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Properties

Label 10002.5971
Modulus $10002$
Conductor $1667$
Order $7$
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(10002, base_ring=CyclotomicField(14)) M = H._module chi = DirichletCharacter(H, M([0,8]))
 
Copy content pari:[g,chi] = znchar(Mod(5971,10002))
 

Basic properties

Modulus: \(10002\)
Conductor: \(1667\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(7\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1667}(970,\cdot)\)
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 10002.e

\(\chi_{10002}(1843,\cdot)\) \(\chi_{10002}(2353,\cdot)\) \(\chi_{10002}(5503,\cdot)\) \(\chi_{10002}(5713,\cdot)\) \(\chi_{10002}(5971,\cdot)\) \(\chi_{10002}(6955,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 7.7.21459203535665809369.1

Values on generators

\((3335,1669)\) → \((1,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 10002 }(5971, a) \) \(1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 10002 }(5971,a) \;\) at \(\;a = \) e.g. 2