Properties

Label 177.4
Modulus $177$
Conductor $59$
Order $29$
Real no
Primitive no
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(177)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(4,177))
 

Basic properties

Modulus: \(177\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(29\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{59}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 177.e

\(\chi_{177}(4,\cdot)\) \(\chi_{177}(7,\cdot)\) \(\chi_{177}(16,\cdot)\) \(\chi_{177}(19,\cdot)\) \(\chi_{177}(22,\cdot)\) \(\chi_{177}(25,\cdot)\) \(\chi_{177}(28,\cdot)\) \(\chi_{177}(46,\cdot)\) \(\chi_{177}(49,\cdot)\) \(\chi_{177}(64,\cdot)\) \(\chi_{177}(76,\cdot)\) \(\chi_{177}(79,\cdot)\) \(\chi_{177}(85,\cdot)\) \(\chi_{177}(88,\cdot)\) \(\chi_{177}(94,\cdot)\) \(\chi_{177}(100,\cdot)\) \(\chi_{177}(112,\cdot)\) \(\chi_{177}(121,\cdot)\) \(\chi_{177}(127,\cdot)\) \(\chi_{177}(130,\cdot)\) \(\chi_{177}(133,\cdot)\) \(\chi_{177}(139,\cdot)\) \(\chi_{177}(145,\cdot)\) \(\chi_{177}(154,\cdot)\) \(\chi_{177}(163,\cdot)\) \(\chi_{177}(166,\cdot)\) \(\chi_{177}(169,\cdot)\) \(\chi_{177}(175,\cdot)\)

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

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{1}{29}\right)\)\(e\left(\frac{2}{29}\right)\)\(e\left(\frac{6}{29}\right)\)\(e\left(\frac{18}{29}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{7}{29}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{16}{29}\right)\)\(e\left(\frac{19}{29}\right)\)\(e\left(\frac{4}{29}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{29})$
Fixed field: \(\Q(\zeta_{59})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 177 }(4,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{177}(4,\cdot)) = \sum_{r\in \Z/177\Z} \chi_{177}(4,r) e\left(\frac{2r}{177}\right) = 3.9955898359+6.5601266652i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 177 }(4,·),\chi_{ 177 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{177}(4,\cdot),\chi_{177}(1,\cdot)) = \sum_{r\in \Z/177\Z} \chi_{177}(4,r) \chi_{177}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 177 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{177}(4,·)) = \sum_{r \in \Z/177\Z} \chi_{177}(4,r) e\left(\frac{1 r + 2 r^{-1}}{177}\right) = 13.9572178344+1.5179389151i \)