Properties

Conductor 349
Order 29
Real No
Primitive Yes
Parity Even
Orbit Label 349.g

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[31]
 
pari: [g,chi] = znchar(Mod(31,349))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 349
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 29
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.g
Orbit index = 7

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 ]
 

\(\chi_{349}(31,\cdot)\) \(\chi_{349}(41,\cdot)\) \(\chi_{349}(66,\cdot)\) \(\chi_{349}(67,\cdot)\) \(\chi_{349}(88,\cdot)\) \(\chi_{349}(110,\cdot)\) \(\chi_{349}(118,\cdot)\) \(\chi_{349}(126,\cdot)\) \(\chi_{349}(168,\cdot)\) \(\chi_{349}(171,\cdot)\) \(\chi_{349}(210,\cdot)\) \(\chi_{349}(224,\cdot)\) \(\chi_{349}(228,\cdot)\) \(\chi_{349}(234,\cdot)\) \(\chi_{349}(249,\cdot)\) \(\chi_{349}(257,\cdot)\) \(\chi_{349}(263,\cdot)\) \(\chi_{349}(269,\cdot)\) \(\chi_{349}(274,\cdot)\) \(\chi_{349}(280,\cdot)\) \(\chi_{349}(285,\cdot)\) \(\chi_{349}(289,\cdot)\) \(\chi_{349}(301,\cdot)\) \(\chi_{349}(304,\cdot)\) \(\chi_{349}(312,\cdot)\) \(\chi_{349}(313,\cdot)\) \(\chi_{349}(322,\cdot)\) \(\chi_{349}(332,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{13}{29}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{13}{29}\right)\)\(e\left(\frac{19}{29}\right)\)\(e\left(\frac{26}{29}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{1}{29}\right)\)\(e\left(\frac{10}{29}\right)\)\(e\left(\frac{9}{29}\right)\)\(e\left(\frac{6}{29}\right)\)\(e\left(\frac{12}{29}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 349 }(31,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{349}(31,\cdot)) = \sum_{r\in \Z/349\Z} \chi_{349}(31,r) e\left(\frac{2r}{349}\right) = 17.343378594+-6.943141864i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 349 }(31,·),\chi_{ 349 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{349}(31,\cdot),\chi_{349}(1,\cdot)) = \sum_{r\in \Z/349\Z} \chi_{349}(31,r) \chi_{349}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 349 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{349}(31,·)) = \sum_{r \in \Z/349\Z} \chi_{349}(31,r) e\left(\frac{1 r + 2 r^{-1}}{349}\right) = 3.0252456152+18.4531818983i \)