Properties

Label 3381.40
Modulus $3381$
Conductor $1127$
Order $462$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3381, base_ring=CyclotomicField(462))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,253,147]))
 
pari: [g,chi] = znchar(Mod(40,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(40,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cf

\(\chi_{3381}(10,\cdot)\) \(\chi_{3381}(40,\cdot)\) \(\chi_{3381}(61,\cdot)\) \(\chi_{3381}(103,\cdot)\) \(\chi_{3381}(136,\cdot)\) \(\chi_{3381}(145,\cdot)\) \(\chi_{3381}(157,\cdot)\) \(\chi_{3381}(199,\cdot)\) \(\chi_{3381}(241,\cdot)\) \(\chi_{3381}(250,\cdot)\) \(\chi_{3381}(283,\cdot)\) \(\chi_{3381}(304,\cdot)\) \(\chi_{3381}(355,\cdot)\) \(\chi_{3381}(388,\cdot)\) \(\chi_{3381}(451,\cdot)\) \(\chi_{3381}(481,\cdot)\) \(\chi_{3381}(493,\cdot)\) \(\chi_{3381}(502,\cdot)\) \(\chi_{3381}(523,\cdot)\) \(\chi_{3381}(544,\cdot)\) \(\chi_{3381}(586,\cdot)\) \(\chi_{3381}(628,\cdot)\) \(\chi_{3381}(640,\cdot)\) \(\chi_{3381}(649,\cdot)\) \(\chi_{3381}(661,\cdot)\) \(\chi_{3381}(682,\cdot)\) \(\chi_{3381}(724,\cdot)\) \(\chi_{3381}(733,\cdot)\) \(\chi_{3381}(787,\cdot)\) \(\chi_{3381}(796,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{23}{42}\right),e\left(\frac{7}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{202}{231}\right)\)\(e\left(\frac{173}{231}\right)\)\(e\left(\frac{46}{231}\right)\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{17}{231}\right)\)\(e\left(\frac{355}{462}\right)\)\(e\left(\frac{81}{154}\right)\)\(e\left(\frac{115}{231}\right)\)\(e\left(\frac{212}{231}\right)\)\(e\left(\frac{31}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(40,a) \;\) at \(\;a = \) e.g. 2