Properties

Label 931.835
Modulus $931$
Conductor $931$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,21]))
 
pari: [g,chi] = znchar(Mod(835,931))
 

Basic properties

Modulus: \(931\)
Conductor: \(931\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 931.bq

\(\chi_{931}(37,\cdot)\) \(\chi_{931}(151,\cdot)\) \(\chi_{931}(170,\cdot)\) \(\chi_{931}(284,\cdot)\) \(\chi_{931}(303,\cdot)\) \(\chi_{931}(417,\cdot)\) \(\chi_{931}(436,\cdot)\) \(\chi_{931}(550,\cdot)\) \(\chi_{931}(683,\cdot)\) \(\chi_{931}(702,\cdot)\) \(\chi_{931}(816,\cdot)\) \(\chi_{931}(835,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((248,344)\) → \((e\left(\frac{13}{21}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 931 }(835, a) \) \(-1\)\(1\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{13}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 931 }(835,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 931 }(835,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 931 }(835,·),\chi_{ 931 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 931 }(835,·)) \;\) at \(\; a,b = \) e.g. 1,2