Properties

Label 173.29
Modulus $173$
Conductor $173$
Order $43$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(173, base_ring=CyclotomicField(86))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([72]))
 
pari: [g,chi] = znchar(Mod(29,173))
 

Basic properties

Modulus: \(173\)
Conductor: \(173\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(43\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 173.d

\(\chi_{173}(6,\cdot)\) \(\chi_{173}(10,\cdot)\) \(\chi_{173}(14,\cdot)\) \(\chi_{173}(16,\cdot)\) \(\chi_{173}(22,\cdot)\) \(\chi_{173}(23,\cdot)\) \(\chi_{173}(29,\cdot)\) \(\chi_{173}(36,\cdot)\) \(\chi_{173}(43,\cdot)\) \(\chi_{173}(47,\cdot)\) \(\chi_{173}(51,\cdot)\) \(\chi_{173}(52,\cdot)\) \(\chi_{173}(57,\cdot)\) \(\chi_{173}(60,\cdot)\) \(\chi_{173}(81,\cdot)\) \(\chi_{173}(83,\cdot)\) \(\chi_{173}(84,\cdot)\) \(\chi_{173}(85,\cdot)\) \(\chi_{173}(95,\cdot)\) \(\chi_{173}(96,\cdot)\) \(\chi_{173}(100,\cdot)\) \(\chi_{173}(106,\cdot)\) \(\chi_{173}(109,\cdot)\) \(\chi_{173}(117,\cdot)\) \(\chi_{173}(118,\cdot)\) \(\chi_{173}(119,\cdot)\) \(\chi_{173}(124,\cdot)\) \(\chi_{173}(132,\cdot)\) \(\chi_{173}(133,\cdot)\) \(\chi_{173}(135,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{43})$
Fixed field: 43.43.9952594992767664919302480397055915291864099597331865326873171797085952964991380209309055259529.1

Values on generators

\(2\) → \(e\left(\frac{36}{43}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{36}{43}\right)\)\(e\left(\frac{26}{43}\right)\)\(e\left(\frac{29}{43}\right)\)\(e\left(\frac{28}{43}\right)\)\(e\left(\frac{19}{43}\right)\)\(e\left(\frac{23}{43}\right)\)\(e\left(\frac{22}{43}\right)\)\(e\left(\frac{9}{43}\right)\)\(e\left(\frac{21}{43}\right)\)\(e\left(\frac{11}{43}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 173 }(29,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{173}(29,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(29,r) e\left(\frac{2r}{173}\right) = -13.105392299+1.1174491883i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 173 }(29,·),\chi_{ 173 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{173}(29,\cdot),\chi_{173}(1,\cdot)) = \sum_{r\in \Z/173\Z} \chi_{173}(29,r) \chi_{173}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 173 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{173}(29,·)) = \sum_{r \in \Z/173\Z} \chi_{173}(29,r) e\left(\frac{1 r + 2 r^{-1}}{173}\right) = 6.5446330797+-3.6730261i \)