Properties

Label 3381.818
Modulus $3381$
Conductor $3381$
Order $154$
Real no
Primitive yes
Minimal yes
Parity even

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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([77,33,98]))
 
pari: [g,chi] = znchar(Mod(818,3381))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.bz

\(\chi_{3381}(41,\cdot)\) \(\chi_{3381}(62,\cdot)\) \(\chi_{3381}(104,\cdot)\) \(\chi_{3381}(167,\cdot)\) \(\chi_{3381}(188,\cdot)\) \(\chi_{3381}(209,\cdot)\) \(\chi_{3381}(335,\cdot)\) \(\chi_{3381}(377,\cdot)\) \(\chi_{3381}(524,\cdot)\) \(\chi_{3381}(545,\cdot)\) \(\chi_{3381}(629,\cdot)\) \(\chi_{3381}(650,\cdot)\) \(\chi_{3381}(671,\cdot)\) \(\chi_{3381}(692,\cdot)\) \(\chi_{3381}(818,\cdot)\) \(\chi_{3381}(860,\cdot)\) \(\chi_{3381}(923,\cdot)\) \(\chi_{3381}(1007,\cdot)\) \(\chi_{3381}(1070,\cdot)\) \(\chi_{3381}(1112,\cdot)\) \(\chi_{3381}(1133,\cdot)\) \(\chi_{3381}(1154,\cdot)\) \(\chi_{3381}(1301,\cdot)\) \(\chi_{3381}(1343,\cdot)\) \(\chi_{3381}(1406,\cdot)\) \(\chi_{3381}(1490,\cdot)\) \(\chi_{3381}(1511,\cdot)\) \(\chi_{3381}(1553,\cdot)\) \(\chi_{3381}(1595,\cdot)\) \(\chi_{3381}(1637,\cdot)\) ...

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

\((2255,346,442)\) → \((-1,e\left(\frac{3}{14}\right),e\left(\frac{7}{11}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(818, a) \) \(1\)\(1\)\(e\left(\frac{53}{154}\right)\)\(e\left(\frac{53}{77}\right)\)\(e\left(\frac{27}{77}\right)\)\(e\left(\frac{5}{154}\right)\)\(e\left(\frac{107}{154}\right)\)\(e\left(\frac{123}{154}\right)\)\(e\left(\frac{151}{154}\right)\)\(e\left(\frac{29}{77}\right)\)\(e\left(\frac{24}{77}\right)\)\(e\left(\frac{1}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(818,a) \;\) at \(\;a = \) e.g. 2