Properties

Label 7744.19
Modulus $7744$
Conductor $7744$
Order $880$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7744, base_ring=CyclotomicField(880))
 
M = H._module
 
chi = DirichletCharacter(H, M([440,385,664]))
 
pari: [g,chi] = znchar(Mod(19,7744))
 

Basic properties

Modulus: \(7744\)
Conductor: \(7744\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(880\)
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 7744.db

\(\chi_{7744}(19,\cdot)\) \(\chi_{7744}(35,\cdot)\) \(\chi_{7744}(51,\cdot)\) \(\chi_{7744}(83,\cdot)\) \(\chi_{7744}(107,\cdot)\) \(\chi_{7744}(123,\cdot)\) \(\chi_{7744}(139,\cdot)\) \(\chi_{7744}(171,\cdot)\) \(\chi_{7744}(195,\cdot)\) \(\chi_{7744}(211,\cdot)\) \(\chi_{7744}(227,\cdot)\) \(\chi_{7744}(259,\cdot)\) \(\chi_{7744}(283,\cdot)\) \(\chi_{7744}(299,\cdot)\) \(\chi_{7744}(315,\cdot)\) \(\chi_{7744}(347,\cdot)\) \(\chi_{7744}(371,\cdot)\) \(\chi_{7744}(387,\cdot)\) \(\chi_{7744}(435,\cdot)\) \(\chi_{7744}(459,\cdot)\) \(\chi_{7744}(491,\cdot)\) \(\chi_{7744}(523,\cdot)\) \(\chi_{7744}(547,\cdot)\) \(\chi_{7744}(563,\cdot)\) \(\chi_{7744}(579,\cdot)\) \(\chi_{7744}(611,\cdot)\) \(\chi_{7744}(635,\cdot)\) \(\chi_{7744}(651,\cdot)\) \(\chi_{7744}(667,\cdot)\) \(\chi_{7744}(739,\cdot)\) ...

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: $\Q(\zeta_{880})$
Fixed field: Number field defined by a degree 880 polynomial (not computed)

Values on generators

\((5567,4357,6657)\) → \((-1,e\left(\frac{7}{16}\right),e\left(\frac{83}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 7744 }(19, a) \) \(1\)\(1\)\(e\left(\frac{17}{80}\right)\)\(e\left(\frac{241}{880}\right)\)\(e\left(\frac{69}{440}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{679}{880}\right)\)\(e\left(\frac{107}{220}\right)\)\(e\left(\frac{49}{220}\right)\)\(e\left(\frac{167}{880}\right)\)\(e\left(\frac{65}{176}\right)\)\(e\left(\frac{39}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7744 }(19,a) \;\) at \(\;a = \) e.g. 2