Properties

Label 7056.439
Modulus $7056$
Conductor $3528$
Order $42$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,28,5]))
 
pari: [g,chi] = znchar(Mod(439,7056))
 

Basic properties

Modulus: \(7056\)
Conductor: \(3528\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3528}(2203,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7056.fu

\(\chi_{7056}(439,\cdot)\) \(\chi_{7056}(535,\cdot)\) \(\chi_{7056}(1447,\cdot)\) \(\chi_{7056}(1543,\cdot)\) \(\chi_{7056}(2455,\cdot)\) \(\chi_{7056}(2551,\cdot)\) \(\chi_{7056}(3463,\cdot)\) \(\chi_{7056}(4471,\cdot)\) \(\chi_{7056}(4567,\cdot)\) \(\chi_{7056}(5479,\cdot)\) \(\chi_{7056}(5575,\cdot)\) \(\chi_{7056}(6583,\cdot)\)

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

\((6175,1765,785,4609)\) → \((-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7056 }(439, a) \) \(1\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7056 }(439,a) \;\) at \(\;a = \) e.g. 2