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Properties

Label 980.89
Modulus $980$
Conductor $245$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,21,23]))
 
Copy content pari:[g,chi] = znchar(Mod(89,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(245\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(42\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(89,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 980.br

\(\chi_{980}(89,\cdot)\) \(\chi_{980}(229,\cdot)\) \(\chi_{980}(269,\cdot)\) \(\chi_{980}(369,\cdot)\) \(\chi_{980}(409,\cdot)\) \(\chi_{980}(549,\cdot)\) \(\chi_{980}(649,\cdot)\) \(\chi_{980}(689,\cdot)\) \(\chi_{980}(789,\cdot)\) \(\chi_{980}(829,\cdot)\) \(\chi_{980}(929,\cdot)\) \(\chi_{980}(969,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.56353276529596271503862578540802938668269419115433656434196014026165008544921875.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(89, a) \) \(-1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 980 }(89,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(89,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(89,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(89,·)) \;\) at \(\; a,b = \) e.g. 1,2