Properties

Label 7744.87
Modulus $7744$
Conductor $3872$
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(7744, base_ring=CyclotomicField(88))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,77,84]))
 
pari: [g,chi] = znchar(Mod(87,7744))
 

Basic properties

Modulus: \(7744\)
Conductor: \(3872\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3872}(3475,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7744.cb

\(\chi_{7744}(87,\cdot)\) \(\chi_{7744}(263,\cdot)\) \(\chi_{7744}(439,\cdot)\) \(\chi_{7744}(615,\cdot)\) \(\chi_{7744}(791,\cdot)\) \(\chi_{7744}(1143,\cdot)\) \(\chi_{7744}(1319,\cdot)\) \(\chi_{7744}(1495,\cdot)\) \(\chi_{7744}(1671,\cdot)\) \(\chi_{7744}(1847,\cdot)\) \(\chi_{7744}(2023,\cdot)\) \(\chi_{7744}(2199,\cdot)\) \(\chi_{7744}(2375,\cdot)\) \(\chi_{7744}(2551,\cdot)\) \(\chi_{7744}(2727,\cdot)\) \(\chi_{7744}(3079,\cdot)\) \(\chi_{7744}(3255,\cdot)\) \(\chi_{7744}(3431,\cdot)\) \(\chi_{7744}(3607,\cdot)\) \(\chi_{7744}(3783,\cdot)\) \(\chi_{7744}(3959,\cdot)\) \(\chi_{7744}(4135,\cdot)\) \(\chi_{7744}(4311,\cdot)\) \(\chi_{7744}(4487,\cdot)\) \(\chi_{7744}(4663,\cdot)\) \(\chi_{7744}(5015,\cdot)\) \(\chi_{7744}(5191,\cdot)\) \(\chi_{7744}(5367,\cdot)\) \(\chi_{7744}(5543,\cdot)\) \(\chi_{7744}(5719,\cdot)\) ...

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

\((5567,4357,6657)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 7744 }(87, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{45}{88}\right)\)\(e\left(\frac{41}{44}\right)\)\(i\)\(e\left(\frac{47}{88}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{75}{88}\right)\)\(e\left(\frac{5}{88}\right)\)\(e\left(\frac{25}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7744 }(87,a) \;\) at \(\;a = \) e.g. 2