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Properties

Label 980.57
Modulus $980$
Conductor $245$
Order $28$
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(28)) M = H._module chi = DirichletCharacter(H, M([0,7,24]))
 
Copy content pari:[g,chi] = znchar(Mod(57,980))
 

Basic properties

Modulus: \(980\)
Conductor: \(245\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(28\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(57,\cdot)\)
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.bh

\(\chi_{980}(57,\cdot)\) \(\chi_{980}(113,\cdot)\) \(\chi_{980}(253,\cdot)\) \(\chi_{980}(337,\cdot)\) \(\chi_{980}(477,\cdot)\) \(\chi_{980}(533,\cdot)\) \(\chi_{980}(617,\cdot)\) \(\chi_{980}(673,\cdot)\) \(\chi_{980}(757,\cdot)\) \(\chi_{980}(813,\cdot)\) \(\chi_{980}(897,\cdot)\) \(\chi_{980}(953,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\((491,197,101)\) → \((1,i,e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 980 }(57, a) \) \(-1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{19}{28}\right)\)\(-1\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 980 }(57,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 980 }(57,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 980 }(57,·),\chi_{ 980 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 980 }(57,·)) \;\) at \(\; a,b = \) e.g. 1,2