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Properties

Label 9576.37
Modulus $9576$
Conductor $1064$
Order $6$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9576, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([0,3,0,2,3]))
 
Copy content pari:[g,chi] = znchar(Mod(37,9576))
 

Basic properties

Modulus: \(9576\)
Conductor: \(1064\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(6\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1064}(37,\cdot)\)
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 9576.nt

\(\chi_{9576}(37,\cdot)\) \(\chi_{9576}(4141,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: 6.0.8431851008.6

Values on generators

\((7183,4789,5321,4105,1009)\) → \((1,-1,1,e\left(\frac{1}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9576 }(37, a) \) \(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 9576 }(37,a) \;\) at \(\;a = \) e.g. 2