Properties

Label 7056.269
Modulus $7056$
Conductor $2352$
Order $84$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,63,42,74]))
 
pari: [g,chi] = znchar(Mod(269,7056))
 

Basic properties

Modulus: \(7056\)
Conductor: \(2352\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2352}(269,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7056.iz

\(\chi_{7056}(269,\cdot)\) \(\chi_{7056}(341,\cdot)\) \(\chi_{7056}(773,\cdot)\) \(\chi_{7056}(845,\cdot)\) \(\chi_{7056}(1277,\cdot)\) \(\chi_{7056}(1349,\cdot)\) \(\chi_{7056}(1781,\cdot)\) \(\chi_{7056}(1853,\cdot)\) \(\chi_{7056}(2357,\cdot)\) \(\chi_{7056}(2789,\cdot)\) \(\chi_{7056}(3293,\cdot)\) \(\chi_{7056}(3365,\cdot)\) \(\chi_{7056}(3797,\cdot)\) \(\chi_{7056}(3869,\cdot)\) \(\chi_{7056}(4301,\cdot)\) \(\chi_{7056}(4373,\cdot)\) \(\chi_{7056}(4805,\cdot)\) \(\chi_{7056}(4877,\cdot)\) \(\chi_{7056}(5309,\cdot)\) \(\chi_{7056}(5381,\cdot)\) \(\chi_{7056}(5885,\cdot)\) \(\chi_{7056}(6317,\cdot)\) \(\chi_{7056}(6821,\cdot)\) \(\chi_{7056}(6893,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((6175,1765,785,4609)\) → \((1,-i,-1,e\left(\frac{37}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7056 }(269, a) \) \(1\)\(1\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{41}{84}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{79}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7056 }(269,a) \;\) at \(\;a = \) e.g. 2