Properties

Label 3311.362
Modulus $3311$
Conductor $3311$
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(3311, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,21,29]))
 
pari: [g,chi] = znchar(Mod(362,3311))
 

Basic properties

Modulus: \(3311\)
Conductor: \(3311\)
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 3311.ej

\(\chi_{3311}(362,\cdot)\) \(\chi_{3311}(549,\cdot)\) \(\chi_{3311}(593,\cdot)\) \(\chi_{3311}(1363,\cdot)\) \(\chi_{3311}(1396,\cdot)\) \(\chi_{3311}(1748,\cdot)\) \(\chi_{3311}(2133,\cdot)\) \(\chi_{3311}(2628,\cdot)\) \(\chi_{3311}(2936,\cdot)\) \(\chi_{3311}(3013,\cdot)\) \(\chi_{3311}(3211,\cdot)\) \(\chi_{3311}(3244,\cdot)\)

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

\((1893,904,2927)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 3311 }(362, a) \) \(-1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3311 }(362,a) \;\) at \(\;a = \) e.g. 2