Properties

Conductor 349
Order 87
Real No
Primitive Yes
Parity Even
Orbit Label 349.i

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[180]
 
pari: [g,chi] = znchar(Mod(180,349))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 349
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 87
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.i
Orbit index = 9

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 ]
 

\(\chi_{349}(9,\cdot)\) \(\chi_{349}(12,\cdot)\) \(\chi_{349}(14,\cdot)\) \(\chi_{349}(15,\cdot)\) \(\chi_{349}(16,\cdot)\) \(\chi_{349}(19,\cdot)\) \(\chi_{349}(20,\cdot)\) \(\chi_{349}(23,\cdot)\) \(\chi_{349}(25,\cdot)\) \(\chi_{349}(26,\cdot)\) \(\chi_{349}(51,\cdot)\) \(\chi_{349}(68,\cdot)\) \(\chi_{349}(77,\cdot)\) \(\chi_{349}(81,\cdot)\) \(\chi_{349}(85,\cdot)\) \(\chi_{349}(87,\cdot)\) \(\chi_{349}(94,\cdot)\) \(\chi_{349}(106,\cdot)\) \(\chi_{349}(108,\cdot)\) \(\chi_{349}(111,\cdot)\) \(\chi_{349}(116,\cdot)\) \(\chi_{349}(135,\cdot)\) \(\chi_{349}(143,\cdot)\) \(\chi_{349}(144,\cdot)\) \(\chi_{349}(145,\cdot)\) \(\chi_{349}(147,\cdot)\) \(\chi_{349}(148,\cdot)\) \(\chi_{349}(151,\cdot)\) \(\chi_{349}(158,\cdot)\) \(\chi_{349}(180,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{5}{87}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{5}{87}\right)\)\(e\left(\frac{43}{87}\right)\)\(e\left(\frac{10}{87}\right)\)\(e\left(\frac{4}{87}\right)\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{74}{87}\right)\)\(e\left(\frac{5}{29}\right)\)\(e\left(\frac{86}{87}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{6}{29}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{87})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 349 }(180,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{349}(180,\cdot)) = \sum_{r\in \Z/349\Z} \chi_{349}(180,r) e\left(\frac{2r}{349}\right) = 17.487366305+6.5720635811i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 349 }(180,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{349}(180,\cdot),\chi_{349}(1,\cdot)) = \sum_{r\in \Z/349\Z} \chi_{349}(180,r) \chi_{349}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(180,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{349}(180,·)) = \sum_{r \in \Z/349\Z} \chi_{349}(180,r) e\left(\frac{1 r + 2 r^{-1}}{349}\right) = 25.7270350754+4.6961909445i \)