Properties

Label 8007.73
Modulus $8007$
Conductor $2669$
Order $624$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, base_ring=CyclotomicField(624))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,195,484]))
 
pari: [g,chi] = znchar(Mod(73,8007))
 

Basic properties

Modulus: \(8007\)
Conductor: \(2669\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(624\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2669}(73,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8007.fk

\(\chi_{8007}(73,\cdot)\) \(\chi_{8007}(163,\cdot)\) \(\chi_{8007}(175,\cdot)\) \(\chi_{8007}(226,\cdot)\) \(\chi_{8007}(244,\cdot)\) \(\chi_{8007}(334,\cdot)\) \(\chi_{8007}(352,\cdot)\) \(\chi_{8007}(367,\cdot)\) \(\chi_{8007}(388,\cdot)\) \(\chi_{8007}(394,\cdot)\) \(\chi_{8007}(397,\cdot)\) \(\chi_{8007}(445,\cdot)\) \(\chi_{8007}(466,\cdot)\) \(\chi_{8007}(505,\cdot)\) \(\chi_{8007}(532,\cdot)\) \(\chi_{8007}(541,\cdot)\) \(\chi_{8007}(556,\cdot)\) \(\chi_{8007}(568,\cdot)\) \(\chi_{8007}(649,\cdot)\) \(\chi_{8007}(691,\cdot)\) \(\chi_{8007}(724,\cdot)\) \(\chi_{8007}(742,\cdot)\) \(\chi_{8007}(805,\cdot)\) \(\chi_{8007}(889,\cdot)\) \(\chi_{8007}(1015,\cdot)\) \(\chi_{8007}(1030,\cdot)\) \(\chi_{8007}(1078,\cdot)\) \(\chi_{8007}(1081,\cdot)\) \(\chi_{8007}(1093,\cdot)\) \(\chi_{8007}(1117,\cdot)\) ...

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: $\Q(\zeta_{624})$
Fixed field: Number field defined by a degree 624 polynomial (not computed)

Values on generators

\((5339,1414,7855)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{121}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 8007 }(73, a) \) \(1\)\(1\)\(e\left(\frac{77}{104}\right)\)\(e\left(\frac{25}{52}\right)\)\(e\left(\frac{211}{624}\right)\)\(e\left(\frac{95}{208}\right)\)\(e\left(\frac{23}{104}\right)\)\(e\left(\frac{49}{624}\right)\)\(e\left(\frac{565}{624}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{41}{208}\right)\)\(e\left(\frac{25}{26}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8007 }(73,a) \;\) at \(\;a = \) e.g. 2