Properties

Label 3312.281
Modulus $3312$
Conductor $1656$
Order $66$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3312, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,33,11,3]))
 
pari: [g,chi] = znchar(Mod(281,3312))
 

Basic properties

Modulus: \(3312\)
Conductor: \(1656\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1656}(1109,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3312.cv

\(\chi_{3312}(281,\cdot)\) \(\chi_{3312}(329,\cdot)\) \(\chi_{3312}(425,\cdot)\) \(\chi_{3312}(569,\cdot)\) \(\chi_{3312}(617,\cdot)\) \(\chi_{3312}(1049,\cdot)\) \(\chi_{3312}(1193,\cdot)\) \(\chi_{3312}(1433,\cdot)\) \(\chi_{3312}(1625,\cdot)\) \(\chi_{3312}(1721,\cdot)\) \(\chi_{3312}(1769,\cdot)\) \(\chi_{3312}(2057,\cdot)\) \(\chi_{3312}(2153,\cdot)\) \(\chi_{3312}(2297,\cdot)\) \(\chi_{3312}(2489,\cdot)\) \(\chi_{3312}(2633,\cdot)\) \(\chi_{3312}(2729,\cdot)\) \(\chi_{3312}(2777,\cdot)\) \(\chi_{3312}(2873,\cdot)\) \(\chi_{3312}(3161,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((415,2485,2945,2305)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3312 }(281, a) \) \(1\)\(1\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{10}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3312 }(281,a) \;\) at \(\;a = \) e.g. 2