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Properties

Label 5328.7
Modulus $5328$
Conductor $2664$
Order $18$
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(5328, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([9,9,12,16]))
 
Copy content pari:[g,chi] = znchar(Mod(7,5328))
 

Basic properties

Modulus: \(5328\)
Conductor: \(2664\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(18\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2664}(1339,\cdot)\)
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 5328.ip

\(\chi_{5328}(7,\cdot)\) \(\chi_{5328}(823,\cdot)\) \(\chi_{5328}(1735,\cdot)\) \(\chi_{5328}(3031,\cdot)\) \(\chi_{5328}(4567,\cdot)\) \(\chi_{5328}(4711,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((1999,1333,2369,1297)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 5328 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 5328 }(7,a) \;\) at \(\;a = \) e.g. 2