Properties

Label 7056.4609
Modulus $7056$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,1]))
 
pari: [g,chi] = znchar(Mod(4609,7056))
 

Basic properties

Modulus: \(7056\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7056.gv

\(\chi_{7056}(145,\cdot)\) \(\chi_{7056}(577,\cdot)\) \(\chi_{7056}(1153,\cdot)\) \(\chi_{7056}(1585,\cdot)\) \(\chi_{7056}(2161,\cdot)\) \(\chi_{7056}(2593,\cdot)\) \(\chi_{7056}(3169,\cdot)\) \(\chi_{7056}(3601,\cdot)\) \(\chi_{7056}(4177,\cdot)\) \(\chi_{7056}(4609,\cdot)\) \(\chi_{7056}(5185,\cdot)\) \(\chi_{7056}(6625,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7056 }(4609, a) \) \(-1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{16}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7056 }(4609,a) \;\) at \(\;a = \) e.g. 2