Properties

Label 199.61
Modulus $199$
Conductor $199$
Order $11$
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(199, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14]))
 
pari: [g,chi] = znchar(Mod(61,199))
 

Basic properties

Modulus: \(199\)
Conductor: \(199\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
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 199.f

\(\chi_{199}(18,\cdot)\) \(\chi_{199}(61,\cdot)\) \(\chi_{199}(62,\cdot)\) \(\chi_{199}(63,\cdot)\) \(\chi_{199}(103,\cdot)\) \(\chi_{199}(114,\cdot)\) \(\chi_{199}(121,\cdot)\) \(\chi_{199}(125,\cdot)\) \(\chi_{199}(139,\cdot)\) \(\chi_{199}(188,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 11.11.97393677359695041798001.1

Values on generators

\(3\) → \(e\left(\frac{7}{11}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{3}{11}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 199 }(61,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{199}(61,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(61,r) e\left(\frac{2r}{199}\right) = -3.2021089493+-13.7385042227i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 199 }(61,·),\chi_{ 199 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{199}(61,\cdot),\chi_{199}(1,\cdot)) = \sum_{r\in \Z/199\Z} \chi_{199}(61,r) \chi_{199}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 199 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{199}(61,·)) = \sum_{r \in \Z/199\Z} \chi_{199}(61,r) e\left(\frac{1 r + 2 r^{-1}}{199}\right) = -0.988265099+-6.8735347427i \)