Properties

Label 7056.43
Modulus $7056$
Conductor $7056$
Order $84$
Real no
Primitive yes
Minimal yes
Parity odd

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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([42,21,56,12]))
 
pari: [g,chi] = znchar(Mod(43,7056))
 

Basic properties

Modulus: \(7056\)
Conductor: \(7056\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 7056.ic

\(\chi_{7056}(43,\cdot)\) \(\chi_{7056}(211,\cdot)\) \(\chi_{7056}(547,\cdot)\) \(\chi_{7056}(715,\cdot)\) \(\chi_{7056}(1051,\cdot)\) \(\chi_{7056}(1219,\cdot)\) \(\chi_{7056}(1555,\cdot)\) \(\chi_{7056}(1723,\cdot)\) \(\chi_{7056}(2227,\cdot)\) \(\chi_{7056}(2563,\cdot)\) \(\chi_{7056}(2731,\cdot)\) \(\chi_{7056}(3067,\cdot)\) \(\chi_{7056}(3571,\cdot)\) \(\chi_{7056}(3739,\cdot)\) \(\chi_{7056}(4075,\cdot)\) \(\chi_{7056}(4243,\cdot)\) \(\chi_{7056}(4579,\cdot)\) \(\chi_{7056}(4747,\cdot)\) \(\chi_{7056}(5083,\cdot)\) \(\chi_{7056}(5251,\cdot)\) \(\chi_{7056}(5755,\cdot)\) \(\chi_{7056}(6091,\cdot)\) \(\chi_{7056}(6259,\cdot)\) \(\chi_{7056}(6595,\cdot)\)

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

\((6175,1765,785,4609)\) → \((-1,i,e\left(\frac{2}{3}\right),e\left(\frac{1}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7056 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{67}{84}\right)\)\(e\left(\frac{4}{7}\right)\)\(i\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7056 }(43,a) \;\) at \(\;a = \) e.g. 2