Properties

Label 980.59
Modulus $980$
Conductor $980$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(980, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,21,13]))
 
pari: [g,chi] = znchar(Mod(59,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 980.bl

\(\chi_{980}(59,\cdot)\) \(\chi_{980}(159,\cdot)\) \(\chi_{980}(199,\cdot)\) \(\chi_{980}(299,\cdot)\) \(\chi_{980}(339,\cdot)\) \(\chi_{980}(439,\cdot)\) \(\chi_{980}(479,\cdot)\) \(\chi_{980}(579,\cdot)\) \(\chi_{980}(719,\cdot)\) \(\chi_{980}(759,\cdot)\) \(\chi_{980}(859,\cdot)\) \(\chi_{980}(899,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.42.247844331230269810885249548811243543716772810770129718520393580847038464000000000000000000000.1

Values on generators

\((491,197,101)\) → \((-1,-1,e\left(\frac{13}{42}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(59, a) \) \(1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 980 }(59,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(59,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(59,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2