}(24,\cdot)\) \(\chi_{293}(26,\cdot)\) \(\chi_{293}(33,\cdot)\) \(\chi_{293}(36,\cdot)\) \(\chi_{293}(38,\cdot)\) \(\chi_{293}(39,\cdot)\) \(\chi_{293}(40,\cdot)\) \(\chi_{293}(46,\cdot)\) \(\chi_{293}(53,\cdot)\) \(\chi_{293}(54,\cdot)\) \(\chi_{293}(55,\cdot)\) \(\chi_{293}(56,\cdot)\) \(\chi_{293}(57,\cdot)\) \(\chi_{293}(59,\cdot)\) \(\chi_{293}(60,\cdot)\) \(\chi_{293}(65,\cdot)\) \(\chi_{293}(69,\cdot)\) \(\chi_{293}(73,\cdot)\) \(\chi_{293}(77,\cdot)\) \(\chi_{293}(81,\cdot)\) \(\chi_{293}(82,\cdot)\) \(\chi_{293}(84,\cdot)\) \(\chi_{293}(90,\cdot)\) \(\chi_{293}(91,\cdot)\) \(\chi_{293}(94,\cdot)\) \(\chi_{293}(95,\cdot)\) \(\chi_{293}(100,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{29}{73}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{29}{73}\right)\)\(e\left(\frac{27}{73}\right)\)\(e\left(\frac{58}{73}\right)\)\(e\left(\frac{53}{73}\right)\)\(e\left(\frac{56}{73}\right)\)\(e\left(\frac{45}{73}\right)\)\(e\left(\frac{14}{73}\right)\)\(e\left(\frac{54}{73}\right)\)\(e\left(\frac{9}{73}\right)\)\(e\left(\frac{50}{73}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{73})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
pari: znchargauss(g,chi,a)
\( \tau_{ a }( \chi_{ 293 }(22,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{293}(22,\cdot)) = \sum_{r\in \Z/293\Z} \chi_{293}(22,r) e\left(\frac{2r}{293}\right) = 16.8515657678+3.0041190344i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
\( J(\chi_{ 293 }(22,·),\chi_{ 293 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{293}(22,\cdot),\chi_{293}(1,\cdot)) = \sum_{r\in \Z/293\Z} \chi_{293}(22,r) \chi_{293}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
\(K(a,b,\chi_{ 293 }(22,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{293}(22,·)) = \sum_{r \in \Z/293\Z} \chi_{293}(22,r) e\left(\frac{1 r + 2 r^{-1}}{293}\right) = 5.0079714681+14.973197263i \)