Properties

Label 6762.61
Modulus $6762$
Conductor $1127$
Order $462$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, base_ring=CyclotomicField(462))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,121,357]))
 
pari: [g,chi] = znchar(Mod(61,6762))
 

Basic properties

Modulus: \(6762\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6762.ch

\(\chi_{6762}(61,\cdot)\) \(\chi_{6762}(103,\cdot)\) \(\chi_{6762}(145,\cdot)\) \(\chi_{6762}(157,\cdot)\) \(\chi_{6762}(199,\cdot)\) \(\chi_{6762}(241,\cdot)\) \(\chi_{6762}(283,\cdot)\) \(\chi_{6762}(355,\cdot)\) \(\chi_{6762}(451,\cdot)\) \(\chi_{6762}(481,\cdot)\) \(\chi_{6762}(493,\cdot)\) \(\chi_{6762}(523,\cdot)\) \(\chi_{6762}(649,\cdot)\) \(\chi_{6762}(661,\cdot)\) \(\chi_{6762}(733,\cdot)\) \(\chi_{6762}(787,\cdot)\) \(\chi_{6762}(871,\cdot)\) \(\chi_{6762}(985,\cdot)\) \(\chi_{6762}(1027,\cdot)\) \(\chi_{6762}(1069,\cdot)\) \(\chi_{6762}(1111,\cdot)\) \(\chi_{6762}(1123,\cdot)\) \(\chi_{6762}(1165,\cdot)\) \(\chi_{6762}(1249,\cdot)\) \(\chi_{6762}(1279,\cdot)\) \(\chi_{6762}(1321,\cdot)\) \(\chi_{6762}(1417,\cdot)\) \(\chi_{6762}(1447,\cdot)\) \(\chi_{6762}(1459,\cdot)\)

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((2255,3727,3823)\) → \((1,e\left(\frac{11}{42}\right),e\left(\frac{17}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6762 }(61, a) \) \(1\)\(1\)\(e\left(\frac{85}{231}\right)\)\(e\left(\frac{199}{462}\right)\)\(e\left(\frac{71}{154}\right)\)\(e\left(\frac{221}{231}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{170}{231}\right)\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{281}{462}\right)\)\(e\left(\frac{31}{154}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6762 }(61,a) \;\) at \(\;a = \) e.g. 2