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Properties

Label 4620.113
Modulus $4620$
Conductor $165$
Order $20$
Real no
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4620, base_ring=CyclotomicField(20)) M = H._module chi = DirichletCharacter(H, M([0,10,15,0,16]))
 
Copy content pari:[g,chi] = znchar(Mod(113,4620))
 

Basic properties

Modulus: \(4620\)
Conductor: \(165\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(20\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{165}(113,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 4620.fi

\(\chi_{4620}(113,\cdot)\) \(\chi_{4620}(533,\cdot)\) \(\chi_{4620}(1037,\cdot)\) \(\chi_{4620}(1373,\cdot)\) \(\chi_{4620}(1457,\cdot)\) \(\chi_{4620}(2297,\cdot)\) \(\chi_{4620}(2633,\cdot)\) \(\chi_{4620}(3557,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((2311,1541,3697,661,2521)\) → \((1,-1,-i,1,e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 4620 }(113, a) \) \(1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(-i\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(i\)\(e\left(\frac{13}{20}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 4620 }(113,a) \;\) at \(\;a = \) e.g. 2