Properties

Label 2888.61
Modulus $2888$
Conductor $2888$
Order $342$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(342))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,171,92]))
 
pari: [g,chi] = znchar(Mod(61,2888))
 

Basic properties

Modulus: \(2888\)
Conductor: \(2888\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(342\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2888.bt

\(\chi_{2888}(5,\cdot)\) \(\chi_{2888}(61,\cdot)\) \(\chi_{2888}(85,\cdot)\) \(\chi_{2888}(93,\cdot)\) \(\chi_{2888}(101,\cdot)\) \(\chi_{2888}(149,\cdot)\) \(\chi_{2888}(157,\cdot)\) \(\chi_{2888}(213,\cdot)\) \(\chi_{2888}(237,\cdot)\) \(\chi_{2888}(253,\cdot)\) \(\chi_{2888}(301,\cdot)\) \(\chi_{2888}(309,\cdot)\) \(\chi_{2888}(365,\cdot)\) \(\chi_{2888}(397,\cdot)\) \(\chi_{2888}(405,\cdot)\) \(\chi_{2888}(453,\cdot)\) \(\chi_{2888}(461,\cdot)\) \(\chi_{2888}(517,\cdot)\) \(\chi_{2888}(541,\cdot)\) \(\chi_{2888}(549,\cdot)\) \(\chi_{2888}(557,\cdot)\) \(\chi_{2888}(605,\cdot)\) \(\chi_{2888}(613,\cdot)\) \(\chi_{2888}(669,\cdot)\) \(\chi_{2888}(693,\cdot)\) \(\chi_{2888}(701,\cdot)\) \(\chi_{2888}(709,\cdot)\) \(\chi_{2888}(757,\cdot)\) \(\chi_{2888}(765,\cdot)\) \(\chi_{2888}(845,\cdot)\) ...

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

\((2167,1445,2529)\) → \((1,-1,e\left(\frac{46}{171}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 2888 }(61, a) \) \(1\)\(1\)\(e\left(\frac{305}{342}\right)\)\(e\left(\frac{311}{342}\right)\)\(e\left(\frac{20}{57}\right)\)\(e\left(\frac{134}{171}\right)\)\(e\left(\frac{107}{114}\right)\)\(e\left(\frac{1}{342}\right)\)\(e\left(\frac{137}{171}\right)\)\(e\left(\frac{136}{171}\right)\)\(e\left(\frac{83}{342}\right)\)\(e\left(\frac{47}{171}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2888 }(61,a) \;\) at \(\;a = \) e.g. 2