Properties

Label 3381.799
Modulus $3381$
Conductor $1127$
Order $154$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,110,49]))
 
pari: [g,chi] = znchar(Mod(799,3381))
 

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 \(\chi_{1127}(799,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3381.cb

\(\chi_{3381}(43,\cdot)\) \(\chi_{3381}(106,\cdot)\) \(\chi_{3381}(274,\cdot)\) \(\chi_{3381}(316,\cdot)\) \(\chi_{3381}(337,\cdot)\) \(\chi_{3381}(379,\cdot)\) \(\chi_{3381}(421,\cdot)\) \(\chi_{3381}(526,\cdot)\) \(\chi_{3381}(631,\cdot)\) \(\chi_{3381}(757,\cdot)\) \(\chi_{3381}(778,\cdot)\) \(\chi_{3381}(799,\cdot)\) \(\chi_{3381}(820,\cdot)\) \(\chi_{3381}(862,\cdot)\) \(\chi_{3381}(904,\cdot)\) \(\chi_{3381}(925,\cdot)\) \(\chi_{3381}(1009,\cdot)\) \(\chi_{3381}(1072,\cdot)\) \(\chi_{3381}(1114,\cdot)\) \(\chi_{3381}(1240,\cdot)\) \(\chi_{3381}(1261,\cdot)\) \(\chi_{3381}(1282,\cdot)\) \(\chi_{3381}(1303,\cdot)\) \(\chi_{3381}(1345,\cdot)\) \(\chi_{3381}(1387,\cdot)\) \(\chi_{3381}(1408,\cdot)\) \(\chi_{3381}(1492,\cdot)\) \(\chi_{3381}(1555,\cdot)\) \(\chi_{3381}(1597,\cdot)\) \(\chi_{3381}(1723,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((2255,346,442)\) → \((1,e\left(\frac{5}{7}\right),e\left(\frac{7}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3381 }(799, a) \) \(-1\)\(1\)\(e\left(\frac{16}{77}\right)\)\(e\left(\frac{32}{77}\right)\)\(e\left(\frac{5}{154}\right)\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{37}{154}\right)\)\(e\left(\frac{67}{154}\right)\)\(e\left(\frac{2}{77}\right)\)\(e\left(\frac{64}{77}\right)\)\(e\left(\frac{13}{154}\right)\)\(e\left(\frac{17}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3381 }(799,a) \;\) at \(\;a = \) e.g. 2