Properties

Label 1007.667
Modulus $1007$
Conductor $1007$
Order $468$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1007, base_ring=CyclotomicField(468))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,297]))
 
pari: [g,chi] = znchar(Mod(667,1007))
 

Basic properties

Modulus: \(1007\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(468\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1007.bi

\(\chi_{1007}(2,\cdot)\) \(\chi_{1007}(3,\cdot)\) \(\chi_{1007}(14,\cdot)\) \(\chi_{1007}(21,\cdot)\) \(\chi_{1007}(22,\cdot)\) \(\chi_{1007}(32,\cdot)\) \(\chi_{1007}(33,\cdot)\) \(\chi_{1007}(34,\cdot)\) \(\chi_{1007}(41,\cdot)\) \(\chi_{1007}(48,\cdot)\) \(\chi_{1007}(51,\cdot)\) \(\chi_{1007}(67,\cdot)\) \(\chi_{1007}(71,\cdot)\) \(\chi_{1007}(72,\cdot)\) \(\chi_{1007}(79,\cdot)\) \(\chi_{1007}(86,\cdot)\) \(\chi_{1007}(98,\cdot)\) \(\chi_{1007}(108,\cdot)\) \(\chi_{1007}(109,\cdot)\) \(\chi_{1007}(124,\cdot)\) \(\chi_{1007}(127,\cdot)\) \(\chi_{1007}(128,\cdot)\) \(\chi_{1007}(147,\cdot)\) \(\chi_{1007}(154,\cdot)\) \(\chi_{1007}(162,\cdot)\) \(\chi_{1007}(167,\cdot)\) \(\chi_{1007}(173,\cdot)\) \(\chi_{1007}(181,\cdot)\) \(\chi_{1007}(185,\cdot)\) \(\chi_{1007}(186,\cdot)\) ...

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: $\Q(\zeta_{468})$
Fixed field: Number field defined by a degree 468 polynomial (not computed)

Values on generators

\((743,267)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{33}{52}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1007 }(667, a) \) \(1\)\(1\)\(e\left(\frac{323}{468}\right)\)\(e\left(\frac{239}{468}\right)\)\(e\left(\frac{89}{234}\right)\)\(e\left(\frac{335}{468}\right)\)\(e\left(\frac{47}{234}\right)\)\(e\left(\frac{17}{78}\right)\)\(e\left(\frac{11}{156}\right)\)\(e\left(\frac{5}{234}\right)\)\(e\left(\frac{95}{234}\right)\)\(e\left(\frac{37}{78}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1007 }(667,a) \;\) at \(\;a = \) e.g. 2