Properties

Label 7938.647
Modulus $7938$
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(7938, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,13]))
 
pari: [g,chi] = znchar(Mod(647,7938))
 

Basic properties

Modulus: \(7938\)
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 \(\chi_{147}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7938.ca

\(\chi_{7938}(647,\cdot)\) \(\chi_{7938}(971,\cdot)\) \(\chi_{7938}(1781,\cdot)\) \(\chi_{7938}(2105,\cdot)\) \(\chi_{7938}(2915,\cdot)\) \(\chi_{7938}(3239,\cdot)\) \(\chi_{7938}(4373,\cdot)\) \(\chi_{7938}(5183,\cdot)\) \(\chi_{7938}(6317,\cdot)\) \(\chi_{7938}(6641,\cdot)\) \(\chi_{7938}(7451,\cdot)\) \(\chi_{7938}(7775,\cdot)\)

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

\((6077,3727)\) → \((-1,e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 7938 }(647, a) \) \(1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{19}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7938 }(647,a) \;\) at \(\;a = \) e.g. 2