Properties

Label 7728.25
Modulus $7728$
Conductor $1288$
Order $66$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7728, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,33,0,44,6]))
 
pari: [g,chi] = znchar(Mod(25,7728))
 

Basic properties

Modulus: \(7728\)
Conductor: \(1288\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1288}(669,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7728.fv

\(\chi_{7728}(25,\cdot)\) \(\chi_{7728}(121,\cdot)\) \(\chi_{7728}(361,\cdot)\) \(\chi_{7728}(1129,\cdot)\) \(\chi_{7728}(1369,\cdot)\) \(\chi_{7728}(1465,\cdot)\) \(\chi_{7728}(1705,\cdot)\) \(\chi_{7728}(2377,\cdot)\) \(\chi_{7728}(2473,\cdot)\) \(\chi_{7728}(2809,\cdot)\) \(\chi_{7728}(3049,\cdot)\) \(\chi_{7728}(3385,\cdot)\) \(\chi_{7728}(3481,\cdot)\) \(\chi_{7728}(3721,\cdot)\) \(\chi_{7728}(4057,\cdot)\) \(\chi_{7728}(4153,\cdot)\) \(\chi_{7728}(4489,\cdot)\) \(\chi_{7728}(4825,\cdot)\) \(\chi_{7728}(5161,\cdot)\) \(\chi_{7728}(6745,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((4831,5797,5153,6625,6721)\) → \((1,-1,1,e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 7728 }(25, a) \) \(1\)\(1\)\(e\left(\frac{61}{66}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7728 }(25,a) \;\) at \(\;a = \) e.g. 2