Properties

Label 2888.2747
Modulus $2888$
Conductor $2888$
Order $114$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,57,112]))
 
pari: [g,chi] = znchar(Mod(2747,2888))
 

Basic properties

Modulus: \(2888\)
Conductor: \(2888\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2888.bm

\(\chi_{2888}(11,\cdot)\) \(\chi_{2888}(83,\cdot)\) \(\chi_{2888}(163,\cdot)\) \(\chi_{2888}(235,\cdot)\) \(\chi_{2888}(315,\cdot)\) \(\chi_{2888}(387,\cdot)\) \(\chi_{2888}(467,\cdot)\) \(\chi_{2888}(539,\cdot)\) \(\chi_{2888}(619,\cdot)\) \(\chi_{2888}(691,\cdot)\) \(\chi_{2888}(771,\cdot)\) \(\chi_{2888}(843,\cdot)\) \(\chi_{2888}(923,\cdot)\) \(\chi_{2888}(995,\cdot)\) \(\chi_{2888}(1075,\cdot)\) \(\chi_{2888}(1147,\cdot)\) \(\chi_{2888}(1227,\cdot)\) \(\chi_{2888}(1299,\cdot)\) \(\chi_{2888}(1379,\cdot)\) \(\chi_{2888}(1451,\cdot)\) \(\chi_{2888}(1531,\cdot)\) \(\chi_{2888}(1603,\cdot)\) \(\chi_{2888}(1683,\cdot)\) \(\chi_{2888}(1755,\cdot)\) \(\chi_{2888}(1835,\cdot)\) \(\chi_{2888}(1907,\cdot)\) \(\chi_{2888}(1987,\cdot)\) \(\chi_{2888}(2059,\cdot)\) \(\chi_{2888}(2139,\cdot)\) \(\chi_{2888}(2211,\cdot)\) ...

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

\((2167,1445,2529)\) → \((-1,-1,e\left(\frac{56}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 2888 }(2747, a) \) \(-1\)\(1\)\(e\left(\frac{32}{57}\right)\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{7}{57}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{83}{114}\right)\)\(e\left(\frac{113}{114}\right)\)\(e\left(\frac{2}{57}\right)\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{107}{114}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2888 }(2747,a) \;\) at \(\;a = \) e.g. 2