Properties

Label 980.27
Modulus $980$
Conductor $980$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(980, base_ring=CyclotomicField(28))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,7,2]))
 
pari: [g,chi] = znchar(Mod(27,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
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.bj

\(\chi_{980}(27,\cdot)\) \(\chi_{980}(83,\cdot)\) \(\chi_{980}(167,\cdot)\) \(\chi_{980}(223,\cdot)\) \(\chi_{980}(307,\cdot)\) \(\chi_{980}(363,\cdot)\) \(\chi_{980}(447,\cdot)\) \(\chi_{980}(503,\cdot)\) \(\chi_{980}(643,\cdot)\) \(\chi_{980}(727,\cdot)\) \(\chi_{980}(867,\cdot)\) \(\chi_{980}(923,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.0.230203525458868754767395883592915116704159872000000000000000000000.1

Values on generators

\((491,197,101)\) → \((-1,i,e\left(\frac{1}{14}\right))\)

Values

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(27,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(27,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(27,·)) \;\) at \(\; a,b = \) e.g. 1,2