Properties

Label 3528.359
Modulus $3528$
Conductor $588$
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(3528, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,0,21,20]))
 
pari: [g,chi] = znchar(Mod(359,3528))
 

Basic properties

Modulus: \(3528\)
Conductor: \(588\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{588}(359,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3528.ep

\(\chi_{3528}(359,\cdot)\) \(\chi_{3528}(431,\cdot)\) \(\chi_{3528}(935,\cdot)\) \(\chi_{3528}(1367,\cdot)\) \(\chi_{3528}(1871,\cdot)\) \(\chi_{3528}(1943,\cdot)\) \(\chi_{3528}(2375,\cdot)\) \(\chi_{3528}(2447,\cdot)\) \(\chi_{3528}(2879,\cdot)\) \(\chi_{3528}(2951,\cdot)\) \(\chi_{3528}(3383,\cdot)\) \(\chi_{3528}(3455,\cdot)\)

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

\((2647,1765,785,1081)\) → \((-1,1,-1,e\left(\frac{10}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3528 }(359, a) \) \(1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3528 }(359,a) \;\) at \(\;a = \) e.g. 2