Properties

Label 177.113
Modulus $177$
Conductor $177$
Order $58$
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(177, base_ring=CyclotomicField(58))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([29,35]))
 
pari: [g,chi] = znchar(Mod(113,177))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 177.f

\(\chi_{177}(2,\cdot)\) \(\chi_{177}(8,\cdot)\) \(\chi_{177}(11,\cdot)\) \(\chi_{177}(14,\cdot)\) \(\chi_{177}(23,\cdot)\) \(\chi_{177}(32,\cdot)\) \(\chi_{177}(38,\cdot)\) \(\chi_{177}(44,\cdot)\) \(\chi_{177}(47,\cdot)\) \(\chi_{177}(50,\cdot)\) \(\chi_{177}(56,\cdot)\) \(\chi_{177}(65,\cdot)\) \(\chi_{177}(77,\cdot)\) \(\chi_{177}(83,\cdot)\) \(\chi_{177}(89,\cdot)\) \(\chi_{177}(92,\cdot)\) \(\chi_{177}(98,\cdot)\) \(\chi_{177}(101,\cdot)\) \(\chi_{177}(113,\cdot)\) \(\chi_{177}(128,\cdot)\) \(\chi_{177}(131,\cdot)\) \(\chi_{177}(149,\cdot)\) \(\chi_{177}(152,\cdot)\) \(\chi_{177}(155,\cdot)\) \(\chi_{177}(158,\cdot)\) \(\chi_{177}(161,\cdot)\) \(\chi_{177}(170,\cdot)\) \(\chi_{177}(173,\cdot)\)

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

\((119,61)\) → \((-1,e\left(\frac{35}{58}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{6}{29}\right)\)\(e\left(\frac{7}{58}\right)\)\(e\left(\frac{25}{29}\right)\)\(e\left(\frac{9}{29}\right)\)\(e\left(\frac{13}{58}\right)\)\(e\left(\frac{17}{29}\right)\)\(e\left(\frac{9}{58}\right)\)\(e\left(\frac{28}{29}\right)\)\(e\left(\frac{12}{29}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 177 }(113,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{177}(113,\cdot)) = \sum_{r\in \Z/177\Z} \chi_{177}(113,r) e\left(\frac{2r}{177}\right) = 10.8393067849+7.7142354399i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 177 }(113,·),\chi_{ 177 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{177}(113,\cdot),\chi_{177}(1,\cdot)) = \sum_{r\in \Z/177\Z} \chi_{177}(113,r) \chi_{177}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 177 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{177}(113,·)) = \sum_{r \in \Z/177\Z} \chi_{177}(113,r) e\left(\frac{1 r + 2 r^{-1}}{177}\right) = 0.0 \)