Properties

Label 3920.241
Modulus $3920$
Conductor $49$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,31]))
 
pari: [g,chi] = znchar(Mod(241,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.ey

\(\chi_{3920}(241,\cdot)\) \(\chi_{3920}(481,\cdot)\) \(\chi_{3920}(801,\cdot)\) \(\chi_{3920}(1041,\cdot)\) \(\chi_{3920}(1361,\cdot)\) \(\chi_{3920}(1601,\cdot)\) \(\chi_{3920}(1921,\cdot)\) \(\chi_{3920}(2161,\cdot)\) \(\chi_{3920}(2721,\cdot)\) \(\chi_{3920}(3041,\cdot)\) \(\chi_{3920}(3281,\cdot)\) \(\chi_{3920}(3601,\cdot)\)

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

\((1471,981,3137,3041)\) → \((1,1,1,e\left(\frac{31}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(241, a) \) \(-1\)\(1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(241,a) \;\) at \(\;a = \) e.g. 2