Properties

Label 177.40
Modulus $177$
Conductor $59$
Order $58$
Real no
Primitive no
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([0,9]))
 
pari: [g,chi] = znchar(Mod(40,177))
 

Basic properties

Modulus: \(177\)
Conductor: \(59\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(58\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{59}(40,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 177.g

\(\chi_{177}(10,\cdot)\) \(\chi_{177}(13,\cdot)\) \(\chi_{177}(31,\cdot)\) \(\chi_{177}(34,\cdot)\) \(\chi_{177}(37,\cdot)\) \(\chi_{177}(40,\cdot)\) \(\chi_{177}(43,\cdot)\) \(\chi_{177}(52,\cdot)\) \(\chi_{177}(55,\cdot)\) \(\chi_{177}(61,\cdot)\) \(\chi_{177}(67,\cdot)\) \(\chi_{177}(70,\cdot)\) \(\chi_{177}(73,\cdot)\) \(\chi_{177}(82,\cdot)\) \(\chi_{177}(91,\cdot)\) \(\chi_{177}(97,\cdot)\) \(\chi_{177}(103,\cdot)\) \(\chi_{177}(106,\cdot)\) \(\chi_{177}(109,\cdot)\) \(\chi_{177}(115,\cdot)\) \(\chi_{177}(124,\cdot)\) \(\chi_{177}(136,\cdot)\) \(\chi_{177}(142,\cdot)\) \(\chi_{177}(148,\cdot)\) \(\chi_{177}(151,\cdot)\) \(\chi_{177}(157,\cdot)\) \(\chi_{177}(160,\cdot)\) \(\chi_{177}(172,\cdot)\)

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

\((119,61)\) → \((1,e\left(\frac{9}{58}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{9}{29}\right)\)\(e\left(\frac{27}{29}\right)\)\(e\left(\frac{23}{29}\right)\)\(e\left(\frac{27}{58}\right)\)\(e\left(\frac{5}{58}\right)\)\(e\left(\frac{51}{58}\right)\)\(e\left(\frac{57}{58}\right)\)\(e\left(\frac{55}{58}\right)\)\(e\left(\frac{18}{29}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 177 }(40,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 177 }(40,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 177 }(40,·),\chi_{ 177 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 177 }(40,·)) \;\) at \(\; a,b = \) e.g. 1,2