Properties

Label 8007.35
Modulus $8007$
Conductor $471$
Order $78$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, base_ring=CyclotomicField(78))
 
M = H._module
 
chi = DirichletCharacter(H, M([39,0,74]))
 
pari: [g,chi] = znchar(Mod(35,8007))
 

Basic properties

Modulus: \(8007\)
Conductor: \(471\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(78\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{471}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8007.ds

\(\chi_{8007}(35,\cdot)\) \(\chi_{8007}(647,\cdot)\) \(\chi_{8007}(749,\cdot)\) \(\chi_{8007}(953,\cdot)\) \(\chi_{8007}(1055,\cdot)\) \(\chi_{8007}(1208,\cdot)\) \(\chi_{8007}(1973,\cdot)\) \(\chi_{8007}(2228,\cdot)\) \(\chi_{8007}(2279,\cdot)\) \(\chi_{8007}(2330,\cdot)\) \(\chi_{8007}(2636,\cdot)\) \(\chi_{8007}(3554,\cdot)\) \(\chi_{8007}(3758,\cdot)\) \(\chi_{8007}(3962,\cdot)\) \(\chi_{8007}(4727,\cdot)\) \(\chi_{8007}(4982,\cdot)\) \(\chi_{8007}(5033,\cdot)\) \(\chi_{8007}(5390,\cdot)\) \(\chi_{8007}(5492,\cdot)\) \(\chi_{8007}(5849,\cdot)\) \(\chi_{8007}(6563,\cdot)\) \(\chi_{8007}(6665,\cdot)\) \(\chi_{8007}(7328,\cdot)\) \(\chi_{8007}(7583,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: Number field defined by a degree 78 polynomial

Values on generators

\((5339,1414,7855)\) → \((-1,1,e\left(\frac{37}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 8007 }(35, a) \) \(-1\)\(1\)\(e\left(\frac{7}{26}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{35}{78}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{21}{26}\right)\)\(e\left(\frac{28}{39}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{19}{26}\right)\)\(e\left(\frac{1}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8007 }(35,a) \;\) at \(\;a = \) e.g. 2