Properties

Label 7056.17
Modulus $7056$
Conductor $147$
Order $42$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,21,25]))
 
pari: [g,chi] = znchar(Mod(17,7056))
 

Basic properties

Modulus: \(7056\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{147}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7056.hi

\(\chi_{7056}(17,\cdot)\) \(\chi_{7056}(593,\cdot)\) \(\chi_{7056}(1025,\cdot)\) \(\chi_{7056}(1601,\cdot)\) \(\chi_{7056}(2033,\cdot)\) \(\chi_{7056}(2609,\cdot)\) \(\chi_{7056}(3041,\cdot)\) \(\chi_{7056}(3617,\cdot)\) \(\chi_{7056}(5057,\cdot)\) \(\chi_{7056}(5633,\cdot)\) \(\chi_{7056}(6065,\cdot)\) \(\chi_{7056}(6641,\cdot)\)

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: \(\Q(\zeta_{147})^+\)

Values on generators

\((6175,1765,785,4609)\) → \((1,1,-1,e\left(\frac{25}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7056 }(17, a) \) \(1\)\(1\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7056 }(17,a) \;\) at \(\;a = \) e.g. 2