Properties

Label 775.133
Modulus $775$
Conductor $775$
Order $60$
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(775, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,4]))
 
pari: [g,chi] = znchar(Mod(133,775))
 

Basic properties

Modulus: \(775\)
Conductor: \(775\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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 775.cv

\(\chi_{775}(133,\cdot)\) \(\chi_{775}(227,\cdot)\) \(\chi_{775}(298,\cdot)\) \(\chi_{775}(338,\cdot)\) \(\chi_{775}(417,\cdot)\) \(\chi_{775}(462,\cdot)\) \(\chi_{775}(472,\cdot)\) \(\chi_{775}(503,\cdot)\) \(\chi_{775}(547,\cdot)\) \(\chi_{775}(567,\cdot)\) \(\chi_{775}(577,\cdot)\) \(\chi_{775}(578,\cdot)\) \(\chi_{775}(638,\cdot)\) \(\chi_{775}(723,\cdot)\) \(\chi_{775}(758,\cdot)\) \(\chi_{775}(762,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((652,251)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{1}{15}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 775 }(133, a) \) \(-1\)\(1\)\(-i\)\(e\left(\frac{7}{60}\right)\)\(-1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(i\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 775 }(133,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 775 }(133,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 775 }(133,·),\chi_{ 775 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 775 }(133,·)) \;\) at \(\; a,b = \) e.g. 1,2