Properties

Label 5054.145
Modulus $5054$
Conductor $2527$
Order $114$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5054, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([95,83]))
 
pari: [g,chi] = znchar(Mod(145,5054))
 

Basic properties

Modulus: \(5054\)
Conductor: \(2527\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2527}(145,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5054.bx

\(\chi_{5054}(145,\cdot)\) \(\chi_{5054}(255,\cdot)\) \(\chi_{5054}(411,\cdot)\) \(\chi_{5054}(521,\cdot)\) \(\chi_{5054}(677,\cdot)\) \(\chi_{5054}(787,\cdot)\) \(\chi_{5054}(943,\cdot)\) \(\chi_{5054}(1053,\cdot)\) \(\chi_{5054}(1209,\cdot)\) \(\chi_{5054}(1319,\cdot)\) \(\chi_{5054}(1475,\cdot)\) \(\chi_{5054}(1585,\cdot)\) \(\chi_{5054}(1741,\cdot)\) \(\chi_{5054}(1851,\cdot)\) \(\chi_{5054}(2007,\cdot)\) \(\chi_{5054}(2117,\cdot)\) \(\chi_{5054}(2273,\cdot)\) \(\chi_{5054}(2383,\cdot)\) \(\chi_{5054}(2539,\cdot)\) \(\chi_{5054}(2649,\cdot)\) \(\chi_{5054}(2805,\cdot)\) \(\chi_{5054}(2915,\cdot)\) \(\chi_{5054}(3071,\cdot)\) \(\chi_{5054}(3337,\cdot)\) \(\chi_{5054}(3447,\cdot)\) \(\chi_{5054}(3603,\cdot)\) \(\chi_{5054}(3713,\cdot)\) \(\chi_{5054}(3869,\cdot)\) \(\chi_{5054}(3979,\cdot)\) \(\chi_{5054}(4135,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((1445,1807)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{83}{114}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 5054 }(145, a) \) \(1\)\(1\)\(e\left(\frac{2}{57}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{4}{57}\right)\)\(e\left(\frac{34}{57}\right)\)\(e\left(\frac{2}{57}\right)\)\(e\left(\frac{13}{114}\right)\)\(e\left(\frac{43}{114}\right)\)\(e\left(\frac{55}{57}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{2}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5054 }(145,a) \;\) at \(\;a = \) e.g. 2