Properties

Label 5776.121
Modulus $5776$
Conductor $2888$
Order $114$
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(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,57,68]))
 
pari: [g,chi] = znchar(Mod(121,5776))
 

Basic properties

Modulus: \(5776\)
Conductor: \(2888\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2888}(1565,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5776.bx

\(\chi_{5776}(121,\cdot)\) \(\chi_{5776}(201,\cdot)\) \(\chi_{5776}(425,\cdot)\) \(\chi_{5776}(505,\cdot)\) \(\chi_{5776}(729,\cdot)\) \(\chi_{5776}(809,\cdot)\) \(\chi_{5776}(1033,\cdot)\) \(\chi_{5776}(1113,\cdot)\) \(\chi_{5776}(1337,\cdot)\) \(\chi_{5776}(1417,\cdot)\) \(\chi_{5776}(1641,\cdot)\) \(\chi_{5776}(1721,\cdot)\) \(\chi_{5776}(1945,\cdot)\) \(\chi_{5776}(2025,\cdot)\) \(\chi_{5776}(2249,\cdot)\) \(\chi_{5776}(2329,\cdot)\) \(\chi_{5776}(2553,\cdot)\) \(\chi_{5776}(2633,\cdot)\) \(\chi_{5776}(2857,\cdot)\) \(\chi_{5776}(2937,\cdot)\) \(\chi_{5776}(3161,\cdot)\) \(\chi_{5776}(3241,\cdot)\) \(\chi_{5776}(3465,\cdot)\) \(\chi_{5776}(3545,\cdot)\) \(\chi_{5776}(3769,\cdot)\) \(\chi_{5776}(3849,\cdot)\) \(\chi_{5776}(4073,\cdot)\) \(\chi_{5776}(4153,\cdot)\) \(\chi_{5776}(4377,\cdot)\) \(\chi_{5776}(4457,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((5055,1445,2529)\) → \((1,-1,e\left(\frac{34}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 5776 }(121, a) \) \(1\)\(1\)\(e\left(\frac{47}{114}\right)\)\(e\left(\frac{101}{114}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{13}{38}\right)\)\(e\left(\frac{85}{114}\right)\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{46}{57}\right)\)\(e\left(\frac{101}{114}\right)\)\(e\left(\frac{5}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5776 }(121,a) \;\) at \(\;a = \) e.g. 2