Properties

Label 3724.1699
Modulus $3724$
Conductor $3724$
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(3724, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,41,7]))
 
pari: [g,chi] = znchar(Mod(1699,3724))
 

Basic properties

Modulus: \(3724\)
Conductor: \(3724\)
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 3724.dw

\(\chi_{3724}(103,\cdot)\) \(\chi_{3724}(563,\cdot)\) \(\chi_{3724}(635,\cdot)\) \(\chi_{3724}(1095,\cdot)\) \(\chi_{3724}(1167,\cdot)\) \(\chi_{3724}(1627,\cdot)\) \(\chi_{3724}(1699,\cdot)\) \(\chi_{3724}(2159,\cdot)\) \(\chi_{3724}(2231,\cdot)\) \(\chi_{3724}(2691,\cdot)\) \(\chi_{3724}(3223,\cdot)\) \(\chi_{3724}(3295,\cdot)\)

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

\((1863,3041,3137)\) → \((-1,e\left(\frac{41}{42}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 3724 }(1699, a) \) \(-1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{13}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3724 }(1699,a) \;\) at \(\;a = \) e.g. 2