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Properties

Label 3520.321
Modulus $3520$
Conductor $11$
Order $10$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(321,3520))
 

Basic properties

Modulus: \(3520\)
Conductor: \(11\)
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 \(\chi_{11}(2,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3520.cl

\(\chi_{3520}(321,\cdot)\) \(\chi_{3520}(1601,\cdot)\) \(\chi_{3520}(1921,\cdot)\) \(\chi_{3520}(2241,\cdot)\)

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: \(\Q(\zeta_{11})\)

Values on generators

\((2751,1541,2817,321)\) → \((1,1,1,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 3520 }(321, a) \) \(-1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(-1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3520 }(321,a) \;\) at \(\;a = \) e.g. 2