Properties

Label 6776.285
Modulus $6776$
Conductor $6776$
Order $66$
Real no
Primitive yes
Minimal yes
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([0,33,55,15]))
 
pari: [g,chi] = znchar(Mod(285,6776))
 

Basic properties

Modulus: \(6776\)
Conductor: \(6776\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 6776.dk

\(\chi_{6776}(285,\cdot)\) \(\chi_{6776}(549,\cdot)\) \(\chi_{6776}(901,\cdot)\) \(\chi_{6776}(1165,\cdot)\) \(\chi_{6776}(1517,\cdot)\) \(\chi_{6776}(1781,\cdot)\) \(\chi_{6776}(2133,\cdot)\) \(\chi_{6776}(2397,\cdot)\) \(\chi_{6776}(2749,\cdot)\) \(\chi_{6776}(3013,\cdot)\) \(\chi_{6776}(3365,\cdot)\) \(\chi_{6776}(3981,\cdot)\) \(\chi_{6776}(4245,\cdot)\) \(\chi_{6776}(4861,\cdot)\) \(\chi_{6776}(5213,\cdot)\) \(\chi_{6776}(5477,\cdot)\) \(\chi_{6776}(5829,\cdot)\) \(\chi_{6776}(6093,\cdot)\) \(\chi_{6776}(6445,\cdot)\) \(\chi_{6776}(6709,\cdot)\)

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

\((1695,3389,969,3753)\) → \((1,-1,e\left(\frac{5}{6}\right),e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 6776 }(285, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6776 }(285,a) \;\) at \(\;a = \) e.g. 2