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Properties

Label 2156.687
Modulus $2156$
Conductor $44$
Order $10$
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(2156, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([5,0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(687,2156))
 

Basic properties

Modulus: \(2156\)
Conductor: \(44\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(10\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{44}(27,\cdot)\)
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 2156.v

\(\chi_{2156}(295,\cdot)\) \(\chi_{2156}(687,\cdot)\) \(\chi_{2156}(883,\cdot)\) \(\chi_{2156}(1863,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.0.219503494144.1

Values on generators

\((1079,1277,981)\) → \((-1,1,e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 2156 }(687, a) \) \(-1\)\(1\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(-1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2156 }(687,a) \;\) at \(\;a = \) e.g. 2