Properties

Label 3381.20
Modulus $3381$
Conductor $3381$
Order $154$
Real no
Primitive yes
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([77,143,35]))
 
pari: [g,chi] = znchar(Mod(20,3381))
 

Basic properties

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

Galois orbit 3381.by

\(\chi_{3381}(20,\cdot)\) \(\chi_{3381}(83,\cdot)\) \(\chi_{3381}(125,\cdot)\) \(\chi_{3381}(251,\cdot)\) \(\chi_{3381}(272,\cdot)\) \(\chi_{3381}(314,\cdot)\) \(\chi_{3381}(356,\cdot)\) \(\chi_{3381}(398,\cdot)\) \(\chi_{3381}(419,\cdot)\) \(\chi_{3381}(503,\cdot)\) \(\chi_{3381}(566,\cdot)\) \(\chi_{3381}(608,\cdot)\) \(\chi_{3381}(755,\cdot)\) \(\chi_{3381}(776,\cdot)\) \(\chi_{3381}(797,\cdot)\) \(\chi_{3381}(839,\cdot)\) \(\chi_{3381}(902,\cdot)\) \(\chi_{3381}(986,\cdot)\) \(\chi_{3381}(1049,\cdot)\) \(\chi_{3381}(1091,\cdot)\) \(\chi_{3381}(1217,\cdot)\) \(\chi_{3381}(1238,\cdot)\) \(\chi_{3381}(1259,\cdot)\) \(\chi_{3381}(1280,\cdot)\) \(\chi_{3381}(1364,\cdot)\) \(\chi_{3381}(1385,\cdot)\) \(\chi_{3381}(1532,\cdot)\) \(\chi_{3381}(1574,\cdot)\) \(\chi_{3381}(1700,\cdot)\) \(\chi_{3381}(1721,\cdot)\) ...

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

\((2255,346,442)\) → \((-1,e\left(\frac{13}{14}\right),e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{15}{154}\right)\)\(e\left(\frac{15}{77}\right)\)\(e\left(\frac{101}{154}\right)\)\(e\left(\frac{45}{154}\right)\)\(e\left(\frac{58}{77}\right)\)\(e\left(\frac{53}{77}\right)\)\(e\left(\frac{127}{154}\right)\)\(e\left(\frac{30}{77}\right)\)\(e\left(\frac{47}{154}\right)\)\(e\left(\frac{10}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(20,a) \;\) at \(\;a = \) e.g. 2