Properties

Label 7360.87
Modulus $7360$
Conductor $3680$
Order $88$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7360, base_ring=CyclotomicField(88))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,77,22,48]))
 
pari: [g,chi] = znchar(Mod(87,7360))
 

Basic properties

Modulus: \(7360\)
Conductor: \(3680\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3680}(2387,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7360.ew

\(\chi_{7360}(87,\cdot)\) \(\chi_{7360}(407,\cdot)\) \(\chi_{7360}(423,\cdot)\) \(\chi_{7360}(583,\cdot)\) \(\chi_{7360}(887,\cdot)\) \(\chi_{7360}(903,\cdot)\) \(\chi_{7360}(1047,\cdot)\) \(\chi_{7360}(1223,\cdot)\) \(\chi_{7360}(1383,\cdot)\) \(\chi_{7360}(1527,\cdot)\) \(\chi_{7360}(1543,\cdot)\) \(\chi_{7360}(1687,\cdot)\) \(\chi_{7360}(2007,\cdot)\) \(\chi_{7360}(2327,\cdot)\) \(\chi_{7360}(2487,\cdot)\) \(\chi_{7360}(2647,\cdot)\) \(\chi_{7360}(2663,\cdot)\) \(\chi_{7360}(2983,\cdot)\) \(\chi_{7360}(3463,\cdot)\) \(\chi_{7360}(3623,\cdot)\) \(\chi_{7360}(3767,\cdot)\) \(\chi_{7360}(4087,\cdot)\) \(\chi_{7360}(4103,\cdot)\) \(\chi_{7360}(4263,\cdot)\) \(\chi_{7360}(4567,\cdot)\) \(\chi_{7360}(4583,\cdot)\) \(\chi_{7360}(4727,\cdot)\) \(\chi_{7360}(4903,\cdot)\) \(\chi_{7360}(5063,\cdot)\) \(\chi_{7360}(5207,\cdot)\) ...

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: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

Values on generators

\((1151,5061,4417,6721)\) → \((-1,e\left(\frac{7}{8}\right),i,e\left(\frac{6}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 7360 }(87, a) \) \(1\)\(1\)\(e\left(\frac{53}{88}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{69}{88}\right)\)\(e\left(\frac{45}{88}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{27}{88}\right)\)\(e\left(\frac{41}{88}\right)\)\(e\left(\frac{71}{88}\right)\)\(e\left(\frac{83}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7360 }(87,a) \;\) at \(\;a = \) e.g. 2