Properties

Label 3381.31
Modulus $3381$
Conductor $161$
Order $66$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3381, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,18]))
 
pari: [g,chi] = znchar(Mod(31,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bu

\(\chi_{3381}(31,\cdot)\) \(\chi_{3381}(325,\cdot)\) \(\chi_{3381}(472,\cdot)\) \(\chi_{3381}(607,\cdot)\) \(\chi_{3381}(754,\cdot)\) \(\chi_{3381}(901,\cdot)\) \(\chi_{3381}(913,\cdot)\) \(\chi_{3381}(1048,\cdot)\) \(\chi_{3381}(1060,\cdot)\) \(\chi_{3381}(1342,\cdot)\) \(\chi_{3381}(1501,\cdot)\) \(\chi_{3381}(1636,\cdot)\) \(\chi_{3381}(1783,\cdot)\) \(\chi_{3381}(2224,\cdot)\) \(\chi_{3381}(2371,\cdot)\) \(\chi_{3381}(2677,\cdot)\) \(\chi_{3381}(2812,\cdot)\) \(\chi_{3381}(2824,\cdot)\) \(\chi_{3381}(2971,\cdot)\) \(\chi_{3381}(3118,\cdot)\)

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

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{3}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{5}{66}\right)\)\(e\left(\frac{61}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(31,a) \;\) at \(\;a = \) e.g. 2