Properties

Conductor 349
Order 58
Real No
Primitive Yes
Parity Even
Orbit Label 349.h

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(349)
 
sage: chi = H[17]
 
pari: [g,chi] = znchar(Mod(17,349))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 349
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 58
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 = 349.h
Orbit index = 8

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 ]
 

\(\chi_{349}(17,\cdot)\) \(\chi_{349}(27,\cdot)\) \(\chi_{349}(36,\cdot)\) \(\chi_{349}(37,\cdot)\) \(\chi_{349}(45,\cdot)\) \(\chi_{349}(48,\cdot)\) \(\chi_{349}(60,\cdot)\) \(\chi_{349}(64,\cdot)\) \(\chi_{349}(69,\cdot)\) \(\chi_{349}(75,\cdot)\) \(\chi_{349}(80,\cdot)\) \(\chi_{349}(86,\cdot)\) \(\chi_{349}(92,\cdot)\) \(\chi_{349}(100,\cdot)\) \(\chi_{349}(115,\cdot)\) \(\chi_{349}(121,\cdot)\) \(\chi_{349}(125,\cdot)\) \(\chi_{349}(139,\cdot)\) \(\chi_{349}(178,\cdot)\) \(\chi_{349}(181,\cdot)\) \(\chi_{349}(223,\cdot)\) \(\chi_{349}(231,\cdot)\) \(\chi_{349}(239,\cdot)\) \(\chi_{349}(261,\cdot)\) \(\chi_{349}(282,\cdot)\) \(\chi_{349}(283,\cdot)\) \(\chi_{349}(308,\cdot)\) \(\chi_{349}(318,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{43}{58}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{43}{58}\right)\)\(e\left(\frac{8}{29}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{1}{58}\right)\)\(e\left(\frac{39}{58}\right)\)\(e\left(\frac{13}{58}\right)\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{31}{58}\right)\)\(e\left(\frac{33}{58}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{29})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 349 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{349}(17,\cdot)) = \sum_{r\in \Z/349\Z} \chi_{349}(17,r) e\left(\frac{2r}{349}\right) = -8.3508168106+-16.7111896224i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 349 }(17,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{349}(17,\cdot),\chi_{349}(1,\cdot)) = \sum_{r\in \Z/349\Z} \chi_{349}(17,r) \chi_{349}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{349}(17,·)) = \sum_{r \in \Z/349\Z} \chi_{349}(17,r) e\left(\frac{1 r + 2 r^{-1}}{349}\right) = 0.8765013323+-0.9253112064i \)