Properties

Label 3360.221
Modulus $3360$
Conductor $672$
Order $24$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,12,0,16]))
 
pari: [g,chi] = znchar(Mod(221,3360))
 

Basic properties

Modulus: \(3360\)
Conductor: \(672\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{672}(221,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3360.hl

\(\chi_{3360}(221,\cdot)\) \(\chi_{3360}(821,\cdot)\) \(\chi_{3360}(1061,\cdot)\) \(\chi_{3360}(1661,\cdot)\) \(\chi_{3360}(1901,\cdot)\) \(\chi_{3360}(2501,\cdot)\) \(\chi_{3360}(2741,\cdot)\) \(\chi_{3360}(3341,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((1471,421,1121,2017,1921)\) → \((1,e\left(\frac{3}{8}\right),-1,1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3360 }(221, a) \) \(-1\)\(1\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{17}{24}\right)\)\(-i\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3360 }(221,a) \;\) at \(\;a = \) e.g. 2