Properties

Label 5776.9
Modulus $5776$
Conductor $2888$
Order $342$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5776, base_ring=CyclotomicField(342))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,171,278]))
 
pari: [g,chi] = znchar(Mod(9,5776))
 

Basic properties

Modulus: \(5776\)
Conductor: \(2888\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(342\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2888}(1453,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5776.cn

\(\chi_{5776}(9,\cdot)\) \(\chi_{5776}(25,\cdot)\) \(\chi_{5776}(73,\cdot)\) \(\chi_{5776}(137,\cdot)\) \(\chi_{5776}(169,\cdot)\) \(\chi_{5776}(233,\cdot)\) \(\chi_{5776}(313,\cdot)\) \(\chi_{5776}(329,\cdot)\) \(\chi_{5776}(377,\cdot)\) \(\chi_{5776}(441,\cdot)\) \(\chi_{5776}(473,\cdot)\) \(\chi_{5776}(537,\cdot)\) \(\chi_{5776}(617,\cdot)\) \(\chi_{5776}(633,\cdot)\) \(\chi_{5776}(681,\cdot)\) \(\chi_{5776}(745,\cdot)\) \(\chi_{5776}(777,\cdot)\) \(\chi_{5776}(841,\cdot)\) \(\chi_{5776}(921,\cdot)\) \(\chi_{5776}(937,\cdot)\) \(\chi_{5776}(985,\cdot)\) \(\chi_{5776}(1049,\cdot)\) \(\chi_{5776}(1081,\cdot)\) \(\chi_{5776}(1225,\cdot)\) \(\chi_{5776}(1241,\cdot)\) \(\chi_{5776}(1289,\cdot)\) \(\chi_{5776}(1353,\cdot)\) \(\chi_{5776}(1385,\cdot)\) \(\chi_{5776}(1449,\cdot)\) \(\chi_{5776}(1529,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{171})$
Fixed field: Number field defined by a degree 342 polynomial (not computed)

Values on generators

\((5055,1445,2529)\) → \((1,-1,e\left(\frac{139}{171}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 5776 }(9, a) \) \(1\)\(1\)\(e\left(\frac{167}{342}\right)\)\(e\left(\frac{29}{342}\right)\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{167}{171}\right)\)\(e\left(\frac{47}{114}\right)\)\(e\left(\frac{319}{342}\right)\)\(e\left(\frac{98}{171}\right)\)\(e\left(\frac{121}{171}\right)\)\(e\left(\frac{143}{342}\right)\)\(e\left(\frac{116}{171}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5776 }(9,a) \;\) at \(\;a = \) e.g. 2