Properties

Label 684.61
Modulus $684$
Conductor $171$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(684, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,12,2]))
 
pari: [g,chi] = znchar(Mod(61,684))
 

Basic properties

Modulus: \(684\)
Conductor: \(171\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{171}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 684.bp

\(\chi_{684}(25,\cdot)\) \(\chi_{684}(61,\cdot)\) \(\chi_{684}(157,\cdot)\) \(\chi_{684}(301,\cdot)\) \(\chi_{684}(313,\cdot)\) \(\chi_{684}(625,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 9.9.9025761726072081.1

Values on generators

\((343,533,325)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{1}{9}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 684 }(61, a) \) \(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(1\)\(e\left(\frac{4}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 684 }(61,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 684 }(61,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 684 }(61,·),\chi_{ 684 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 684 }(61,·)) \;\) at \(\; a,b = \) e.g. 1,2