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Properties

Label 3240.379
Modulus $3240$
Conductor $360$
Order $6$
Real no
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([3,3,4,3]))
 
Copy content pari:[g,chi] = znchar(Mod(379,3240))
 

Basic properties

Modulus: \(3240\)
Conductor: \(360\)
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 \(\chi_{360}(259,\cdot)\)
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 3240.z

\(\chi_{3240}(379,\cdot)\) \(\chi_{3240}(2539,\cdot)\)

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: 6.0.419904000.3

Values on generators

\((2431,1621,3161,1297)\) → \((-1,-1,e\left(\frac{2}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3240 }(379, a) \) \(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3240 }(379,a) \;\) at \(\;a = \) e.g. 2