Properties

Label 311.279
Modulus $311$
Conductor $311$
Order $62$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(311, base_ring=CyclotomicField(62))
 
M = H._module
 
chi = DirichletCharacter(H, M([53]))
 
pari: [g,chi] = znchar(Mod(279,311))
 

Basic properties

Modulus: \(311\)
Conductor: \(311\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(62\)
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 311.f

\(\chi_{311}(11,\cdot)\) \(\chi_{311}(41,\cdot)\) \(\chi_{311}(46,\cdot)\) \(\chi_{311}(51,\cdot)\) \(\chi_{311}(61,\cdot)\) \(\chi_{311}(68,\cdot)\) \(\chi_{311}(77,\cdot)\) \(\chi_{311}(86,\cdot)\) \(\chi_{311}(87,\cdot)\) \(\chi_{311}(116,\cdot)\) \(\chi_{311}(142,\cdot)\) \(\chi_{311}(143,\cdot)\) \(\chi_{311}(165,\cdot)\) \(\chi_{311}(171,\cdot)\) \(\chi_{311}(185,\cdot)\) \(\chi_{311}(190,\cdot)\) \(\chi_{311}(198,\cdot)\) \(\chi_{311}(206,\cdot)\) \(\chi_{311}(220,\cdot)\) \(\chi_{311}(222,\cdot)\) \(\chi_{311}(228,\cdot)\) \(\chi_{311}(262,\cdot)\) \(\chi_{311}(264,\cdot)\) \(\chi_{311}(279,\cdot)\) \(\chi_{311}(287,\cdot)\) \(\chi_{311}(291,\cdot)\) \(\chi_{311}(293,\cdot)\) \(\chi_{311}(296,\cdot)\) \(\chi_{311}(298,\cdot)\) \(\chi_{311}(304,\cdot)\)

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: $\Q(\zeta_{31})$
Fixed field: Number field defined by a degree 62 polynomial

Values on generators

\(17\) → \(e\left(\frac{53}{62}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 311 }(279, a) \) \(-1\)\(1\)\(e\left(\frac{25}{31}\right)\)\(e\left(\frac{6}{31}\right)\)\(e\left(\frac{19}{31}\right)\)\(e\left(\frac{17}{31}\right)\)\(1\)\(e\left(\frac{21}{31}\right)\)\(e\left(\frac{13}{31}\right)\)\(e\left(\frac{12}{31}\right)\)\(e\left(\frac{11}{31}\right)\)\(e\left(\frac{25}{62}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 311 }(279,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 311 }(279,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 311 }(279,·),\chi_{ 311 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 311 }(279,·)) \;\) at \(\; a,b = \) e.g. 1,2