Properties

Label 3528.229
Modulus $3528$
Conductor $3528$
Order $42$
Real no
Primitive yes
Minimal yes
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([0,21,14,41]))
 
pari: [g,chi] = znchar(Mod(229,3528))
 

Basic properties

Modulus: \(3528\)
Conductor: \(3528\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3528.fi

\(\chi_{3528}(229,\cdot)\) \(\chi_{3528}(493,\cdot)\) \(\chi_{3528}(733,\cdot)\) \(\chi_{3528}(997,\cdot)\) \(\chi_{3528}(1237,\cdot)\) \(\chi_{3528}(1741,\cdot)\) \(\chi_{3528}(2005,\cdot)\) \(\chi_{3528}(2245,\cdot)\) \(\chi_{3528}(2509,\cdot)\) \(\chi_{3528}(2749,\cdot)\) \(\chi_{3528}(3013,\cdot)\) \(\chi_{3528}(3517,\cdot)\)

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: 42.0.570465089486557408425029121334452517240510961404409196418927617797130755171930161583171029486327333686268133376.2

Values on generators

\((2647,1765,785,1081)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{41}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3528 }(229, a) \) \(-1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(-1\)\(e\left(\frac{31}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3528 }(229,a) \;\) at \(\;a = \) e.g. 2