Properties

Label 3381.55
Modulus $3381$
Conductor $1127$
Order $154$
Real no
Primitive no
Minimal yes
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(154))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,99,70]))
 
pari: [g,chi] = znchar(Mod(55,3381))
 

Basic properties

Modulus: \(3381\)
Conductor: \(1127\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1127}(55,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cc

\(\chi_{3381}(13,\cdot)\) \(\chi_{3381}(55,\cdot)\) \(\chi_{3381}(118,\cdot)\) \(\chi_{3381}(202,\cdot)\) \(\chi_{3381}(223,\cdot)\) \(\chi_{3381}(265,\cdot)\) \(\chi_{3381}(307,\cdot)\) \(\chi_{3381}(328,\cdot)\) \(\chi_{3381}(349,\cdot)\) \(\chi_{3381}(370,\cdot)\) \(\chi_{3381}(496,\cdot)\) \(\chi_{3381}(601,\cdot)\) \(\chi_{3381}(706,\cdot)\) \(\chi_{3381}(748,\cdot)\) \(\chi_{3381}(790,\cdot)\) \(\chi_{3381}(811,\cdot)\) \(\chi_{3381}(853,\cdot)\) \(\chi_{3381}(1021,\cdot)\) \(\chi_{3381}(1084,\cdot)\) \(\chi_{3381}(1168,\cdot)\) \(\chi_{3381}(1189,\cdot)\) \(\chi_{3381}(1231,\cdot)\) \(\chi_{3381}(1294,\cdot)\) \(\chi_{3381}(1315,\cdot)\) \(\chi_{3381}(1336,\cdot)\) \(\chi_{3381}(1462,\cdot)\) \(\chi_{3381}(1504,\cdot)\) \(\chi_{3381}(1651,\cdot)\) \(\chi_{3381}(1672,\cdot)\) \(\chi_{3381}(1756,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((2255,346,442)\) → \((1,e\left(\frac{9}{14}\right),e\left(\frac{5}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{19}{77}\right)\)\(e\left(\frac{15}{154}\right)\)\(e\left(\frac{67}{77}\right)\)\(e\left(\frac{111}{154}\right)\)\(e\left(\frac{62}{77}\right)\)\(e\left(\frac{89}{154}\right)\)\(e\left(\frac{38}{77}\right)\)\(e\left(\frac{39}{154}\right)\)\(e\left(\frac{7}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(55,a) \;\) at \(\;a = \) e.g. 2