Properties

Label 3528.395
Modulus $3528$
Conductor $1176$
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(3528, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,21,1]))
 
pari: [g,chi] = znchar(Mod(395,3528))
 

Basic properties

Modulus: \(3528\)
Conductor: \(1176\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1176}(395,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3528.ee

\(\chi_{3528}(395,\cdot)\) \(\chi_{3528}(467,\cdot)\) \(\chi_{3528}(899,\cdot)\) \(\chi_{3528}(971,\cdot)\) \(\chi_{3528}(1475,\cdot)\) \(\chi_{3528}(1907,\cdot)\) \(\chi_{3528}(2411,\cdot)\) \(\chi_{3528}(2483,\cdot)\) \(\chi_{3528}(2915,\cdot)\) \(\chi_{3528}(2987,\cdot)\) \(\chi_{3528}(3419,\cdot)\) \(\chi_{3528}(3491,\cdot)\)

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: 42.0.11402108177106104552822037830207017370882719938852769609842060849495301710065603498954056531968.1

Values on generators

\((2647,1765,785,1081)\) → \((-1,-1,-1,e\left(\frac{1}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3528 }(395, a) \) \(-1\)\(1\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3528 }(395,a) \;\) at \(\;a = \) e.g. 2