Properties

Label 161.66
Modulus $161$
Conductor $161$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(161, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,15]))
 
pari: [g,chi] = znchar(Mod(66,161))
 

Basic properties

Modulus: \(161\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 161.o

\(\chi_{161}(5,\cdot)\) \(\chi_{161}(10,\cdot)\) \(\chi_{161}(17,\cdot)\) \(\chi_{161}(19,\cdot)\) \(\chi_{161}(33,\cdot)\) \(\chi_{161}(38,\cdot)\) \(\chi_{161}(40,\cdot)\) \(\chi_{161}(61,\cdot)\) \(\chi_{161}(66,\cdot)\) \(\chi_{161}(80,\cdot)\) \(\chi_{161}(89,\cdot)\) \(\chi_{161}(103,\cdot)\) \(\chi_{161}(122,\cdot)\) \(\chi_{161}(129,\cdot)\) \(\chi_{161}(136,\cdot)\) \(\chi_{161}(143,\cdot)\) \(\chi_{161}(145,\cdot)\) \(\chi_{161}(152,\cdot)\) \(\chi_{161}(157,\cdot)\) \(\chi_{161}(159,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((24,120)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{25}{66}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 161 }(66,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{161}(66,\cdot)) = \sum_{r\in \Z/161\Z} \chi_{161}(66,r) e\left(\frac{2r}{161}\right) = 4.1219137753+-12.0004094442i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 161 }(66,·),\chi_{ 161 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{161}(66,\cdot),\chi_{161}(1,\cdot)) = \sum_{r\in \Z/161\Z} \chi_{161}(66,r) \chi_{161}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 161 }(66,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{161}(66,·)) = \sum_{r \in \Z/161\Z} \chi_{161}(66,r) e\left(\frac{1 r + 2 r^{-1}}{161}\right) = 0.7711008347+-0.6064003989i \)