Properties

Label 311.15
Modulus $311$
Conductor $311$
Order $31$
Real no
Primitive yes
Minimal yes
Parity even

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([10]))
 
pari: [g,chi] = znchar(Mod(15,311))
 

Basic properties

Modulus: \(311\)
Conductor: \(311\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(31\)
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.e

\(\chi_{311}(7,\cdot)\) \(\chi_{311}(13,\cdot)\) \(\chi_{311}(15,\cdot)\) \(\chi_{311}(18,\cdot)\) \(\chi_{311}(20,\cdot)\) \(\chi_{311}(24,\cdot)\) \(\chi_{311}(32,\cdot)\) \(\chi_{311}(47,\cdot)\) \(\chi_{311}(49,\cdot)\) \(\chi_{311}(83,\cdot)\) \(\chi_{311}(89,\cdot)\) \(\chi_{311}(91,\cdot)\) \(\chi_{311}(105,\cdot)\) \(\chi_{311}(113,\cdot)\) \(\chi_{311}(121,\cdot)\) \(\chi_{311}(126,\cdot)\) \(\chi_{311}(140,\cdot)\) \(\chi_{311}(146,\cdot)\) \(\chi_{311}(168,\cdot)\) \(\chi_{311}(169,\cdot)\) \(\chi_{311}(195,\cdot)\) \(\chi_{311}(224,\cdot)\) \(\chi_{311}(225,\cdot)\) \(\chi_{311}(234,\cdot)\) \(\chi_{311}(243,\cdot)\) \(\chi_{311}(250,\cdot)\) \(\chi_{311}(260,\cdot)\) \(\chi_{311}(265,\cdot)\) \(\chi_{311}(270,\cdot)\) \(\chi_{311}(300,\cdot)\)

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 31 polynomial

Values on generators

\(17\) → \(e\left(\frac{5}{31}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 311 }(15, a) \) \(1\)\(1\)\(e\left(\frac{17}{31}\right)\)\(e\left(\frac{14}{31}\right)\)\(e\left(\frac{3}{31}\right)\)\(e\left(\frac{19}{31}\right)\)\(1\)\(e\left(\frac{18}{31}\right)\)\(e\left(\frac{20}{31}\right)\)\(e\left(\frac{28}{31}\right)\)\(e\left(\frac{5}{31}\right)\)\(e\left(\frac{24}{31}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 311 }(15,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 311 }(15,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 311 }(15,·),\chi_{ 311 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 311 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2