Properties

Label 3381.34
Modulus $3381$
Conductor $1127$
Order $154$
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(154))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,33,63]))
 
pari: [g,chi] = znchar(Mod(34,3381))
 

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 \(\chi_{1127}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cd

\(\chi_{3381}(34,\cdot)\) \(\chi_{3381}(76,\cdot)\) \(\chi_{3381}(181,\cdot)\) \(\chi_{3381}(286,\cdot)\) \(\chi_{3381}(412,\cdot)\) \(\chi_{3381}(433,\cdot)\) \(\chi_{3381}(454,\cdot)\) \(\chi_{3381}(475,\cdot)\) \(\chi_{3381}(517,\cdot)\) \(\chi_{3381}(559,\cdot)\) \(\chi_{3381}(580,\cdot)\) \(\chi_{3381}(664,\cdot)\) \(\chi_{3381}(727,\cdot)\) \(\chi_{3381}(769,\cdot)\) \(\chi_{3381}(895,\cdot)\) \(\chi_{3381}(916,\cdot)\) \(\chi_{3381}(937,\cdot)\) \(\chi_{3381}(958,\cdot)\) \(\chi_{3381}(1000,\cdot)\) \(\chi_{3381}(1042,\cdot)\) \(\chi_{3381}(1063,\cdot)\) \(\chi_{3381}(1147,\cdot)\) \(\chi_{3381}(1210,\cdot)\) \(\chi_{3381}(1252,\cdot)\) \(\chi_{3381}(1378,\cdot)\) \(\chi_{3381}(1399,\cdot)\) \(\chi_{3381}(1441,\cdot)\) \(\chi_{3381}(1483,\cdot)\) \(\chi_{3381}(1525,\cdot)\) \(\chi_{3381}(1546,\cdot)\) ...

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

\((2255,346,442)\) → \((1,e\left(\frac{3}{14}\right),e\left(\frac{9}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(1\)\(1\)\(e\left(\frac{30}{77}\right)\)\(e\left(\frac{60}{77}\right)\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{13}{77}\right)\)\(e\left(\frac{1}{77}\right)\)\(e\left(\frac{39}{154}\right)\)\(e\left(\frac{123}{154}\right)\)\(e\left(\frac{43}{77}\right)\)\(e\left(\frac{17}{77}\right)\)\(e\left(\frac{7}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(34,a) \;\) at \(\;a = \) e.g. 2