Properties

Label 5415.221
Modulus $5415$
Conductor $1083$
Order $114$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5415, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([57,0,71]))
 
pari: [g,chi] = znchar(Mod(221,5415))
 

Basic properties

Modulus: \(5415\)
Conductor: \(1083\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1083}(221,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5415.cb

\(\chi_{5415}(221,\cdot)\) \(\chi_{5415}(236,\cdot)\) \(\chi_{5415}(506,\cdot)\) \(\chi_{5415}(521,\cdot)\) \(\chi_{5415}(806,\cdot)\) \(\chi_{5415}(1076,\cdot)\) \(\chi_{5415}(1091,\cdot)\) \(\chi_{5415}(1361,\cdot)\) \(\chi_{5415}(1646,\cdot)\) \(\chi_{5415}(1661,\cdot)\) \(\chi_{5415}(1931,\cdot)\) \(\chi_{5415}(1946,\cdot)\) \(\chi_{5415}(2216,\cdot)\) \(\chi_{5415}(2231,\cdot)\) \(\chi_{5415}(2501,\cdot)\) \(\chi_{5415}(2516,\cdot)\) \(\chi_{5415}(2786,\cdot)\) \(\chi_{5415}(2801,\cdot)\) \(\chi_{5415}(3071,\cdot)\) \(\chi_{5415}(3086,\cdot)\) \(\chi_{5415}(3356,\cdot)\) \(\chi_{5415}(3371,\cdot)\) \(\chi_{5415}(3641,\cdot)\) \(\chi_{5415}(3656,\cdot)\) \(\chi_{5415}(3926,\cdot)\) \(\chi_{5415}(3941,\cdot)\) \(\chi_{5415}(4211,\cdot)\) \(\chi_{5415}(4226,\cdot)\) \(\chi_{5415}(4496,\cdot)\) \(\chi_{5415}(4511,\cdot)\) ...

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

\((3611,2167,5056)\) → \((-1,1,e\left(\frac{71}{114}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 5415 }(221, a) \) \(1\)\(1\)\(e\left(\frac{7}{57}\right)\)\(e\left(\frac{14}{57}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{1}{38}\right)\)\(e\left(\frac{103}{114}\right)\)\(e\left(\frac{31}{57}\right)\)\(e\left(\frac{28}{57}\right)\)\(e\left(\frac{29}{114}\right)\)\(e\left(\frac{17}{114}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5415 }(221,a) \;\) at \(\;a = \) e.g. 2