Properties

Label 3381.764
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([77,66,7]))
 
pari: [g,chi] = znchar(Mod(764,3381))
 

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.ca

\(\chi_{3381}(113,\cdot)\) \(\chi_{3381}(134,\cdot)\) \(\chi_{3381}(155,\cdot)\) \(\chi_{3381}(176,\cdot)\) \(\chi_{3381}(218,\cdot)\) \(\chi_{3381}(260,\cdot)\) \(\chi_{3381}(281,\cdot)\) \(\chi_{3381}(365,\cdot)\) \(\chi_{3381}(428,\cdot)\) \(\chi_{3381}(470,\cdot)\) \(\chi_{3381}(596,\cdot)\) \(\chi_{3381}(617,\cdot)\) \(\chi_{3381}(659,\cdot)\) \(\chi_{3381}(701,\cdot)\) \(\chi_{3381}(743,\cdot)\) \(\chi_{3381}(764,\cdot)\) \(\chi_{3381}(848,\cdot)\) \(\chi_{3381}(911,\cdot)\) \(\chi_{3381}(953,\cdot)\) \(\chi_{3381}(1100,\cdot)\) \(\chi_{3381}(1121,\cdot)\) \(\chi_{3381}(1142,\cdot)\) \(\chi_{3381}(1184,\cdot)\) \(\chi_{3381}(1247,\cdot)\) \(\chi_{3381}(1331,\cdot)\) \(\chi_{3381}(1394,\cdot)\) \(\chi_{3381}(1436,\cdot)\) \(\chi_{3381}(1562,\cdot)\) \(\chi_{3381}(1583,\cdot)\) \(\chi_{3381}(1604,\cdot)\) ...

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

\((2255,346,442)\) → \((-1,e\left(\frac{3}{7}\right),e\left(\frac{1}{22}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(764, a) \) \(1\)\(1\)\(e\left(\frac{113}{154}\right)\)\(e\left(\frac{36}{77}\right)\)\(e\left(\frac{75}{77}\right)\)\(e\left(\frac{31}{154}\right)\)\(e\left(\frac{109}{154}\right)\)\(e\left(\frac{4}{77}\right)\)\(e\left(\frac{60}{77}\right)\)\(e\left(\frac{72}{77}\right)\)\(e\left(\frac{41}{77}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(764,a) \;\) at \(\;a = \) e.g. 2