Properties

Label 229.214
Modulus $229$
Conductor $229$
Order $19$
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(229, base_ring=CyclotomicField(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([32]))
 
pari: [g,chi] = znchar(Mod(214,229))
 

Basic properties

Modulus: \(229\)
Conductor: \(229\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
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 229.g

\(\chi_{229}(16,\cdot)\) \(\chi_{229}(17,\cdot)\) \(\chi_{229}(27,\cdot)\) \(\chi_{229}(42,\cdot)\) \(\chi_{229}(43,\cdot)\) \(\chi_{229}(44,\cdot)\) \(\chi_{229}(53,\cdot)\) \(\chi_{229}(57,\cdot)\) \(\chi_{229}(60,\cdot)\) \(\chi_{229}(61,\cdot)\) \(\chi_{229}(104,\cdot)\) \(\chi_{229}(121,\cdot)\) \(\chi_{229}(161,\cdot)\) \(\chi_{229}(165,\cdot)\) \(\chi_{229}(203,\cdot)\) \(\chi_{229}(214,\cdot)\) \(\chi_{229}(218,\cdot)\) \(\chi_{229}(225,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 19.19.2999429662895796650415561622892044448017561.1

Values on generators

\(6\) → \(e\left(\frac{16}{19}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{10}{19}\right)\)\(e\left(\frac{16}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{8}{19}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 229 }(214,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{229}(214,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(214,r) e\left(\frac{2r}{229}\right) = -10.3432477832+-11.0461407422i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 229 }(214,·),\chi_{ 229 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{229}(214,\cdot),\chi_{229}(1,\cdot)) = \sum_{r\in \Z/229\Z} \chi_{229}(214,r) \chi_{229}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 229 }(214,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{229}(214,·)) = \sum_{r \in \Z/229\Z} \chi_{229}(214,r) e\left(\frac{1 r + 2 r^{-1}}{229}\right) = -2.2349853046+3.4208996751i \)