Properties

Label 6008.265
Modulus $6008$
Conductor $751$
Order $375$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6008, base_ring=CyclotomicField(750))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,46]))
 
pari: [g,chi] = znchar(Mod(265,6008))
 

Basic properties

Modulus: \(6008\)
Conductor: \(751\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(375\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{751}(265,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6008.ce

\(\chi_{6008}(9,\cdot)\) \(\chi_{6008}(25,\cdot)\) \(\chi_{6008}(33,\cdot)\) \(\chi_{6008}(65,\cdot)\) \(\chi_{6008}(81,\cdot)\) \(\chi_{6008}(89,\cdot)\) \(\chi_{6008}(97,\cdot)\) \(\chi_{6008}(105,\cdot)\) \(\chi_{6008}(169,\cdot)\) \(\chi_{6008}(201,\cdot)\) \(\chi_{6008}(217,\cdot)\) \(\chi_{6008}(225,\cdot)\) \(\chi_{6008}(233,\cdot)\) \(\chi_{6008}(265,\cdot)\) \(\chi_{6008}(289,\cdot)\) \(\chi_{6008}(297,\cdot)\) \(\chi_{6008}(361,\cdot)\) \(\chi_{6008}(385,\cdot)\) \(\chi_{6008}(401,\cdot)\) \(\chi_{6008}(441,\cdot)\) \(\chi_{6008}(465,\cdot)\) \(\chi_{6008}(529,\cdot)\) \(\chi_{6008}(553,\cdot)\) \(\chi_{6008}(585,\cdot)\) \(\chi_{6008}(593,\cdot)\) \(\chi_{6008}(601,\cdot)\) \(\chi_{6008}(625,\cdot)\) \(\chi_{6008}(641,\cdot)\) \(\chi_{6008}(689,\cdot)\) \(\chi_{6008}(697,\cdot)\) ...

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: $\Q(\zeta_{375})$
Fixed field: Number field defined by a degree 375 polynomial (not computed)

Values on generators

\((1503,3005,1505)\) → \((1,1,e\left(\frac{23}{375}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 6008 }(265, a) \) \(1\)\(1\)\(e\left(\frac{23}{375}\right)\)\(e\left(\frac{53}{375}\right)\)\(e\left(\frac{116}{125}\right)\)\(e\left(\frac{46}{375}\right)\)\(e\left(\frac{64}{75}\right)\)\(e\left(\frac{59}{375}\right)\)\(e\left(\frac{76}{375}\right)\)\(e\left(\frac{67}{375}\right)\)\(e\left(\frac{164}{375}\right)\)\(e\left(\frac{371}{375}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6008 }(265,a) \;\) at \(\;a = \) e.g. 2