Properties

Label 7056.337
Modulus $7056$
Conductor $441$
Order $21$
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([0,0,14,6]))
 
pari: [g,chi] = znchar(Mod(337,7056))
 

Basic properties

Modulus: \(7056\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(337,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7056.fi

\(\chi_{7056}(337,\cdot)\) \(\chi_{7056}(673,\cdot)\) \(\chi_{7056}(1345,\cdot)\) \(\chi_{7056}(1681,\cdot)\) \(\chi_{7056}(2689,\cdot)\) \(\chi_{7056}(3361,\cdot)\) \(\chi_{7056}(3697,\cdot)\) \(\chi_{7056}(4369,\cdot)\) \(\chi_{7056}(5377,\cdot)\) \(\chi_{7056}(5713,\cdot)\) \(\chi_{7056}(6385,\cdot)\) \(\chi_{7056}(6721,\cdot)\)

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 21 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7056 }(337, a) \) \(1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(1\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7056 }(337,a) \;\) at \(\;a = \) e.g. 2