Properties

Label 980.383
Modulus $980$
Conductor $980$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(980, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([42,63,46]))
 
pari: [g,chi] = znchar(Mod(383,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 980.bt

\(\chi_{980}(3,\cdot)\) \(\chi_{980}(47,\cdot)\) \(\chi_{980}(87,\cdot)\) \(\chi_{980}(103,\cdot)\) \(\chi_{980}(143,\cdot)\) \(\chi_{980}(187,\cdot)\) \(\chi_{980}(243,\cdot)\) \(\chi_{980}(283,\cdot)\) \(\chi_{980}(327,\cdot)\) \(\chi_{980}(367,\cdot)\) \(\chi_{980}(383,\cdot)\) \(\chi_{980}(467,\cdot)\) \(\chi_{980}(507,\cdot)\) \(\chi_{980}(523,\cdot)\) \(\chi_{980}(563,\cdot)\) \(\chi_{980}(647,\cdot)\) \(\chi_{980}(663,\cdot)\) \(\chi_{980}(703,\cdot)\) \(\chi_{980}(747,\cdot)\) \(\chi_{980}(787,\cdot)\) \(\chi_{980}(843,\cdot)\) \(\chi_{980}(887,\cdot)\) \(\chi_{980}(927,\cdot)\) \(\chi_{980}(943,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((491,197,101)\) → \((-1,-i,e\left(\frac{23}{42}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{47}{84}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 980 }(383,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(383,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(383,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(383,·)) \;\) at \(\; a,b = \) e.g. 1,2