Properties

Conductor 59
Order 29
Real No
Primitive No
Parity Even
Orbit Label 354.e

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(354)
 
sage: chi = H[7]
 
pari: [g,chi] = znchar(Mod(7,354))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 59
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 = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 354.e
Orbit index = 5

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 ]
 

\(\chi_{354}(7,\cdot)\) \(\chi_{354}(19,\cdot)\) \(\chi_{354}(25,\cdot)\) \(\chi_{354}(49,\cdot)\) \(\chi_{354}(79,\cdot)\) \(\chi_{354}(85,\cdot)\) \(\chi_{354}(121,\cdot)\) \(\chi_{354}(127,\cdot)\) \(\chi_{354}(133,\cdot)\) \(\chi_{354}(139,\cdot)\) \(\chi_{354}(145,\cdot)\) \(\chi_{354}(163,\cdot)\) \(\chi_{354}(169,\cdot)\) \(\chi_{354}(175,\cdot)\) \(\chi_{354}(181,\cdot)\) \(\chi_{354}(193,\cdot)\) \(\chi_{354}(199,\cdot)\) \(\chi_{354}(205,\cdot)\) \(\chi_{354}(223,\cdot)\) \(\chi_{354}(241,\cdot)\) \(\chi_{354}(253,\cdot)\) \(\chi_{354}(265,\cdot)\) \(\chi_{354}(271,\cdot)\) \(\chi_{354}(277,\cdot)\) \(\chi_{354}(289,\cdot)\) \(\chi_{354}(307,\cdot)\) \(\chi_{354}(331,\cdot)\) \(\chi_{354}(343,\cdot)\)

Inducing primitive character

\(\chi_{59}(7,\cdot)\)

Values on generators

\((119,61)\) → \((1,e\left(\frac{9}{29}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{17}{29}\right)\)\(e\left(\frac{22}{29}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{12}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{19}{29}\right)\)\(e\left(\frac{21}{29}\right)\)\(e\left(\frac{20}{29}\right)\)\(e\left(\frac{6}{29}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 354 }(7,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{354}(7,\cdot)) = \sum_{r\in \Z/354\Z} \chi_{354}(7,r) e\left(\frac{r}{177}\right) = 7.1420739747+-2.8267966569i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 354 }(7,·),\chi_{ 354 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{354}(7,\cdot),\chi_{354}(1,\cdot)) = \sum_{r\in \Z/354\Z} \chi_{354}(7,r) \chi_{354}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 354 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{354}(7,·)) = \sum_{r \in \Z/354\Z} \chi_{354}(7,r) e\left(\frac{1 r + 2 r^{-1}}{354}\right) = -4.5836679605+-6.7604044646i \)