Properties

Label 3381.361
Modulus $3381$
Conductor $161$
Order $33$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,44,24]))
 
pari: [g,chi] = znchar(Mod(361,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(39,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bh

\(\chi_{3381}(361,\cdot)\) \(\chi_{3381}(508,\cdot)\) \(\chi_{3381}(814,\cdot)\) \(\chi_{3381}(949,\cdot)\) \(\chi_{3381}(961,\cdot)\) \(\chi_{3381}(1108,\cdot)\) \(\chi_{3381}(1255,\cdot)\) \(\chi_{3381}(1549,\cdot)\) \(\chi_{3381}(1843,\cdot)\) \(\chi_{3381}(1990,\cdot)\) \(\chi_{3381}(2125,\cdot)\) \(\chi_{3381}(2272,\cdot)\) \(\chi_{3381}(2419,\cdot)\) \(\chi_{3381}(2431,\cdot)\) \(\chi_{3381}(2566,\cdot)\) \(\chi_{3381}(2578,\cdot)\) \(\chi_{3381}(2860,\cdot)\) \(\chi_{3381}(3019,\cdot)\) \(\chi_{3381}(3154,\cdot)\) \(\chi_{3381}(3301,\cdot)\)

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: 33.33.277966181338944111003326058293667039541136678070715028736001.1

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{4}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(361, a) \) \(1\)\(1\)\(e\left(\frac{2}{33}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{26}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(361,a) \;\) at \(\;a = \) e.g. 2