Properties

Label 2888.49
Modulus $2888$
Conductor $361$
Order $57$
Real no
Primitive no
Minimal yes
Parity even

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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([0,0,100]))
 
pari: [g,chi] = znchar(Mod(49,2888))
 

Basic properties

Modulus: \(2888\)
Conductor: \(361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{361}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2888.bg

\(\chi_{2888}(49,\cdot)\) \(\chi_{2888}(121,\cdot)\) \(\chi_{2888}(201,\cdot)\) \(\chi_{2888}(273,\cdot)\) \(\chi_{2888}(353,\cdot)\) \(\chi_{2888}(425,\cdot)\) \(\chi_{2888}(505,\cdot)\) \(\chi_{2888}(577,\cdot)\) \(\chi_{2888}(657,\cdot)\) \(\chi_{2888}(729,\cdot)\) \(\chi_{2888}(809,\cdot)\) \(\chi_{2888}(881,\cdot)\) \(\chi_{2888}(961,\cdot)\) \(\chi_{2888}(1033,\cdot)\) \(\chi_{2888}(1113,\cdot)\) \(\chi_{2888}(1185,\cdot)\) \(\chi_{2888}(1265,\cdot)\) \(\chi_{2888}(1337,\cdot)\) \(\chi_{2888}(1417,\cdot)\) \(\chi_{2888}(1489,\cdot)\) \(\chi_{2888}(1569,\cdot)\) \(\chi_{2888}(1641,\cdot)\) \(\chi_{2888}(1721,\cdot)\) \(\chi_{2888}(1793,\cdot)\) \(\chi_{2888}(1945,\cdot)\) \(\chi_{2888}(2025,\cdot)\) \(\chi_{2888}(2177,\cdot)\) \(\chi_{2888}(2249,\cdot)\) \(\chi_{2888}(2329,\cdot)\) \(\chi_{2888}(2401,\cdot)\) ...

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 57 polynomial

Values on generators

\((2167,1445,2529)\) → \((1,1,e\left(\frac{50}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 2888 }(49, a) \) \(1\)\(1\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{29}{57}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{25}{57}\right)\)\(e\left(\frac{14}{57}\right)\)\(e\left(\frac{29}{57}\right)\)\(e\left(\frac{4}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2888 }(49,a) \;\) at \(\;a = \) e.g. 2