Properties

Label 3381.1013
Modulus $3381$
Conductor $147$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,41,0]))
 
pari: [g,chi] = znchar(Mod(1013,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(147\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{147}(131,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bk

\(\chi_{3381}(47,\cdot)\) \(\chi_{3381}(185,\cdot)\) \(\chi_{3381}(530,\cdot)\) \(\chi_{3381}(1013,\cdot)\) \(\chi_{3381}(1151,\cdot)\) \(\chi_{3381}(1496,\cdot)\) \(\chi_{3381}(1634,\cdot)\) \(\chi_{3381}(2117,\cdot)\) \(\chi_{3381}(2462,\cdot)\) \(\chi_{3381}(2600,\cdot)\) \(\chi_{3381}(2945,\cdot)\) \(\chi_{3381}(3083,\cdot)\)

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: \(\Q(\zeta_{147})^+\)

Values on generators

\((2255,346,442)\) → \((-1,e\left(\frac{41}{42}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(1013, a) \) \(1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(1013,a) \;\) at \(\;a = \) e.g. 2