Properties

Label 177.137
Modulus $177$
Conductor $177$
Order $58$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(177, base_ring=CyclotomicField(58))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([29,38]))
 
pari: [g,chi] = znchar(Mod(137,177))
 

Basic properties

Modulus: \(177\)
Conductor: \(177\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 177.h

\(\chi_{177}(5,\cdot)\) \(\chi_{177}(17,\cdot)\) \(\chi_{177}(20,\cdot)\) \(\chi_{177}(26,\cdot)\) \(\chi_{177}(29,\cdot)\) \(\chi_{177}(35,\cdot)\) \(\chi_{177}(41,\cdot)\) \(\chi_{177}(53,\cdot)\) \(\chi_{177}(62,\cdot)\) \(\chi_{177}(68,\cdot)\) \(\chi_{177}(71,\cdot)\) \(\chi_{177}(74,\cdot)\) \(\chi_{177}(80,\cdot)\) \(\chi_{177}(86,\cdot)\) \(\chi_{177}(95,\cdot)\) \(\chi_{177}(104,\cdot)\) \(\chi_{177}(107,\cdot)\) \(\chi_{177}(110,\cdot)\) \(\chi_{177}(116,\cdot)\) \(\chi_{177}(122,\cdot)\) \(\chi_{177}(125,\cdot)\) \(\chi_{177}(134,\cdot)\) \(\chi_{177}(137,\cdot)\) \(\chi_{177}(140,\cdot)\) \(\chi_{177}(143,\cdot)\) \(\chi_{177}(146,\cdot)\) \(\chi_{177}(164,\cdot)\) \(\chi_{177}(167,\cdot)\)

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: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((119,61)\) → \((-1,e\left(\frac{19}{29}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{9}{29}\right)\)\(e\left(\frac{25}{58}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{27}{58}\right)\)\(e\left(\frac{17}{29}\right)\)\(e\left(\frac{51}{58}\right)\)\(e\left(\frac{14}{29}\right)\)\(e\left(\frac{55}{58}\right)\)\(e\left(\frac{18}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 177 }(137,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 177 }(137,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 177 }(137,·),\chi_{ 177 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 177 }(137,·)) \;\) at \(\; a,b = \) e.g. 1,2