Properties

Label 185.181
Modulus $185$
Conductor $37$
Order $9$
Real no
Primitive no
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(185, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10]))
 
pari: [g,chi] = znchar(Mod(181,185))
 

Basic properties

Modulus: \(185\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(33,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 185.o

\(\chi_{185}(16,\cdot)\) \(\chi_{185}(46,\cdot)\) \(\chi_{185}(71,\cdot)\) \(\chi_{185}(81,\cdot)\) \(\chi_{185}(86,\cdot)\) \(\chi_{185}(181,\cdot)\)

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

\((112,76)\) → \((1,e\left(\frac{5}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(1\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{9}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: 9.9.3512479453921.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 185 }(181,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{185}(181,\cdot)) = \sum_{r\in \Z/185\Z} \chi_{185}(181,r) e\left(\frac{2r}{185}\right) = -6.0242446876+-0.8417101305i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 185 }(181,·),\chi_{ 185 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{185}(181,\cdot),\chi_{185}(1,\cdot)) = \sum_{r\in \Z/185\Z} \chi_{185}(181,r) \chi_{185}(1,1-r) = -3 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 185 }(181,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{185}(181,·)) = \sum_{r \in \Z/185\Z} \chi_{185}(181,r) e\left(\frac{1 r + 2 r^{-1}}{185}\right) = -1.0553479212+5.9851754789i \)