Properties

Label 4334.43
Modulus $4334$
Conductor $2167$
Order $98$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, base_ring=CyclotomicField(98))
 
M = H._module
 
chi = DirichletCharacter(H, M([49,39]))
 
pari: [g,chi] = znchar(Mod(43,4334))
 

Basic properties

Modulus: \(4334\)
Conductor: \(2167\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(98\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2167}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4334.z

\(\chi_{4334}(43,\cdot)\) \(\chi_{4334}(65,\cdot)\) \(\chi_{4334}(109,\cdot)\) \(\chi_{4334}(219,\cdot)\) \(\chi_{4334}(549,\cdot)\) \(\chi_{4334}(703,\cdot)\) \(\chi_{4334}(725,\cdot)\) \(\chi_{4334}(813,\cdot)\) \(\chi_{4334}(835,\cdot)\) \(\chi_{4334}(945,\cdot)\) \(\chi_{4334}(989,\cdot)\) \(\chi_{4334}(1011,\cdot)\) \(\chi_{4334}(1077,\cdot)\) \(\chi_{4334}(1121,\cdot)\) \(\chi_{4334}(1319,\cdot)\) \(\chi_{4334}(1363,\cdot)\) \(\chi_{4334}(1495,\cdot)\) \(\chi_{4334}(1517,\cdot)\) \(\chi_{4334}(1539,\cdot)\) \(\chi_{4334}(1583,\cdot)\) \(\chi_{4334}(1869,\cdot)\) \(\chi_{4334}(1979,\cdot)\) \(\chi_{4334}(2067,\cdot)\) \(\chi_{4334}(2133,\cdot)\) \(\chi_{4334}(2177,\cdot)\) \(\chi_{4334}(2419,\cdot)\) \(\chi_{4334}(2485,\cdot)\) \(\chi_{4334}(2507,\cdot)\) \(\chi_{4334}(2705,\cdot)\) \(\chi_{4334}(3101,\cdot)\) ...

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: $\Q(\zeta_{49})$
Fixed field: Number field defined by a degree 98 polynomial

Values on generators

\((1971,199)\) → \((-1,e\left(\frac{39}{98}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 4334 }(43, a) \) \(-1\)\(1\)\(e\left(\frac{3}{98}\right)\)\(e\left(\frac{41}{98}\right)\)\(e\left(\frac{59}{98}\right)\)\(e\left(\frac{3}{49}\right)\)\(e\left(\frac{22}{49}\right)\)\(e\left(\frac{22}{49}\right)\)\(e\left(\frac{38}{49}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{31}{49}\right)\)\(e\left(\frac{37}{49}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4334 }(43,a) \;\) at \(\;a = \) e.g. 2