Properties

Conductor 173
Order 172
Real no
Primitive yes
Minimal yes
Parity odd
Orbit label 173.f

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(173)
 
sage: chi = H[65]
 
pari: [g,chi] = znchar(Mod(65,173))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 173
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 172
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = odd
Orbit label = 173.f
Orbit index = 6

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 ]
 

\(\chi_{173}(2,\cdot)\) \(\chi_{173}(3,\cdot)\) \(\chi_{173}(5,\cdot)\) \(\chi_{173}(7,\cdot)\) \(\chi_{173}(8,\cdot)\) \(\chi_{173}(11,\cdot)\) \(\chi_{173}(12,\cdot)\) \(\chi_{173}(17,\cdot)\) \(\chi_{173}(18,\cdot)\) \(\chi_{173}(19,\cdot)\) \(\chi_{173}(20,\cdot)\) \(\chi_{173}(26,\cdot)\) \(\chi_{173}(27,\cdot)\) \(\chi_{173}(28,\cdot)\) \(\chi_{173}(30,\cdot)\) \(\chi_{173}(32,\cdot)\) \(\chi_{173}(39,\cdot)\) \(\chi_{173}(42,\cdot)\) \(\chi_{173}(44,\cdot)\) \(\chi_{173}(45,\cdot)\) \(\chi_{173}(46,\cdot)\) \(\chi_{173}(48,\cdot)\) \(\chi_{173}(50,\cdot)\) \(\chi_{173}(53,\cdot)\) \(\chi_{173}(58,\cdot)\) \(\chi_{173}(59,\cdot)\) \(\chi_{173}(61,\cdot)\) \(\chi_{173}(62,\cdot)\) \(\chi_{173}(63,\cdot)\) \(\chi_{173}(65,\cdot)\) ...

Values on generators

\(2\) → \(e\left(\frac{169}{172}\right)\)

Values

-11234567891011
\(-1\)\(1\)\(e\left(\frac{169}{172}\right)\)\(e\left(\frac{91}{172}\right)\)\(e\left(\frac{83}{86}\right)\)\(e\left(\frac{55}{172}\right)\)\(e\left(\frac{22}{43}\right)\)\(e\left(\frac{59}{172}\right)\)\(e\left(\frac{163}{172}\right)\)\(e\left(\frac{5}{86}\right)\)\(e\left(\frac{13}{43}\right)\)\(e\left(\frac{103}{172}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{172})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 173 }(65,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{173}(65,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(65,r) e\left(\frac{2r}{173}\right) = 6.7100698171+-11.3126019576i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 173 }(65,·),\chi_{ 173 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{173}(65,\cdot),\chi_{173}(1,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(65,r) \chi_{173}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 173 }(65,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{173}(65,·)) = \sum_{r \in \Z/173\Z} \chi_{173}(65,r) e\left(\frac{1 r + 2 r^{-1}}{173}\right) = 0.6126837859+11.1701432645i \)