Properties

Label 311.2
Modulus $311$
Conductor $311$
Order $155$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(311, base_ring=CyclotomicField(310))
 
M = H._module
 
chi = DirichletCharacter(H, M([22]))
 
pari: [g,chi] = znchar(Mod(2,311))
 

Basic properties

Modulus: \(311\)
Conductor: \(311\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(155\)
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 311.g

\(\chi_{311}(2,\cdot)\) \(\chi_{311}(3,\cdot)\) \(\chi_{311}(4,\cdot)\) \(\chi_{311}(5,\cdot)\) \(\chi_{311}(8,\cdot)\) \(\chi_{311}(9,\cdot)\) \(\chi_{311}(10,\cdot)\) \(\chi_{311}(12,\cdot)\) \(\chi_{311}(14,\cdot)\) \(\chi_{311}(16,\cdot)\) \(\chi_{311}(21,\cdot)\) \(\chi_{311}(25,\cdot)\) \(\chi_{311}(26,\cdot)\) \(\chi_{311}(27,\cdot)\) \(\chi_{311}(28,\cdot)\) \(\chi_{311}(30,\cdot)\) \(\chi_{311}(35,\cdot)\) \(\chi_{311}(39,\cdot)\) \(\chi_{311}(40,\cdot)\) \(\chi_{311}(42,\cdot)\) \(\chi_{311}(45,\cdot)\) \(\chi_{311}(48,\cdot)\) \(\chi_{311}(50,\cdot)\) \(\chi_{311}(53,\cdot)\) \(\chi_{311}(54,\cdot)\) \(\chi_{311}(56,\cdot)\) \(\chi_{311}(60,\cdot)\) \(\chi_{311}(63,\cdot)\) \(\chi_{311}(64,\cdot)\) \(\chi_{311}(65,\cdot)\) ...

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: $\Q(\zeta_{155})$
Fixed field: Number field defined by a degree 155 polynomial (not computed)

Values on generators

\(17\) → \(e\left(\frac{11}{155}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 311 }(2, a) \) \(1\)\(1\)\(e\left(\frac{87}{155}\right)\)\(e\left(\frac{99}{155}\right)\)\(e\left(\frac{19}{155}\right)\)\(e\left(\frac{141}{155}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{29}{31}\right)\)\(e\left(\frac{106}{155}\right)\)\(e\left(\frac{43}{155}\right)\)\(e\left(\frac{73}{155}\right)\)\(e\left(\frac{18}{31}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 311 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

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

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 311 }(2,·),\chi_{ 311 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

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