Properties

Label 6776.263
Modulus $6776$
Conductor $3388$
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(6776, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,0,44,57]))
 
pari: [g,chi] = znchar(Mod(263,6776))
 

Basic properties

Modulus: \(6776\)
Conductor: \(3388\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3388}(263,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6776.dr

\(\chi_{6776}(263,\cdot)\) \(\chi_{6776}(527,\cdot)\) \(\chi_{6776}(879,\cdot)\) \(\chi_{6776}(1143,\cdot)\) \(\chi_{6776}(1495,\cdot)\) \(\chi_{6776}(1759,\cdot)\) \(\chi_{6776}(2111,\cdot)\) \(\chi_{6776}(2375,\cdot)\) \(\chi_{6776}(2727,\cdot)\) \(\chi_{6776}(2991,\cdot)\) \(\chi_{6776}(3343,\cdot)\) \(\chi_{6776}(3607,\cdot)\) \(\chi_{6776}(3959,\cdot)\) \(\chi_{6776}(4223,\cdot)\) \(\chi_{6776}(4575,\cdot)\) \(\chi_{6776}(5191,\cdot)\) \(\chi_{6776}(5455,\cdot)\) \(\chi_{6776}(6071,\cdot)\) \(\chi_{6776}(6423,\cdot)\) \(\chi_{6776}(6687,\cdot)\)

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

\((1695,3389,969,3753)\) → \((-1,1,e\left(\frac{2}{3}\right),e\left(\frac{19}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 6776 }(263, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{16}{33}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6776 }(263,a) \;\) at \(\;a = \) e.g. 2