Properties

Label 1007.4
Modulus $1007$
Conductor $1007$
Order $234$
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(234))
 
M = H._module
 
chi = DirichletCharacter(H, M([26,9]))
 
pari: [g,chi] = znchar(Mod(4,1007))
 

Basic properties

Modulus: \(1007\)
Conductor: \(1007\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(234\)
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.bg

\(\chi_{1007}(4,\cdot)\) \(\chi_{1007}(6,\cdot)\) \(\chi_{1007}(9,\cdot)\) \(\chi_{1007}(17,\cdot)\) \(\chi_{1007}(25,\cdot)\) \(\chi_{1007}(43,\cdot)\) \(\chi_{1007}(62,\cdot)\) \(\chi_{1007}(82,\cdot)\) \(\chi_{1007}(93,\cdot)\) \(\chi_{1007}(112,\cdot)\) \(\chi_{1007}(123,\cdot)\) \(\chi_{1007}(131,\cdot)\) \(\chi_{1007}(149,\cdot)\) \(\chi_{1007}(168,\cdot)\) \(\chi_{1007}(176,\cdot)\) \(\chi_{1007}(188,\cdot)\) \(\chi_{1007}(196,\cdot)\) \(\chi_{1007}(199,\cdot)\) \(\chi_{1007}(218,\cdot)\) \(\chi_{1007}(237,\cdot)\) \(\chi_{1007}(252,\cdot)\) \(\chi_{1007}(271,\cdot)\) \(\chi_{1007}(272,\cdot)\) \(\chi_{1007}(282,\cdot)\) \(\chi_{1007}(290,\cdot)\) \(\chi_{1007}(294,\cdot)\) \(\chi_{1007}(302,\cdot)\) \(\chi_{1007}(308,\cdot)\) \(\chi_{1007}(327,\cdot)\) \(\chi_{1007}(329,\cdot)\) ...

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: $\Q(\zeta_{117})$
Fixed field: Number field defined by a degree 234 polynomial (not computed)

Values on generators

\((743,267)\) → \((e\left(\frac{1}{9}\right),e\left(\frac{1}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1007 }(4, a) \) \(1\)\(1\)\(e\left(\frac{35}{234}\right)\)\(e\left(\frac{23}{234}\right)\)\(e\left(\frac{35}{117}\right)\)\(e\left(\frac{137}{234}\right)\)\(e\left(\frac{29}{117}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{35}{78}\right)\)\(e\left(\frac{23}{117}\right)\)\(e\left(\frac{86}{117}\right)\)\(e\left(\frac{22}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1007 }(4,a) \;\) at \(\;a = \) e.g. 2