Properties

Label 167.122
Modulus $167$
Conductor $167$
Order $83$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(167)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([53]))
 
pari: [g,chi] = znchar(Mod(122,167))
 

Basic properties

Modulus: \(167\)
Conductor: \(167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(83\)
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 167.c

\(\chi_{167}(2,\cdot)\) \(\chi_{167}(3,\cdot)\) \(\chi_{167}(4,\cdot)\) \(\chi_{167}(6,\cdot)\) \(\chi_{167}(7,\cdot)\) \(\chi_{167}(8,\cdot)\) \(\chi_{167}(9,\cdot)\) \(\chi_{167}(11,\cdot)\) \(\chi_{167}(12,\cdot)\) \(\chi_{167}(14,\cdot)\) \(\chi_{167}(16,\cdot)\) \(\chi_{167}(18,\cdot)\) \(\chi_{167}(19,\cdot)\) \(\chi_{167}(21,\cdot)\) \(\chi_{167}(22,\cdot)\) \(\chi_{167}(24,\cdot)\) \(\chi_{167}(25,\cdot)\) \(\chi_{167}(27,\cdot)\) \(\chi_{167}(28,\cdot)\) \(\chi_{167}(29,\cdot)\) \(\chi_{167}(31,\cdot)\) \(\chi_{167}(32,\cdot)\) \(\chi_{167}(33,\cdot)\) \(\chi_{167}(36,\cdot)\) \(\chi_{167}(38,\cdot)\) \(\chi_{167}(42,\cdot)\) \(\chi_{167}(44,\cdot)\) \(\chi_{167}(47,\cdot)\) \(\chi_{167}(48,\cdot)\) \(\chi_{167}(49,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\(5\) → \(e\left(\frac{53}{83}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{45}{83}\right)\)\(e\left(\frac{2}{83}\right)\)\(e\left(\frac{7}{83}\right)\)\(e\left(\frac{53}{83}\right)\)\(e\left(\frac{47}{83}\right)\)\(e\left(\frac{29}{83}\right)\)\(e\left(\frac{52}{83}\right)\)\(e\left(\frac{4}{83}\right)\)\(e\left(\frac{15}{83}\right)\)\(e\left(\frac{73}{83}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 83 polynomial

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 167 }(122,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{167}(122,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(122,r) e\left(\frac{2r}{167}\right) = -8.3487380329+9.8640039161i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 167 }(122,·),\chi_{ 167 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{167}(122,\cdot),\chi_{167}(1,\cdot)) = \sum_{r\in \Z/167\Z} \chi_{167}(122,r) \chi_{167}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 167 }(122,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{167}(122,·)) = \sum_{r \in \Z/167\Z} \chi_{167}(122,r) e\left(\frac{1 r + 2 r^{-1}}{167}\right) = 2.5800621364+-19.3615110137i \)