Properties

Label 3328.51
Modulus $3328$
Conductor $3328$
Order $64$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3328, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,63,32]))
 
pari: [g,chi] = znchar(Mod(51,3328))
 

Basic properties

Modulus: \(3328\)
Conductor: \(3328\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3328.dh

\(\chi_{3328}(51,\cdot)\) \(\chi_{3328}(155,\cdot)\) \(\chi_{3328}(259,\cdot)\) \(\chi_{3328}(363,\cdot)\) \(\chi_{3328}(467,\cdot)\) \(\chi_{3328}(571,\cdot)\) \(\chi_{3328}(675,\cdot)\) \(\chi_{3328}(779,\cdot)\) \(\chi_{3328}(883,\cdot)\) \(\chi_{3328}(987,\cdot)\) \(\chi_{3328}(1091,\cdot)\) \(\chi_{3328}(1195,\cdot)\) \(\chi_{3328}(1299,\cdot)\) \(\chi_{3328}(1403,\cdot)\) \(\chi_{3328}(1507,\cdot)\) \(\chi_{3328}(1611,\cdot)\) \(\chi_{3328}(1715,\cdot)\) \(\chi_{3328}(1819,\cdot)\) \(\chi_{3328}(1923,\cdot)\) \(\chi_{3328}(2027,\cdot)\) \(\chi_{3328}(2131,\cdot)\) \(\chi_{3328}(2235,\cdot)\) \(\chi_{3328}(2339,\cdot)\) \(\chi_{3328}(2443,\cdot)\) \(\chi_{3328}(2547,\cdot)\) \(\chi_{3328}(2651,\cdot)\) \(\chi_{3328}(2755,\cdot)\) \(\chi_{3328}(2859,\cdot)\) \(\chi_{3328}(2963,\cdot)\) \(\chi_{3328}(3067,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((1535,261,769)\) → \((-1,e\left(\frac{63}{64}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 3328 }(51, a) \) \(-1\)\(1\)\(e\left(\frac{61}{64}\right)\)\(e\left(\frac{31}{64}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{43}{64}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{41}{64}\right)\)\(e\left(\frac{51}{64}\right)\)\(e\left(\frac{9}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3328 }(51,a) \;\) at \(\;a = \) e.g. 2