Properties

Label 3528.277
Modulus $3528$
Conductor $3528$
Order $42$
Real no
Primitive yes
Minimal yes
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([0,21,28,4]))
 
pari: [g,chi] = znchar(Mod(277,3528))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3528.eh

\(\chi_{3528}(277,\cdot)\) \(\chi_{3528}(781,\cdot)\) \(\chi_{3528}(877,\cdot)\) \(\chi_{3528}(1285,\cdot)\) \(\chi_{3528}(1381,\cdot)\) \(\chi_{3528}(1789,\cdot)\) \(\chi_{3528}(1885,\cdot)\) \(\chi_{3528}(2293,\cdot)\) \(\chi_{3528}(2389,\cdot)\) \(\chi_{3528}(2797,\cdot)\) \(\chi_{3528}(2893,\cdot)\) \(\chi_{3528}(3397,\cdot)\)

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

\((2647,1765,785,1081)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{2}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3528 }(277, a) \) \(1\)\(1\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(1\)\(e\left(\frac{23}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3528 }(277,a) \;\) at \(\;a = \) e.g. 2