Properties

Label 3528.125
Modulus $3528$
Conductor $1176$
Order $14$
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(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,7,1]))
 
pari: [g,chi] = znchar(Mod(125,3528))
 

Basic properties

Modulus: \(3528\)
Conductor: \(1176\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1176}(125,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3528.dk

\(\chi_{3528}(125,\cdot)\) \(\chi_{3528}(629,\cdot)\) \(\chi_{3528}(1133,\cdot)\) \(\chi_{3528}(1637,\cdot)\) \(\chi_{3528}(2141,\cdot)\) \(\chi_{3528}(3149,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: 14.14.6150772901711899500090300039168.1

Values on generators

\((2647,1765,785,1081)\) → \((1,-1,-1,e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3528 }(125, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(-1\)\(e\left(\frac{11}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3528 }(125,a) \;\) at \(\;a = \) e.g. 2