Properties

Label 775.443
Modulus $775$
Conductor $155$
Order $60$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,4]))
 
pari: [g,chi] = znchar(Mod(443,775))
 

Basic properties

Modulus: \(775\)
Conductor: \(155\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{155}(133,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 775.cw

\(\chi_{775}(7,\cdot)\) \(\chi_{775}(18,\cdot)\) \(\chi_{775}(82,\cdot)\) \(\chi_{775}(107,\cdot)\) \(\chi_{775}(143,\cdot)\) \(\chi_{775}(193,\cdot)\) \(\chi_{775}(257,\cdot)\) \(\chi_{775}(268,\cdot)\) \(\chi_{775}(293,\cdot)\) \(\chi_{775}(307,\cdot)\) \(\chi_{775}(382,\cdot)\) \(\chi_{775}(443,\cdot)\) \(\chi_{775}(493,\cdot)\) \(\chi_{775}(568,\cdot)\) \(\chi_{775}(607,\cdot)\) \(\chi_{775}(732,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((652,251)\) → \((-i,e\left(\frac{1}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 775 }(443, a) \) \(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{59}{60}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 775 }(443,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 775 }(443,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 775 }(443,·),\chi_{ 775 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 775 }(443,·)) \;\) at \(\; a,b = \) e.g. 1,2