Properties

Label 6012.481
Modulus $6012$
Conductor $1503$
Order $249$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6012, base_ring=CyclotomicField(498))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,166,492]))
 
pari: [g,chi] = znchar(Mod(481,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(1503\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(249\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1503}(481,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6012.y

\(\chi_{6012}(25,\cdot)\) \(\chi_{6012}(49,\cdot)\) \(\chi_{6012}(61,\cdot)\) \(\chi_{6012}(85,\cdot)\) \(\chi_{6012}(97,\cdot)\) \(\chi_{6012}(121,\cdot)\) \(\chi_{6012}(133,\cdot)\) \(\chi_{6012}(157,\cdot)\) \(\chi_{6012}(169,\cdot)\) \(\chi_{6012}(205,\cdot)\) \(\chi_{6012}(229,\cdot)\) \(\chi_{6012}(265,\cdot)\) \(\chi_{6012}(337,\cdot)\) \(\chi_{6012}(409,\cdot)\) \(\chi_{6012}(421,\cdot)\) \(\chi_{6012}(481,\cdot)\) \(\chi_{6012}(517,\cdot)\) \(\chi_{6012}(529,\cdot)\) \(\chi_{6012}(565,\cdot)\) \(\chi_{6012}(589,\cdot)\) \(\chi_{6012}(601,\cdot)\) \(\chi_{6012}(625,\cdot)\) \(\chi_{6012}(697,\cdot)\) \(\chi_{6012}(733,\cdot)\) \(\chi_{6012}(745,\cdot)\) \(\chi_{6012}(805,\cdot)\) \(\chi_{6012}(841,\cdot)\) \(\chi_{6012}(853,\cdot)\) \(\chi_{6012}(877,\cdot)\) \(\chi_{6012}(889,\cdot)\) ...

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: $\Q(\zeta_{249})$
Fixed field: Number field defined by a degree 249 polynomial (not computed)

Values on generators

\((3007,3341,4681)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{82}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 6012 }(481, a) \) \(1\)\(1\)\(e\left(\frac{163}{249}\right)\)\(e\left(\frac{227}{249}\right)\)\(e\left(\frac{248}{249}\right)\)\(e\left(\frac{106}{249}\right)\)\(e\left(\frac{30}{83}\right)\)\(e\left(\frac{25}{83}\right)\)\(e\left(\frac{118}{249}\right)\)\(e\left(\frac{77}{249}\right)\)\(e\left(\frac{131}{249}\right)\)\(e\left(\frac{145}{249}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6012 }(481,a) \;\) at \(\;a = \) e.g. 2